# Existence and uniqueness of classical solution to the initial-boundary value problem for the unsteady general planar Broadwell model with four velocities

Koudzo Togbévi Selom Sobah, Amah Séna d'Almeida

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Koudzo Togbévi Selom Sobah, Amah Séna d'Almeida. Existence and uniqueness of classical solution to the initial-boundary value problem for the unsteady general planar Broadwell model with four velocities. 2025. hal-05197590

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# Existence and uniqueness of classical solution to the initial-boundary value problem for the unsteady general planar Broadwell model with four velocities

Koudzo Togbévi Selom SOBAH¹* and Amah Séna D'ALMEIDA²

¹,²Department of Mathematics, Faculty of Sciences and Laboratory of Mathematics and Applications, University of Lomé, Lomé, TOGO

* corresponding email: deselium@gmail.com

## Abstract

We consider the unsteady problem for the general planar Broadwell model with four velocities in a rectangular spatial domain over a finite time interval. We impose a class of non-negative initial and Dirichlet boundary data that are bounded and continuous, along with their first-order partial derivatives. We then prove the existence and uniqueness of a non-negative continuous solution, bounded together with its first-order partial derivatives, to the initial-boundary value problem.

**Key words and phrases:** discrete velocity (Boltzmann) models, discrete kinetic equations, initial-boundary value problems, existence, uniqueness, fixed point theorems.

**2020 Mathematics Subject Classification:** 76A02, 76M28

## Note of the authors

This manuscript is accepted for publication in the Open Journal of Mathematical Analysis https://pisrt.org/psr-press/journals/oma/

## 1 Introduction

The discrete Boltzmann equation (DBE) [17] has long served as a fundamental tool for approximating the behavior of dilute gases, offering a simplified alternative to the full Boltzmann equation. Pionner works started with Carleman (1957) [19], Broadwell (1964) [1, 2] and was generalized by Gatignol (1970) [3]; the model replaces the continuous velocity space by a finite set of discrete velocities, leading to hyperbolic systems of conservation laws with
nonlinear collision terms. These systems have since been instrumental in the study of various physical and mathematical phenomena, such as shock formation, entropy dissipation, and boundary layer behavior.

Historically, the initial-boundary value problem (IBVP) for discrete velocity models has been intensively studied in one-dimensional settings [4, 5, 6, 9, 10, 15, 7, 18]. In the multi-dimensional steady case, the boundary value problem for discrete velocity models is investigated, for example, in [8, 12, 13, 14].

In contrast, the multi-dimensional theory of the IBVP for DBE models remains significantly less developed. To our knowledge, no result had been established for the existence and uniqueness of classical solutions to these multi-dimensional IBVPs in a bounded domain with Dirichlet boundary conditions - even in simple geometries such as rectangles - prior to the recent work [16]. In [16], we addressed this gap by providing the proof of existence and uniqueness of classical solutions for the two-dimensional Broadwell model in a rectangular domain with Dirichlet boundary conditions.

The goal of the present paper is to generalize these results to the general four-velocity Broadwell model in the plane, one of the canonical multi-dimensional discrete models. This work represents a key step toward a broader mathematical understanding of discrete kinetic models in higher dimensions.

The paper is organized as follows. In Section 2, the initial-boundary value problem and our main result are presented. In Section 3, we begin the proof of our result by the positivity of the solution and in Section 4, we end the proof by the existence and uniqueness.

# 2 The initial-boundary value problem

## 2.1 The discrete Boltzmann equations

Let $d = 1, 2, 3$ and $p$ be a non-negative integer, $\vec{u}_i = (u_i^\alpha)_{\alpha=1}^d \in \mathbb{R}^d$, $(i = 1, \dots, p)$, $A_{ij}^{kl} \ge 0$ $(i, j, k, l = 1, \dots, p)$. The discrete $p$ velocity model describes a system of particles with velocities $\vec{u}_i$. Let $N_i(t, (x_\alpha)_{\alpha=1}^d)$ denote the number density of the particles with velocity $\vec{u}_i$, $i = 1, \dots, p$ at time $t$ and position $M(x_\alpha)_{\alpha=1}^d$ (with $(x_\alpha)_{\alpha=1}^d \in \mathbb{R}^d$). The corresponding discrete Boltzmann (or kinetic) equations are given by:

$$ \begin{gathered} \forall i = 1, \dots, p, \\ \frac{\partial N_i}{\partial t} + \sum_{\alpha=1}^{d} u_i^{\alpha} \frac{\partial N_i}{\partial x_{\alpha}} = \frac{1}{2} \underbrace{\sum_{j,k,l \neq i} A_{ij}^{kl} (N_k N_l - N_i N_j)}_{\equiv Q_i(N)} \quad (2.1) \\ \equiv Q_i(N) \end{gathered} $$
In the case of the general four-velocity Broadwell model in the plane, we have $N_i \equiv N_i(t, x, y)$ and the kinetic equations are given by:

$$ \left\{ \begin{aligned} \frac{\partial N_1}{\partial t} + c \cos \theta \frac{\partial N_1}{\partial x} + c \sin \theta \frac{\partial N_1}{\partial y} &= Q \\ \frac{\partial N_2}{\partial t} - c \sin \theta \frac{\partial N_2}{\partial x} + c \cos \theta \frac{\partial N_2}{\partial y} &= -Q \\ \frac{\partial N_3}{\partial t} + c \sin \theta \frac{\partial N_3}{\partial x} - c \cos \theta \frac{\partial N_3}{\partial y} &= -Q \\ \frac{\partial N_4}{\partial t} - c \cos \theta \frac{\partial N_4}{\partial x} - c \sin \theta \frac{\partial N_4}{\partial y} &= Q \end{aligned} \right. \qquad (2.2) $$

$$ Q = 2cS(N_2N_3 - N_1N_4) $$

$c, S > 0$ constants and $0 \le \theta < \frac{\pi}{2}$.

## 2.2 Initial-boundary value problem

Let $[a_1, b_1] \times [a_2, b_2]$ be a rectangular spatial domain and $[0; T]$ a finite time intervall. We define the space-time domain $\mathcal{P} = [0; T] \times [a_1; b_1] \times [a_2; b_2]$. In [16] we considered the case $\theta = 0$. In the sequel, $\theta \neq 0$.

The initial-boundary value problem we consider is the system (2.3)-
(2.27):

$$ \frac{\partial N_1}{\partial t} + c \cos \theta \frac{\partial N_1}{\partial x} + c \sin \theta \frac{\partial N_1}{\partial y} = Q(N), \quad (t, x, y) \in \mathcal{P} \qquad (2.3) $$

$$ \frac{\partial N_2}{\partial t} - c \sin \theta \frac{\partial N_2}{\partial x} + c \cos \theta \frac{\partial N_2}{\partial y} = -Q(N), \quad (t, x, y) \in \mathcal{P} \qquad (2.4) $$

$$ \frac{\partial N_3}{\partial t} + c \sin \theta \frac{\partial N_3}{\partial x} - c \cos \theta \frac{\partial N_3}{\partial y} = -Q(N), \quad (t, x, y) \in \mathcal{P} \qquad (2.5) $$

$$ \frac{\partial N_4}{\partial t} - c \cos \theta \frac{\partial N_4}{\partial x} - c \sin \theta \frac{\partial N_4}{\partial y} = Q(N), \quad (t, x, y) \in \mathcal{P} \qquad (2.6) $$

$$ N_i(0, x, y) = N_i^0(x, y), \quad (x, y) \in [a_1; b_1] \times [a_2; b_2], \quad i = 1, \dots, 4 \qquad (2.7) $$

$$ N_1(t, a_1, y) = N_1^-(t, y), \quad (t, y) \in [0; T] \times [a_2; b_2] \qquad (2.8) $$

$$ N_1(t, x, a_2) = N_1^{--}(t, x), \quad (t, x) \in [0; T] \times [a_1; b_1] \qquad (2.9) $$

$$ N_2(t, b_1, y) = N_2^+(t, y), \quad (t, y) \in [0; T] \times [a_2; b_2] \qquad (2.10) $$

$$ N_2(t, x, a_2) = N_2^{--}(t, x), \quad (t, x) \in [0; T] \times [a_1; b_1] \qquad (2.11) $$

$$ N_3(t, a_1, y) = N_3^-(t, y), \quad (t, y) \in [0; T] \times [a_2; b_2] \qquad (2.12) $$

$$ N_3(t, x, b_2) = N_3^{++}(t, x), \quad (t, x) \in [0; T] \times [a_1; b_1] \qquad (2.13) $$

$$ N_4(t, b_1, y) = N_4^+(t, y), \quad (t, y) \in [0; T] \times [a_2; b_2] \qquad (2.14) $$

$$ N_4(t, x, b_2) = N_4^{++}(t, x), \quad (t, x) \in [0; T] \times [a_1; b_1] \qquad (2.15) $$

$$ N_1^0(a_1, y) = N_1^-(0, y), \quad y \in [a_2; b_2] \qquad (2.16) $$

$$ N_1^0(x, a_2) = N_1^{--}(0, x), \quad x \in [a_1; b_1] \qquad (2.17) $$

$$ N_1^-(t, a_2) = N_1^{--}(t, a_1), \quad t \in [0; T] \qquad (2.18) $$

$$ N_2^0(b_1, y) = N_2^+(0, y), \quad y \in [a_2; b_2] \qquad (2.19) $$

$$ N_2^0(x, a_2) = N_2^{--}(0, x), \quad x \in [a_1; b_1] \qquad (2.20) $$

$$ N_2^+(t, a_2) = N_2^{--}(t, b_1), \quad t \in [0; T] \qquad (2.21) $$

$$ N_3^0(a_1, y) = N_3^-(0, y), \quad y \in [a_2; b_2] \qquad (2.22) $$

$$ N_3^0(x, b_2) = N_3^{++}(0, x), \quad x \in [a_1; b_1] \qquad (2.23) $$

$$ N_3^-(t, b_2) = N_3^{++}(t, a_1), \quad t \in [0; T] \qquad (2.24) $$

$$ N_4^0(b_1, y) = N_4^+(0, y), \quad y \in [a_2; b_2] \qquad (2.25) $$

$$ N_4^0(x, b_2) = N_4^{++}(0, x), \quad x \in [a_1; b_1] \qquad (2.26) $$

$$ N_4^+(t, b_2) = N_4^{++}(t, b_1), \quad t \in [0; T] \qquad (2.27) $$

The initial conditions are given by $N_i^0$, for $i = 1, \dots, 4$, while the boundary conditions are specified by $N_1^-, N_1^{--}$, $N_2^+$, $N_2^{--}$, $N_3^-$, $N_3^{++}$, $N_4^+$ and $N_4^{++}$. All these functions are assumed to be non-negative and continuous, and their first-order partial derivatives to be continuous and bounded. (2.16)-(2.27) are compatibility conditions on the data.
Our problem is to prove the existence and uniqueness of a positive continuous solution to the system Σ.

We shall use the uniform norm for bounded real-valued functions, and $\|f\| = \max_{1 \le i \le 4} \|f_i\|_{\infty}$ for $f = (f_i)_{i=1}^4$ with bounded components; for bounded real functions of two variables $g \equiv g(x, y)$ with bounded partial derivatives, we adopt the $C^1$ norm $\|g\|_1 = \max \{ \|g\|_{\infty}, \left\| \frac{\partial g}{\partial x} \right\|_{\infty}, \left\| \frac{\partial g}{\partial y} \right\|_{\infty} \}$.

We prove in the sequel the following result

**Theorem 2.1.** There exist two positive parameters $p$ and $q$, with $p$ depending on the dimensions of domain $\mathcal{P}$, and $q$ on the data such that if $pq \le \frac{1}{4}$ then the system $\Sigma$ admits a unique non-negative continuous solution which is differentiable except possibly on finitely many planes, with bounded derivatives, and for which explicit bounds on the solution and its derivatives are available.

## 3 Non-negativity of the solution

**Theorem 3.1.** If they exist the solutions of the problem $\Sigma$ (eq.(2.3)-(2.27)) are non-negative.

*Proof.* The proof follows classical arguments (in [16]). Let $\sigma > 0$. Let's put

$$ \rho(N) = \sum_{i=1}^{4} N_i \qquad (3.1) $$

and

$$ \left\{ \begin{array}{l} Q_1^\sigma (N) = \sigma \rho (N) N_1 + Q (N) \\ Q_2^\sigma (N) = \sigma \rho (N) N_2 - Q (N) \\ Q_3^\sigma (N) = \sigma \rho (N) N_3 - Q (N) \\ Q_4^\sigma (N) = \sigma \rho (N) N_4 + Q (N) \end{array} \right. . \qquad (3.2) $$
Then for all $\sigma > 0$, $\Sigma$ is equivalent to system (Σ$_{σ}$)

$$ \frac{\partial N_1}{\partial t} + c \cos \theta \frac{\partial N_1}{\partial x} + c \sin \theta \frac{\partial N_1}{\partial y} + \sigma \rho (N) N_1 = Q_1^\sigma (N), \quad (t, x, y) \in \mathcal{P} \quad (3.3) $$

$$ \frac{\partial N_2}{\partial t} - c \sin \theta \frac{\partial N_2}{\partial x} + c \cos \theta \frac{\partial N_2}{\partial y} + \sigma \rho (N) N_2 = Q_2^\sigma (N), \quad (t, x, y) \in \mathcal{P} \quad (3.4) $$

$$ \frac{\partial N_3}{\partial t} + c \sin \theta \frac{\partial N_3}{\partial x} - c \cos \theta \frac{\partial N_3}{\partial y} + \sigma \rho (N) N_3 = Q_3^\sigma (N), \quad (t, x, y) \in \mathcal{P} \quad (3.5) $$

$$ \frac{\partial N_4}{\partial t} - c \cos \theta \frac{\partial N_4}{\partial x} - c \sin \theta \frac{\partial N_4}{\partial y} + \sigma \rho (N) N_4 = Q_4^\sigma (N), \quad (t, x, y) \in \mathcal{P} \quad (3.6) $$

with the conditions (2.7)-(2.27).

For $M = (M_1, M_2, M_3, M_4)$ fixed 4-tuple of continuous functions defined from $\mathcal{P}$ to $\mathbb{R}$, let's put $|M| = (|M_1|, |M_2|, |M_3|, |M_4|)$. Let's consider the following linear system (Σ$_{σ,M}$) (3.7)-(3.10)

$$ \frac{\partial N_1}{\partial t} + c \cos \theta \frac{\partial N_1}{\partial x} + c \sin \theta \frac{\partial N_1}{\partial y} + \sigma \rho (|M|) N_1 = Q_1^\sigma (|M|), \quad (t, x, y) \in \mathcal{P} \quad (3.7) $$

$$ \frac{\partial N_2}{\partial t} - c \sin \theta \frac{\partial N_2}{\partial x} + c \cos \theta \frac{\partial N_2}{\partial y} + \sigma \rho (|M|) N_2 = Q_2^\sigma (|M|), \quad (t, x, y) \in \mathcal{P} \quad (3.8) $$

$$ \frac{\partial N_3}{\partial t} + c \sin \theta \frac{\partial N_3}{\partial x} - c \cos \theta \frac{\partial N_3}{\partial y} + \sigma \rho (|M|) N_3 = Q_3^\sigma (|M|), \quad (t, x, y) \in \mathcal{P} \quad (3.9) $$

$$ \frac{\partial N_4}{\partial t} - c \cos \theta \frac{\partial N_4}{\partial x} - c \sin \theta \frac{\partial N_4}{\partial y} + \sigma \rho (|M|) N_4 = Q_4^\sigma (|M|), \quad (t, x, y) \in \mathcal{P} \quad (3.10) $$

with the conditions (2.7)-(2.27). It is easy to see that (Σ$_{σ,M}$) has an unique solution $N_M = (N_1, N_2, N_3, N_4)$ defined as followed:

for inequalities $f(t, x, y) \le 0$ and $g(t, x, y) \le 0$ defined on $\mathcal{P}$, let denote the identity function of the set
$\{(t, x, y) \in \mathcal{P} : f(t, x, y) \le 0 \text{ and } g(t, x, y) \le 0\}; \text{ then}$

$$
\begin{align}
N_1(t, x, y) = N_1^A(t, x, y) \cdot \mathbb{I} & \begin{cases} x - ct \cos \theta \ge a_1 \\ y - ct \sin \theta \ge a_2 \end{cases} (t, x, y) + \nonumber \\
& N_1^B(t, x, y) \cdot \mathbb{I} \begin{cases} x - ct \cos \theta \le a_1 \\ x \sin \theta - y \cos \theta \le a_1 \sin \theta - a_2 \cos \theta \end{cases} (t, x, y) + \nonumber \\
& N_1^C(t, x, y) \cdot \mathbb{I} \begin{cases} y - ct \sin \theta \le a_2 \\ x \sin \theta - y \cos \theta \ge a_1 \sin \theta - a_2 \cos \theta \end{cases} (t, x, y) \tag{3.11}
\end{align}
$$

where

$$
\begin{equation}
\begin{split}
N_1^A (t, x, y) = \left( \int_0^t e^{\sigma \int_0^s \rho(|M|) (r, x+c(r-t)\cos\theta, y+c(r-t)\sin\theta) dr} \cdot Q_1^\sigma (|M|) \right. \\
\left. \qquad \qquad (s, x+c(s-t)\cos\theta, y+c(s-t)\sin\theta) ds + N_1^0 (x-ct\cos\theta, y-ct\sin\theta) \right) \\
\qquad -\sigma \int_0^t \rho(|M|) (s,x+c(s-t)\cos\theta, y+c(s-t)\sin\theta) ds
\end{split}
\tag{3.12}
\end{equation}
$$

$$
\begin{align*}
N_1^B (t, x, y) ={}& \left( \int_0^{\frac{1}{c \cos \theta} x - \frac{a_1}{c \cos \theta}} e^{\sigma \int_0^s \rho(|M|) \left( r + t - \frac{1}{c \cos \theta} x + \frac{a_1}{c \cos \theta} \right) \left( rc \cos \theta + a_1, rc \sin \theta - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} a_1 \right) dr} \right) \\
& \cdot Q_1^\sigma (|M|) \left( s + t - \frac{1}{c \cos \theta} x + \frac{a_1}{c \cos \theta}, sc \cos \theta + a_1, sc \sin \theta - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} a_1 \right) ds + \\
& N_1^- \left( t - \frac{1}{c \cos \theta} x + \frac{a_1}{c \cos \theta}, -\frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} a_1 \right) \\
& \cdot e^{-\sigma \int_0^t \rho(|M|) \left( s + t - \frac{1}{c \cos \theta} x + \frac{a_1}{c \cos \theta}, sc \cos \theta + a_1, sc \sin \theta - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} a_1 \right) ds} \tag{3.13}
\end{align*}
$$
$$N_1^C (t, x, y) = \left( \int_0^{\frac{1}{c \sin \theta} y - \frac{a_2}{c \sin \theta}} e^{\sigma \int_0^s \rho(|M|) \left( r + t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta} \right.}} \left( r + t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta}, rc \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} a_2, rc \sin \theta + a_2 \right) dr \right).$$

$$Q_1^\sigma(|M|) \left(s + t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta}, sc \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} a_2, sc \sin \theta + a_2 \right) ds + N_1^{--} \left( t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta}, x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} a_2 \right).$$

$$e^{-\sigma} \int_{0}^{\frac{1}{c \sin \theta} y - \frac{a_2}{c \sin \theta}} \rho(|M|) \left( s + t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta}, sc \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} a_2, sc \sin \theta + a_2 \right) ds \quad (3.14)$$

$$N_2(t, x, y) = N_2^A(t, x, y) \cdot \mathbb{I} \begin{cases} x - ct \cos \theta \ge a_1 \\ y - ct \sin \theta \ge a_2 \end{cases} (t, x, y) +$$

$$N_2^B (t, x, y) \cdot \mathbb{I} \begin{cases} x - ct \cos \theta \le a_1 \\ x \sin \theta - y \cos \theta \le a_1 \sin \theta - a_2 \cos \theta \end{cases} (t, x, y) +$$

$$N_2^C (t, x, y) \cdot \mathbb{I} \begin{cases} y - ct \sin \theta \le a_2 \\ x \sin \theta - y \cos \theta \ge a_1 \sin \theta - a_2 \cos \theta \end{cases} (t, x, y) \quad (3.15)$$

where

$$N_2^A (t, x, y) = \left( \int_0^t e^{\sigma \int_0^s \rho(|M|) (r, x - c(r-t) \sin \theta, y + c(r-t) \cos \theta)} dr \right).$$

$$Q_2^\sigma(|M|) (s, x - c(s - t) \sin \theta, y + c(s - t) \cos \theta) ds + N_2^0 (x + ct \sin \theta, y - ct \cos \theta).$$

$$e^{-\sigma} \int_{0}^{t} \rho(|M|) (s, x - c(s - t) \sin \theta, y + c(s - t) \cos \theta) ds \quad (3.16)$$
$$
\begin{align}
N_2^B (t, x, y) &= \left( \int_0^{-\frac{1}{c \sin \theta} x + \frac{b_1}{c \sin \theta}} e^{\sigma \int_0^s \rho(|M|) \left( r + t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, -rc \sin \theta + b_1, rc \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) dr} \right. \nonumber \\
&\qquad \left. Q_2^\sigma (|M|) \left( s + t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, -sc \sin \theta + b_1, sc \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) ds + \right. \nonumber \\
&\qquad \left. N_2^+ \left( t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) \right). \nonumber \\
&\qquad -\sigma \int_0^{-\frac{1}{c \sin \theta} x + \frac{b_1}{c \sin \theta}} e^{\rho(|M|) \left( s + t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, -sc \sin \theta + b_1, sc \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) ds} \tag{3.17}
\end{align}
$$

$$
\begin{align}
N_2^C (t, x, y) &= \left( \int_0^{\frac{1}{c \cos \theta} y - \frac{a_2}{c \cos \theta}} e^{\sigma \int_0^s \rho(|M|) \left( r + t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, -rc \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2, rc \cos \theta + a_2 \right) dr} \right. \nonumber \\
&\qquad \left. Q_2^\sigma (|M|) \left( s + t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, -sc \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2, sc \cos \theta + a_2 \right) ds + \right. \nonumber \\
&\qquad \left. N_2^{-} \left( t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2 \right) \right). \nonumber \\
&\qquad -\sigma \int_0^{\frac{1}{c \cos \theta} y - \frac{a_2}{c \cos \theta}} e^{\rho(|M|) \left( s + t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, -sc \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2, sc \cos \theta + a_2 \right) ds} \tag{3.18}
\end{align}
$$

$$
\begin{align}
N_3 (t, x, y) &= N_3^A (t, x, y) \cdot \mathbb{I} \begin{cases} x - ct \sin \theta \ge a_1 \\ y + ct \cos \theta \le b_2 \end{cases} (t, x, y) + \nonumber \\
&\quad N_3^B (t, x, y) \cdot \mathbb{I} \begin{cases} x - ct \sin \theta \le a_1 \\ x \cos \theta + y \sin \theta \le a_1 \cos \theta + b_2 \sin \theta \end{cases} (t, x, y) + \nonumber \\
&\quad N_3^C (t, x, y) \cdot \mathbb{I} \begin{cases} y + ct \cos \theta \ge b_2 \\ x \cos \theta + y \sin \theta \ge a_1 \cos \theta + b_2 \sin \theta \end{cases} (t, x, y) \tag{3.19}
\end{align}
$$
where

$$
\begin{equation}
\begin{aligned}
N_3^A (t, x, y) = & \left( \int_0^t e^{\sigma \int_0^s \rho(|M|) (r, x+c(r-t)\sin\theta, y-c(r-t)\cos\theta) dr} \cdot Q_3^\sigma (|M|) \right. \\
& \left. \qquad (s, x+c(s-t)\sin\theta, y-c(s-t)\cos\theta) ds + N_3^0 (x-ct\sin\theta, y+ct\cos\theta) \right) \\
& - \sigma \int_0^t \frac{\rho(|M|)(s, x+c(s-t)\sin\theta, y-c(s-t)\cos\theta) ds}{e}.
\end{aligned}
\tag{3.20}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
N_3^B (t, x, y) = & \left( \int_0^{\frac{1}{c \sin \theta} x - \frac{a_1}{c \sin \theta}} e^{\sigma \int_0^s \rho(|M|) \left( r + t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta} \right.} \right. \\
& \left. \left. \quad rc \sin \theta + a_1, -rc \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right) dr \right) \\
& Q_3^\sigma (|M|) \left( s + t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta}, sc \sin \theta + a_1, -sc \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right) ds \\
& + N_3^- \left( t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta}, \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right) \\
& - \sigma \int_0^{\frac{1}{c \sin \theta} x - \frac{a_1}{c \sin \theta}} \rho(|M|) \left( s + t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta}, sc \sin \theta + a_1, -sc \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right) ds
\end{aligned}
\tag{3.21}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
N_3^C (t, x, y) = & \left( \int_0^{-\frac{1}{c \cos \theta} y + \frac{b_2}{c \cos \theta}} e^{\sigma \int_0^s \rho(|M|) \left( r + t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta} \right.} \right. \\
& \left. \left. \quad rc \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2, -rc \cos \theta + b_2 \right) dr \right) \\
& Q_3^\sigma (|M|) \left( s + t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta}, sc \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2, -sc \cos \theta + b_2 \right) ds \\
& + N_3^{++} \left( t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta}, x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2 \right) \\
& - \sigma \int_0^{-\frac{1}{c \cos \theta} y + \frac{b_2}{c \cos \theta}} \rho(|M|) \left( s + t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta}, sc \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2, -sc \cos \theta + b_2 \right) ds
\end{aligned}
\tag{3.22}
\end{equation}
$$
$$
\begin{align}
N_4 (t, x, y) = N_4^A (t, x, y) \cdot \mathbb{I} & \begin{cases} x + ct \cos \theta \le b_1 \\ y + ct \sin \theta \le b_2 \end{cases} (t, x, y) + \nonumber \\
& N_4^B (t, x, y) \cdot \mathbb{I} \begin{cases} x + ct \cos \theta \ge b_1 \\ x \sin \theta - y \cos \theta \ge b_1 \sin \theta - b_2 \cos \theta \end{cases} (t, x, y) + \nonumber \\
& N_4^C (t, x, y) \cdot \mathbb{I} \begin{cases} y + ct \sin \theta \ge b_2 \\ x \sin \theta - y \cos \theta \le b_1 \sin \theta - b_2 \cos \theta \end{cases} (t, x, y) \tag{3.23}
\end{align}
$$

where

$$
\begin{equation}
\begin{split}
N_4^A (t, x, y) = \left( \int_0^t e^{\sigma \int_0^s \rho(|M|) (r, x-c(r-t) \cos\theta, y-c(r-t) \sin\theta) dr} \cdot Q_4^\sigma (|M|) \right. \\
\left. \qquad (s, x-c(s-t) \cos\theta, y-c(s-t) \sin\theta) ds + N_4^0 (x+ct \cos\theta, y+ct \sin\theta) \right) \\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad e^{\sigma \int_0^t \rho(|M|) (s, x-c(s-t) \cos\theta, y-c(s-t) \sin\theta) ds}
\end{split}
\tag{3.24}
\end{equation}
$$

$$
\begin{align}
N_4^B (t, x, y) = & \left( \int_0^{-\frac{1}{c \cos \theta} x + \frac{b_1}{c \cos \theta}} e^{\sigma \int_0^s \rho(|M|) \left( r + t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, -rc \cos \theta + b_1, -rc \sin \theta - \frac{\sin \theta}{c \cos \theta} x + y + \frac{\sin \theta}{c \cos \theta} b_1 \right) dr} \right) \nonumber \\
& \cdot Q_4^\sigma (|M|) \left( s + t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, -sc \cos \theta + b_1, -sc \sin \theta - \frac{\sin \theta}{c \cos \theta} x + y + \frac{\sin \theta}{c \cos \theta} b_1 \right) ds + \nonumber \\
& N_4^+ \left( t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, -\frac{\sin \theta}{c \cos \theta} x + y + \frac{\sin \theta}{c \cos \theta} b_1 \right) \nonumber \\
& \cdot e^{-\sigma \int_0^{-\frac{1}{c \cos \theta} x + \frac{b_1}{c \cos \theta}} \rho(|M|) \left( s + t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, -sc \cos \theta + b_1, -sc \sin \theta - \frac{\sin \theta}{c \cos \theta} x + y + \frac{\sin \theta}{c \cos \theta} b_1 \right) ds} \tag{3.25}
\end{align}
$$
$$
\begin{align*}
N_4^C (t, x, y) &= \left( \int_0^{-\frac{1}{c \sin \theta} y + \frac{b_2}{c \sin \theta}} e^{\sigma \int_0^s \rho(|M|) \left( r+t+\frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, -rc \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2, -rc \sin \theta + b_2 \right) dr} \right. \\
&\qquad \left. Q_4^\sigma (|M|) \left( s+t+\frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, -sc \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2, -sc \sin \theta + b_2 \right) ds + \right. \\
&\qquad \left. N_4^{++} \left( t+\frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2 \right) \right). \\
&\qquad -\sigma \int_0^{-\frac{1}{c \sin \theta} y + \frac{b_2}{c \sin \theta}} e^{\rho(|M|) \left( s+t+\frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, -sc \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2, -sc \sin \theta + b_2 \right) ds} . \tag{3.26}
\end{align*}
$$

Now we have

$$
Q_1^\sigma (|M|) = \sigma (|M_1| + |M_2| + |M_3|) |M_1| + 2cS |M_2| |M_3| + (\sigma - 2cS) |M_1| |M_4|
$$

$$
Q_2^\sigma (|M|) = \sigma (|M_1| + |M_2| + |M_4|) |M_2| + 2cS |M_1| |M_4| + (\sigma - 2cS) |M_2| |M_3|
$$

$$
Q_3^\sigma (|M|) = \sigma (|M_1| + |M_3| + |M_4|) |M_3| + 2cS |M_1| |M_4| + (\sigma - 2cS) |M_2| |M_3|
$$

$$
Q_4^\sigma (|M|) = \sigma (|M_2| + |M_3| + |M_4|) |M_4| + 2cS |M_2| |M_3| + (\sigma - 2cS) |M_1| |M_4|.
$$

and we conclude that for $\sigma \ge 2cS$, the solution $N = (N_1, N_2, N_3, N_4)$ of $(\Sigma_{\sigma,M})$ is non-negative.

Let us consider the operator $\mathcal{T}^\sigma : M \to N_M$ where $N_M$ is the unique non-negative solution of the problem $(\Sigma_{\sigma,M})$ for sufficiently large $\sigma$.

We easily verify from the statements of $(\Sigma_\sigma)$ (3.3)-(3.6) and $(\Sigma_{\sigma,M})$ (3.7)-(3.10) that for $M$ possessing all first-order partial derivatives, $\mathcal{T}^\sigma(M) = M$ iff $M$ is a solution of $(\Sigma_\sigma)$ for sufficiently large $\sigma$ i.e. $M$ is a solution of $\Sigma$ as $(\Sigma_\sigma)$ is equivalent to $\Sigma$. As $\mathcal{T}^\sigma(M)$ is non-negative for sufficiently large $\sigma$, so is any solution $M$ of $\Sigma$. $\square$
# 4 Existence and uniqueness

## 4.1 Auxilliary operator

Let $M = (M_1, M_2, M_3, M_4)$ be fixed. Let us replace $N$ in the second member of $\Sigma$ (2.3)-(2.11) by $M$. We obtain the linear system $(\Sigma_M)$ defined by

$$ \frac{\partial N_1}{\partial t} + c \cos \theta \frac{\partial N_1}{\partial x} + c \sin \theta \frac{\partial N_1}{\partial y} = Q(M), \quad (t, x, y) \in \mathcal{P} \quad (4.1) $$

$$ \frac{\partial N_2}{\partial t} - c \sin \theta \frac{\partial N_2}{\partial x} + c \cos \theta \frac{\partial N_2}{\partial y} = -Q(M), \quad (t, x, y) \in \mathcal{P} \quad (4.2) $$

$$ \frac{\partial N_3}{\partial t} + c \sin \theta \frac{\partial N_3}{\partial x} - c \cos \theta \frac{\partial N_3}{\partial y} = -Q(M), \quad (t, x, y) \in \mathcal{P} \quad (4.3) $$

$$ \frac{\partial N_4}{\partial t} - c \cos \theta \frac{\partial N_4}{\partial x} - c \sin \theta \frac{\partial N_4}{\partial y} = Q(M), \quad (t, x, y) \in \mathcal{P} \quad (4.4) $$

with the conditions 2.7-2.27. $(\Sigma_M)$ has an unique continuous solution $\mathcal{T}(M) = (\mathcal{T}_i(M))_{i=1}^4$ defined in the appendix (A) by expressions (A.1)-(A.15).

It is immediate that the solutions of $\Sigma$ are the fixed points, possessing all first-order partial derivatives, of the operator

$$ \mathcal{T}: C(\mathcal{P}; \mathbb{R}^4) \to C(\mathcal{P}; \mathbb{R}^4) \\ M \mapsto \mathcal{T}(M) = (\mathcal{T}_i(M))_{i=1}^4 \quad (4.5) $$

We easily prove as in [16], that

**Lemma 4.1.** *T is continuous.*

*Proof.* A.1-A.4 give for $M, N \in C(\mathcal{P}, \mathbb{R}^4)$:

$$ \begin{aligned} & \|\mathcal{T}_1(M) - \mathcal{T}_1(N)\|_{\infty} \le \\ & \max \left\{ \sup_{(t,x,y) \in \mathcal{P}} \left| \int_0^t [Q(M) - Q(N)] (s, x + c(s-t)\cos\theta, y + c(s-t)\sin\theta) ds \right|; \right. \\ & \qquad \left. \sup_{(t,x,y) \in \mathcal{P}} \left| \int_0^{\frac{1}{c\cos\theta}x - \frac{a_1}{c\cos\theta}} [Q(M) - Q(N)] \right. \right. \\ & \qquad \left. \left. \left( s + t - \frac{1}{c\cos\theta}x + \frac{a_1}{c\cos\theta}, sc\cos\theta + a_1, sc\sin\theta - \frac{\sin\theta}{\cos\theta}x + y + \frac{\sin\theta}{\cos\theta}a_1 \right) ds \right|; \right. \\ & \qquad \left. \sup_{(t,x,y) \in \mathcal{P}} \left| \int_0^{\frac{1}{c\sin\theta}y - \frac{a_2}{c\sin\theta}} [Q(M) - Q(N)] \right. \right. \\ & \qquad \left. \left. \left( s + t - \frac{1}{c\sin\theta}y + \frac{a_2}{c\sin\theta}, sc\cos\theta + x - \frac{\cos\theta}{\sin\theta}y + \frac{\cos\theta}{\sin\theta}a_2, sc\sin\theta + a_2 \right) ds \right| \right\}. \end{aligned} \quad (4.6) $$
But

$$Q(M) - Q(N) = 2cS(M_2 - N_2)M_3 + 2cSN_2(M_3 - N_3) \\ - 2cS(M_1 - N_1)M_4 - 2cSN_1(M_4 - N_4), \quad (4.7)$$

hence

$$\|Q(M) - Q(N)\|_{\infty} \le 2cS \|M_2 - N_2\|_{\infty} \|M_3\|_{\infty} + 2cS \|N_2\|_{\infty} \|M_3 - N_3\|_{\infty} \\ + 2cS \|M_1 - N_1\|_{\infty} \|M_4\|_{\infty} + 2cS \|N_1\|_{\infty} \|M_4 - N_4\|_{\infty}, \quad (4.8)$$

and

$$\|Q(M) - Q(N)\|_{\infty} \le 4cS \|M - N\| \|M\| + 4cS \|N\| \|M - N\| \\ \|Q(M) - Q(N)\|_{\infty} \le 4cS (\|M\| + \|N\|) \|M - N\|; \quad (4.9)$$

hence 4.6 implies

$$\| \mathcal{T}_1 (M) - \mathcal{T}_1 (N) \|_{\infty} \le \max \left\{ T, \frac{b_1 - a_1}{c \cos \theta}, \frac{b_2 - a_2}{c \sin \theta} \right\} \cdot 4cS (\|M\| + \|N\|) \|M - N\|. \quad (4.10)$$

Similar inequalities hold: from A.5-A.8 we have

$$\| \mathcal{T}_2 (M) - \mathcal{T}_2 (N) \|_{\infty} \le \max \left\{ T, \frac{b_1 - a_1}{c \sin \theta}, \frac{b_2 - a_2}{c \cos \theta} \right\} \cdot 4cS (\|M\| + \|N\|) \|M - N\|; \quad (4.11)$$

from A.9-A.11,

$$\| \mathcal{T}_3 (M) - \mathcal{T}_3 (N) \|_{\infty} \le \max \left\{ T, \frac{b_1 - a_1}{c \sin \theta}, \frac{b_2 - a_2}{c \cos \theta} \right\} \cdot 4cS (\|M\| + \|N\|) \|M - N\|. \quad (4.12)$$

and from (A.12)-(A.15)

$$\| \mathcal{T}_4 (M) - \mathcal{T}_4 (N) \|_{\infty} \le \max \left\{ T, \frac{b_1 - a_1}{c \cos \theta}, \frac{b_2 - a_2}{c \sin \theta} \right\} \cdot 4cS (\|M\| + \|N\|) \|M - N\|. \quad (4.13)$$

Now from (4.10), (4.11), (4.12), (4.13) and
$$
\| \mathcal{T}(M) - \mathcal{T}(N) \| = \max_{1 \le i \le 4} \| \mathcal{T}_i(M) - \mathcal{T}_i(N) \|_{\infty} \text{ we have}
$$

$$
\| \mathcal{T}(M) - \mathcal{T}(N) \| \le \max \left\{ T, \frac{b_1 - a_1}{c \cos \theta}, \frac{b_2 - a_2}{c \sin \theta}, \frac{b_1 - a_1}{c \sin \theta}, \frac{b_2 - a_2}{c \cos \theta} \right\} \cdot 4cS (\|M\| + \|N\|) \|M - N\|. \tag{4.14}
$$

We deduce that $\mathcal{T}$ is continuous. $\square$

**Proposition 4.2.** Suppose $M = (M_1, M_2, M_3, M_4) \in C(\mathcal{P}; \mathbb{R}^4)$ such that $\frac{\partial M_i}{\partial t}$, $\frac{\partial M_i}{\partial x}$, $\frac{\partial M_i}{\partial y}$ are defined in $\mathcal{P}^*$, except possibly on a finite number of planes, and are continuous and bounded for all $i = 1, 2, 3, 4$. Then all the derivatives $\frac{\partial \mathcal{T}_i(M)}{\partial t}$, $\frac{\partial \mathcal{T}_i(M)}{\partial x}$, $\frac{\partial \mathcal{T}_i(M)}{\partial y}$ ($i = 1, 2, 3, 4$) are defined in $\mathcal{P}^*$, except possibly on a finite number of planes, and are continuous and bounded.

In other words, if we denote by $\mathcal{E}$ the sub-space of $C(\mathcal{P}; \mathbb{R})$ consisting
of functions $u$ that are continuous on $\mathcal{P}$ such that $\frac{\partial u}{\partial t}$, $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial y}$ are defined
in $\mathcal{P}^*$ except possibly on a finite number of planes, and are continuous and
bounded, then $\mathcal{T}(\mathcal{E}^4) \subset \mathcal{E}^4$.

**Proof.** This follows from the explicit formula of the derivatives of $\mathcal{T}_i(M)$ ($i=1,2,3,4$) given in the appendix (B) $\square$

For $N = (N_i)_{i=1}^4 \in \mathcal{E}^4$, let $\mathcal{V}(N) = \max \left\{ \|N\|, \left\|\frac{\partial N}{\partial t}\right\|, \left\|\frac{\partial N}{\partial x}\right\|, \left\|\frac{\partial N}{\partial y}\right\| \right\}$
and consider for $R > 0$, $\mathcal{B}_R = \{N \in \mathcal{E}^4 : \mathcal{V}(N) \le R\}$.

**Proposition 4.3.** The sets $\mathcal{B}_R$, ($R > 0$) are non-empty convex subsets of $C(\mathcal{P}; \mathbb{R}^4)$ and are relatively compact in $(C(\mathcal{P}; \mathbb{R}^4), \|\cdot\|)$.

**Proof.** $\mathcal{B}_R$ contains the zero function. $\forall M, N \in \mathcal{B}_R$, and $\forall \lambda \in \mathbb{R}$ such that $0 \le \lambda \le 1$ we have $\lambda M + (1 - \lambda) N \in \mathcal{E}^4$; $\mathcal{V}(\lambda M + (1 - \lambda) N) \le R$ is immediate. $\mathcal{B}_R$ is thus convex.

By definition, $\mathcal{B}_R$ is a bounded set in $(C(\mathcal{P}; \mathbb{R}^4), \|\cdot\|)$ and the real
functions $\frac{\partial M_i}{\partial t}$, $\frac{\partial M_i}{\partial x}$, $\frac{\partial M_i}{\partial y}$, ($i = 1, 2, 3, 4$) are uniformly bounded and contin-
uous on their domain for $M \in \mathcal{B}_R$. Thus there exists a constant $k^i > 0$ inde-
pendant of $M$ such that $\forall M \in \mathcal{B}_R, \forall (t, x, y)$ in the domain of
$\left(\frac{\partial M_i}{\partial t}, \frac{\partial M_i}{\partial x}, \frac{\partial M_i}{\partial y}\right)$,
it holds

$$
\|d(M_i)(t, x, y)\|_{\mathcal{L}(\mathbb{R}^3, \mathbb{R})} \le k^i. \qquad (4.15)
$$
Let $M \in \mathcal{B}_R$ and $(t, x, y), (t', x', y') \in \mathcal{P}$ such that $\left(\frac{\partial M}{\partial t}, \frac{\partial M}{\partial x}, \frac{\partial M}{\partial y}\right)$ is defined on the segment

$[(t, x, y), (t', x', y')] \equiv \{(t, x, y) + \alpha (t' - t, x' - x, y' - y) \mid (0 \le \alpha \le 1)\}.$
As $\mathcal{P}$ is a convex set, then by the mean value inequality, inequality (4.15) implies that for all $i = 1, 2, 3, 4$,

$$|M_i(t, x, y) - M_i(t', x', y')| \le k^i \|(t, x, y) - (t', x', y')\|_{\mathbb{R}^3}. \quad (4.16)$$

For any $\varepsilon > 0$, let $\alpha_\varepsilon = \min_{1 \le i \le 4} \frac{\varepsilon}{k^i}$. It follows from (4.16) that

$$\|(t, x, y) - (t', x', y')\|_{\mathbb{R}^3} < \alpha_\varepsilon \implies \\ \|M(t, x, y) - M(t', x', y')\|_{\mathbb{R}^4} \le \varepsilon \quad (4.17)$$

The points where $\left(\frac{\partial M}{\partial t}, \frac{\partial M}{\partial x}, \frac{\partial M}{\partial y}\right)$ is not defined are in planes. Therefore as $M$ is continuous on $\mathcal{P}$, we can extend (4.17) to $\mathcal{P}$. We have

$$\forall \varepsilon > 0, \exists \alpha_\varepsilon > 0, \forall M \in \mathcal{B}_R, \forall (t, x, y), (t', x', y') \in \mathcal{P}: \\ \|(t, x, y) - (t', x', y')\| < \alpha_\varepsilon \implies \\ \|M(t, x, y) - M(t', x', y')\|_{\mathbb{R}^4} \le \varepsilon \quad (4.18)$$

which means that $\mathcal{B}_R$ is equicontinuous in $(C(\mathcal{P}; \mathbb{R}^4), \|\cdot\|)$. We conclude by Arzelà-Ascoli theorem that $\mathcal{B}_R$ is relatively compact in $(C(\mathcal{P}; \mathbb{R}^4), \|\cdot\|)$. $\square$

**Proposition 4.4.** There exists $p, q > 0$ such that

$$\forall R > 0, \mathcal{T}(\mathcal{B}_R) \subset \mathcal{B}_{pR^2+q}. \quad (4.19)$$

**Proof.** By proposition (4.2), if $M \in \mathcal{B}_R \subset \mathcal{E}^4$ then $\mathcal{T}(M) \in \mathcal{E}^4$. For any $M \in \mathcal{E}^4$, $Q(M) = 2cS(M_2M_3 - M_1M_4)$ implies

$$\|Q(M)\|_{\infty} \le 4cS(\mathcal{V}(M))^2. \quad (4.20)$$

We then have from the expressions A.1-A.15 of the solution of the system $(\Sigma_M)$, the following inequalities for $M \in \mathcal{E}^4$:

$$\|\mathcal{T}_1(M)\|_{\infty} \le 4cS \cdot \max\left\{T, \frac{b_1 - a_1}{c \cos \theta}, \frac{b_2 - a_2}{c \sin \theta}\right\} (\mathcal{V}(M))^2 + \\ \max\left\{\|N_1^0\|_1, \|N_1^-\|_1, \|N_1^{--}\|_1\right\}. \quad (4.21)$$
$$ \|\mathcal{T}_2(M)\|_{\infty} \le 4cS \cdot \max \left\{ T, \frac{b_1 - a_1}{c \sin \theta}, \frac{b_2 - a_2}{c \cos \theta} \right\} (\mathcal{Y}(M))^2 + \\ \max \left\{ \|N_2^0\|_1, \|N_2^+\|_1, \|N_2^{--}\|_1 \right\} \quad (4.22) $$

$$ \|\mathcal{T}_3(M)\|_{\infty} \le 4cS \cdot \max \left\{ T, \frac{b_1 - a_1}{c \sin \theta}, \frac{b_2 - a_2}{c \cos \theta} \right\} (\mathcal{Y}(M))^2 + \\ \max \left\{ \|N_3^0\|_1, \|N_3^-\|_1, \|N_3^{++}\|_1 \right\} \quad (4.23) $$

$$ \|\mathcal{T}_4(M)\|_{\infty} \le 4cS \cdot \max \left\{ T, \frac{b_1 - a_1}{c \cos \theta}, \frac{b_2 - a_2}{c \sin \theta} \right\} (\mathcal{Y}(M))^2 + \\ \max \left\{ \|N_4^0\|_1, \|N_4^+\|_1, \|N_4^{++}\|_1 \right\}. \quad (4.24) $$

and

$$ \|\mathcal{T}(M)\| \le 4cS \cdot \max \left\{ T, \frac{b_1 - a_1}{c \cos \theta}, \frac{b_2 - a_2}{c \sin \theta}, \frac{b_1 - a_1}{c \sin \theta}, \frac{b_2 - a_2}{c \cos \theta} \right\} (\mathcal{Y}(M))^2 + \\ \max_{1 \le i \le 4} \left\{ \|N_i^0\|_1, \|N_1^-\|_1, \|N_1^{--}\|_1, \|N_2^+\|_1, \|N_2^{--}\|_1, \right. \\ \left. \|N_3^-\|_1, \|N_3^{++}\|_1, \|N_4^+\|_1, \|N_4^{++}\|_1 \right\}. \quad (4.25) $$

We have $\frac{\partial Q(M)}{\partial t} = 2cS \left( \frac{\partial M_2}{\partial t} M_3 + M_2 \frac{\partial M_3}{\partial t} - \frac{\partial M_1}{\partial t} M_4 - M_1 \frac{\partial M_4}{\partial t} \right)$ and similar expressions for $\frac{\partial Q(M)}{\partial x}$, $\frac{\partial Q(M)}{\partial y}$. Therefore

$$ \left\| \frac{\partial Q(M)}{\partial t} \right\|_{\infty}, \left\| \frac{\partial Q(M)}{\partial x} \right\|_{\infty}, \left\| \frac{\partial Q(M)}{\partial y} \right\|_{\infty} \le 2cS \cdot 4 (\mathcal{Y}(M))^2 \le 8cS (\mathcal{Y}(M))^2. \quad (4.26) $$

(4.26) with explicit formula of the derivatives
$\frac{\partial \mathcal{T}_i(M)}{\partial t}$, $\frac{\partial \mathcal{T}_i(M)}{\partial x}$, $\frac{\partial \mathcal{T}_i(M)}{\partial y}$, ($i=1,2,3,4$) in the appendix (B) imply:

$$ \left\| \frac{\partial \mathcal{T}_1(M)}{\partial t} \right\|_{\infty} \le 4cS \max \left\{ 1 + 2T(c \cos \theta + c \sin \theta); 2 \frac{b_1 - a_1}{c \cos \theta}; 2 \frac{b_2 - a_2}{c \sin \theta} \right\} (\mathcal{Y}(M))^2 \\ \max \left\{ (c \cos \theta + c \sin \theta) \|N_1^0\|_1; \|N_1^-\|_1; \|N_1^{--}\|_1 \right\}; \quad (4.27) $$

$$ \left\| \frac{\partial \mathcal{T}_2(M)}{\partial t} \right\|_{\infty} \le 4cS \max \left\{ 1 + 2T(c \cos \theta + c \sin \theta); 2 \frac{b_1 - a_1}{c \sin \theta}; 2 \frac{b_2 - a_2}{c \cos \theta} \right\} (\mathcal{Y}(M))^2 \\ \max \left\{ (c \cos \theta + c \sin \theta) \|N_2^0\|_1; \|N_2^+\|_1; \|N_2^{--}\|_1 \right\}; \quad (4.28) $$

$$ \left\| \frac{\partial \mathcal{T}_3(M)}{\partial t} \right\|_{\infty} \le 4cS \max \left\{ 1 + 2T(c \cos \theta + c \sin \theta); 2 \frac{b_1 - a_1}{c \sin \theta}; 2 \frac{b_2 - a_2}{c \cos \theta} \right\} (\mathcal{Y}(M))^2 \\ \max \left\{ (c \cos \theta + c \sin \theta) \|N_3^0\|_1; \|N_3^-\|_1; \|N_3^{++}\|_1 \right\}; \quad (4.29) $$

$$ \left\| \frac{\partial \mathcal{T}_4(M)}{\partial t} \right\|_{\infty} \le 4cS \max \left\{ 1 + 2T(c \cos \theta + c \sin \theta); 2 \frac{b_1 - a_1}{c \cos \theta}; 2 \frac{b_2 - a_2}{c \sin \theta} \right\} (\mathcal{Y}(M))^2 \\ \max \left\{ (c \cos \theta + c \sin \theta) \|N_4^0\|_1; \|N_4^+\|_1; \|N_4^{++}\|_1 \right\}; \quad (4.30) $$

$$ \left\| \frac{\partial \mathcal{T}_1(M)}{\partial x} \right\|_{\infty} \le 4cS \max \left\{ 2T; \frac{1}{c \cos \theta} + 2 \frac{b_1 - a_1}{c \cos \theta} \left( \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right); 2 \frac{b_2 - a_2}{c \sin \theta} \right\} (\mathcal{Y}(M))^2 \\ \max \left\{ \|N_1^0\|_1; \left( \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right) \|N_1^-\|_1; \|N_1^{--}\|_1 \right\}; \quad (4.31) $$

$$ \left\| \frac{\partial \mathcal{T}_2(M)}{\partial x} \right\|_{\infty} \le 4cS \max \left\{ 2T; \frac{1}{c \sin \theta} + 2 \frac{b_1 - a_1}{c \sin \theta} \left( \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right); 2 \frac{b_2 - a_2}{c \cos \theta} \right\} (\mathcal{Y}(M))^2 \\ \max \left\{ \|N_2^0\|_1; \left( \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right) \|N_2^+\|_1; \|N_2^{--}\|_1 \right\}; \quad (4.32) $$
$$ \left\| \frac{\partial \mathcal{T}_3 (M)}{\partial x} \right\|_{\infty} \le 4cS \max \left\{ 2T; \frac{1}{c \sin \theta} + 2 \frac{b_1 - a_1}{c \sin \theta} \left( \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right) ; 2 \frac{b_2 - a_2}{c \cos \theta} \right\} (\mathcal{Y}(M))^2 \\ \max \left\{ \| N_3^0 \|_1 ; \left( \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right) \| N_3^- \|_1 ; \| N_3^{++} \|_1 \right\}; \quad (4.33) $$

$$ \left\| \frac{\partial \mathcal{T}_4 (M)}{\partial x} \right\|_{\infty} \le 4cS \max \left\{ 2T; \frac{1}{c \cos \theta} + 2 \frac{b_1 - a_1}{c \cos \theta} \left( \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right) ; 2 \frac{b_2 - a_2}{c \sin \theta} \right\} (\mathcal{Y}(M))^2 \\ \max \left\{ \| N_4^0 \|_1 ; \left( \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right) \| N_4^+ \|_1 ; \| N_4^{++} \|_1 \right\}. \quad (4.34) $$

4.27-4.30 imply

$$ \left\| \frac{\partial \mathcal{T} (M)}{\partial t} \right\| \le 4cS \max \left\{ 1 + 2T (c \cos \theta + c \sin \theta) ; 2 \max \left\{ \frac{1}{c \cos \theta} ; \frac{1}{c \sin \theta} \right\} (b_1 - a_1) ; \right. \\ 2 \max \left\{ \frac{1}{c \cos \theta} ; \frac{1}{c \sin \theta} \right\} (b_2 - a_2) \left\} (\mathcal{Y}(M))^2 + \right. \\ \max_{1 \le i \le 4} \left\{ (c \cos \theta + c \sin \theta) \| N_i^0 \|_1, \| N_i^- \|_1, \| N_i^{--} \|_1, \right. \\ \left. \| N_2^+ \|_1, \| N_2^{--} \|_1, \| N_3^- \|_1, \| N_3^{++} \|_1, \| N_4^+ \|_1, \| N_4^{++} \|_1 \right\}; \quad (4.35) $$
4.31-4.34 imply

$$
\begin{equation}
\begin{aligned}
& \left\| \frac{\partial \mathcal{T}(M)}{\partial x} \right\| \le \\
& 4cS \max \left\{ 2T; \max \left\{ \frac{1}{c \cos \theta}; \frac{1}{c \sin \theta} \right\} + 2(b_1 - a_1) \max \left\{ \frac{1}{c \cos \theta} \left( \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right); \right. \right. \\
& \qquad \left. \left( \frac{1}{c \sin \theta} \left( \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right) \right) \right\} + 2(b_2 - a_2) \max \left\{ \frac{1}{c \cos \theta}; \frac{1}{c \sin \theta} \right\} \right\} (\mathcal{Y}(M))^2 + \\
& \max_{1 \le i \le 4} \left\{ \|N_i^0\|_1, \left(\frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta}\right) \|N_1^-\|_1, \|N_1^{--}\|_1, \left(\frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta}\right) \|N_2^+\|_1, \|N_2^{--}\|_1, \right. \\
& \qquad \left. \left(\frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta}\right) \|N_3^-\|_1, \|N_3^{++}\|_1, \left(\frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta}\right) \|N_4^+\|_1, \|N_4^{++}\|_1 \right\}.
\end{aligned}
\tag{4.36}
\end{equation}
$$

We similarly obtain

$$
\begin{equation}
\begin{aligned}
& \left\| \frac{\partial \mathcal{T}(M)}{\partial y} \right\| \le \\
& 4cS \max \left\{ 2T; 2(b_1 - a_1) \max \left\{ \frac{1}{c \cos \theta}; \frac{1}{c \sin \theta} \right\}; \right. \\
& \qquad \left. \max \left\{ \frac{1}{c \cos \theta}; \frac{1}{c \sin \theta} \right\} + 2(b_2 - a_2) \max \left\{ \frac{1}{c \cos \theta} \left( \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right); \right. \right. \\
& \qquad \left. \left. \frac{1}{c \sin \theta} \left( \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right) \right\} \right\} (\mathcal{Y}(M))^2 + \\
& \max_{1 \le i \le 4} \left\{ \|N_i^0\|_1, \|N_1^-\|_1, \left(\frac{1}{c \cos \theta} + \frac{\cos \theta}{\sin \theta}\right) \|N_1^{--}\|_1, \|N_2^+\|_1, \left(\frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta}\right) \|N_2^{--}\|_1, \right. \\
& \qquad \left. \left(\frac{1}{c \sin \theta} + \frac{\sin \theta}{\cos \theta}\right) \|N_3^{++}\|_1, \|N_4^+\|_1, \left(\frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta}\right) \|N_4^{++}\|_1 \right\}.
\end{aligned}
\tag{4.37}
\end{equation}
$$
4.25, 4.35,4.36 and 4.37 imply that for $M \in \mathcal{E}^4$,

$$
\begin{align*}
\mathcal{V}(\mathcal{T}(M)) &\le 4cS \max \left\{ \begin{aligned}[t]
& \max \{1 + 2T(c \cos \theta + c \sin \theta); 2T\}; \\
& \max \left\{ \frac{1}{c \cos \theta}; \frac{1}{c \sin \theta} \right\} + 2 \max \left\{ \frac{1}{c \cos \theta}; \frac{1}{c \sin \theta}; \right. \\
& \qquad \left. \frac{1}{c \cos \theta} \left( \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right); \frac{1}{c \sin \theta} \left( \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right) \right\} \max \{b_1 - a_1; b_2 - a_2\}
\end{aligned} \right\} \\
&\qquad + (\mathcal{V}(M))^2 + \\
&\qquad \max_{1 \le i \le 4} \left\{ \begin{aligned}[t]
& \max \{1; c \cos \theta + c \sin \theta\} \|N_i^0\|_1; \\
& \max \left\{ 1; \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right\} \|N_1^-\|_1; \max \left\{ 1; \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right\} \|N_1^{--}\|_1; \\
& \max \left\{ 1; \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right\} \|N_2^+\|_1; \max \left\{ 1; \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right\} \|N_2^{--}\|_1; \\
& \max \left\{ 1; \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right\} \|N_3^-\|_1; \max \left\{ 1; \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right\} \|N_3^{++}\|_1; \\
& \max \left\{ 1; \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right\} \|N_4^+\|_1; \max \left\{ 1; \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right\} \|N_4^{++}\|_1
\end{aligned} \right\} \tag{4.38}
\end{align*}
$$

that is $\mathcal{V}(\mathcal{T}(M)) \le p(\mathcal{V}(M))^2 + q$ where

$$
p = 4cs \max \left\{ \begin{array}{@{}l@{}}
  \max \{1 + 2T(c \cos \theta + c \sin \theta); 2T\}; \\
  \max \left\{ \frac{1}{c \cos \theta}; \frac{1}{c \sin \theta} \right\} + 2 \max \left\{ \frac{1}{c \cos \theta}; \frac{1}{c \sin \theta}; \right. \\
  \qquad \left. \frac{1}{c \cos \theta} \left( \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right); \frac{1}{c \sin \theta} \left( \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right) \right\} \max \{b_1 - a_1; b_2 - a_2\}
\end{array} \right\} \tag{4.39}
$$
$$
\begin{equation}
\begin{aligned}
& q \equiv \max_{1 \le i \le 4} \left\{ \begin{array}{@{}l@{}}
    \displaystyle \max \{1; c \cos \theta + c \sin \theta\} \|N_i^0\|_1; \\
    \displaystyle \max \left\{1; \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta}\right\} \|N_1^-\|_1; \max \left\{1; \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta}\right\} \|N_1^{--}\|_1; \\
    \displaystyle \max \left\{1; \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta}\right\} \|N_2^+\|_1; \max \left\{1; \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta}\right\} \|N_2^{--}\|_1; \\
    \displaystyle \max \left\{1; \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta}\right\} \|N_3^-\|_1; \max \left\{1; \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta}\right\} \|N_3^{++}\|_1; \\
    \displaystyle \max \left\{1; \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta}\right\} \|N_4^+\|_1; \max \left\{1; \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta}\right\} \|N_4^{++}\|_1
  \end{array} \right\}
\end{aligned}
\tag{4.40}
\end{equation}
$$

If $M \in \mathcal{B}_R$ then $\mathcal{V}(\mathcal{T}(M)) \le pR^2 + q$. So $\mathcal{T}(\mathcal{B}_R) \subset \mathcal{B}_{pR^2+q}$. $\square$

**Lemma 4.5.** The operator $\mathcal{T}$ is compact on $\mathcal{B}_R$ for all $R > 0$.

**Proof.** $\mathcal{T}(\mathcal{B}_R) \subset \mathcal{B}_{pR^2+q}$ and $\mathcal{B}_{pR^2+q}$ is relatively compact in $(C(\mathcal{P}; \mathbb{R}^4), \|\cdot\|)$. Thus $\mathcal{T}$ is compact on $\mathcal{B}_R$. $\square$

**Lemma 4.6.** If $pq \le \frac{1}{4}$ and $\frac{1 - \sqrt{1 - 4pq}}{2p} \le R \le \frac{1 + \sqrt{1 - 4pq}}{2p}$ then $\mathcal{T}(\mathcal{B}_R) \subset \mathcal{B}_R$.

**Proof.** We have $pR^2 - R + q \le 0$. Hence $pR^2 + q \le R$. But $\mathcal{T}(\mathcal{B}_R) \subset \mathcal{B}_{pR^2+q}$ therefore $\mathcal{T}(\mathcal{B}_R) \subset \mathcal{B}_R$. $\square$

**Theorem 4.7.** (Schauder [11])

Let $\mathcal{M}$ be a non-empty convex subset of a normed space $\mathcal{X}$ and $\mathcal{T}$ be a continuous compact mapping from $\mathcal{M}$ into $\mathcal{M}$. Then $\mathcal{T}$ has a fixed point.

**4.2 Proof of the main theorem**

**Proof.** Let $\frac{1 - \sqrt{1 - 4pq}}{2p} \le R \le \frac{1 + \sqrt{1 - 4pq}}{2p}$. From proposition (4.3), $\mathcal{B}_R$ is a non-empty convex subset of $(C(\mathcal{P}; \mathbb{R}^4), \|\cdot\|)$. From lemmas (4.1), (4.5) and (4.6), $\mathcal{T}$ is continuous compact from $\mathcal{B}_R$ into $\mathcal{B}_R$. Thus according to Schauder's theorem 4.7, $\mathcal{T}$ has a fixed point $N \in \mathcal{B}_R$ which is thus a solution of problem $\Sigma$ and hence non-negative from theorem (3.1). $N \in \mathcal{B}_R$ also implies that $N \in \mathcal{E}^4$ and $\mathcal{V}(N) \le R \le \frac{1 + \sqrt{1 - 4pq}}{2p}$. Thus $N \in C(\mathcal{P}; \mathbb{R}^4)$ and $\frac{\partial N_i}{\partial t}, \frac{\partial N_i}{\partial x}, \frac{\partial N_i}{\partial y}$ ($i = 1, 2, 3, 4$) are defined in $\mathcal{P}$, except
possibly on a finite number of planes, and continuous and bounded with
$$ \|N\|, \left\| \frac{\partial N}{\partial t} \right\|, \left\| \frac{\partial N}{\partial x} \right\|, \left\| \frac{\partial N}{\partial y} \right\| \le \frac{1 + \sqrt{1 - 4pq}}{2p}. $$

Now, suppose that the problem Σ has two solutions M and N satisfying
$$ \|M\|, \|N\| \le \frac{1 + \sqrt{1 - 4pq}}{2p}. $$
From relation (4.14) it holds

$$
\begin{align}
\| \mathcal{T}(M) - \mathcal{T}(N) \| &\le p' \cdot 2 \cdot \frac{1 + \sqrt{1 - 4pq}}{2p} \|M - N\| \nonumber \\
&\le \frac{p'}{p} \cdot (1 + \sqrt{1 - 4pq}) \|M - N\|. \tag{4.41}
\end{align}
$$

As M and N are fixed points of T, (4.41) rewrites

$$
\|M - N\| \le \frac{p'}{p} \cdot \left(1 + \sqrt{1 - 4pq}\right) \|M - N\| \quad (4.42)
$$

i.e.

$$
\left(1 - \frac{p'}{p} \cdot \left(1 + \sqrt{1 - 4pq}\right)\right) \|M - N\| \le 0. \quad (4.43)
$$

From (4.14) and (4.39) we have

$$
\begin{align*}
\frac{p'}{p} &= \max \left\{ T, \frac{b_1 - a_1}{c \cos \theta}, \frac{b_2 - a_2}{c \sin \theta}, \frac{b_1 - a_1}{c \sin \theta}, \frac{b_2 - a_2}{c \cos \theta} \right\} \times \\
&\quad \left( \max \left\{ \max \{1 + 2T(c \cos \theta + c \sin \theta) ; 2T\} ; \right. \right. \\
&\quad \left. \left. \max \left\{ \frac{1}{c \cos \theta} ; \frac{1}{c \sin \theta} \right\} + 2 \max \left\{ \frac{1}{c \cos \theta} ; \frac{1}{c \sin \theta} ; \frac{1}{c \cos \theta} \left( \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right) ; \right. \right. \\
&\quad \left. \left. \frac{1}{c \sin \theta} \left( \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right) \right\} \max \{b_1 - a_1 ; b_2 - a_2\} \right)^{-1} \\
&\quad \equiv \frac{\max \{T, \alpha\}}{\max \{\beta_T, \beta\}} \tag{4.44}
\end{align*}
$$

where

$$
\begin{align}
\alpha &= \max \left\{ \frac{b_1 - a_1}{c \cos \theta}, \frac{b_2 - a_2}{c \sin \theta}, \frac{b_1 - a_1}{c \sin \theta}, \frac{b_2 - a_2}{c \cos \theta} \right\}; \nonumber \\
\beta_T &= \max \{1 + 2T(c \cos \theta + c \sin \theta); 2T\} \tag{4.45}
\end{align}
$$
$$
\begin{equation}
\begin{aligned}
\beta = \max & \left\{ \frac{1}{c \cos \theta}; \frac{1}{c \sin \theta} \right\} + 2 \max \left\{ \frac{1}{c \cos \theta}; \frac{1}{c \sin \theta} \right\}; \\
& \frac{1}{c \cos \theta} \left( \frac{1}{c \cos \theta} + \frac{\sin \theta}{\cos \theta} \right); \frac{1}{c \sin \theta} \left( \frac{1}{c \sin \theta} + \frac{\cos \theta}{\sin \theta} \right) \max \{b_1 - a_1; b_2 - a_2\}.
\end{aligned}
\tag{4.46}
\end{equation}
$$

We have

$$
\begin{cases}
\beta_T \ge 2T \\
\beta > 2\alpha
\end{cases}
\tag{4.47}
$$

$$
\max \{\beta_T, \beta\} \geq 2 \max \{T, \alpha\} \qquad (4.48)
$$

and

$$
\frac{p'}{p} = \frac{\max\{T, \alpha\}}{\max\{\beta_T, \beta\}} \le \frac{1}{2}.
$$

As $1 + \sqrt{1 - 4pq} < 2$, we have $\frac{p'}{p} \cdot (1 + \sqrt{1 - 4pq}) < 1$ i.e. $1 - \frac{p'}{p} \cdot (1 + \sqrt{1 - 4pq}) > 0$.

From (4.43) it follows that $||M - N|| \le 0$ therefore $M = N$. Hence the uniqueness. $\square$

**5 Conclusion**

In this paper we generalize the results obtained in [16] to the general four-
velocity Broadwell model in the plane. Precisely, we provide the proof of ex-
istence and uniqueness of classical positive solutions for the initial-boundary
value problem for this model in a rectangular domain with Dirichlet bound-
ary conditions. This work represents a key step toward a broader mathemat-
ical understanding of initial-boundary value problems for unsteady discrete
kinetic models in higher dimensions.
where

$$
\begin{equation}
\begin{aligned}
\mathcal{T}_2^A (M) (t, x, y) &= \int_0^t -Q(M) (s, x - c(s-t)\sin\theta, \\
&\qquad \qquad \qquad \qquad \qquad \qquad y + c(s-t)\cos\theta) \, ds + N_2^0 (x + ct\sin\theta, y - ct\cos\theta) \quad &(\text{A.6})
\end{aligned}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\mathcal{T}_2^B (M) (t, x, y) &= \int_0^{-\frac{1}{c \sin \theta} x + \frac{b_1}{c \sin \theta}} -Q(M) \left(s + t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, \right. \\
&\qquad \left. - sc \sin \theta + b_1, sc \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) ds + \\
&\qquad N_2^+ \left(t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right).
\end{aligned}
\tag{A.7}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\mathcal{T}_2^C (M) (t, x, y) &= \int_0^{\frac{1}{c \cos \theta} y - \frac{a_2}{c \cos \theta}} -Q(M) \left(s + t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, \right. \\
&\qquad \left. - sc \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2, sc \cos \theta + a_2 \right) ds + \\
&\qquad N_2^{-} \left(t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2 \right).
\end{aligned}
\tag{A.8}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\mathcal{T}_3 (M) (t, x, y) &= \mathcal{T}_3^A (M) (t, x, y) \cdot \mathbb{I} \begin{cases} x - ct \sin \theta \ge a_1 \\ y + ct \cos \theta \le b_2 \end{cases} (t, x, y) + \\
&\quad \mathcal{T}_3^B (M) (t, x, y) \cdot \mathbb{I} \begin{cases} x - ct \sin \theta \le a_1 \\ x \cos \theta + y \sin \theta \le a_1 \cos \theta + b_2 \sin \theta \end{cases} (t, x, y) + \\
&\quad \mathcal{T}_3^C (M) (t, x, y) \cdot \mathbb{I} \begin{cases} y + ct \cos \theta \ge b_2 \\ x \cos \theta + y \sin \theta \ge a_1 \cos \theta + b_2 \sin \theta \end{cases} (t, x, y)
\end{aligned}
\tag{A.9}
\end{equation}
$$

where

$$
\begin{equation}
\begin{aligned}
\mathcal{T}_3^A (M) (t, x, y) &= \int_0^t -Q(M) (s, x + c(s-t)\sin\theta, y - c(s-t)\cos\theta) \, ds + \\
&\qquad N_3^0 (x - ct\sin\theta, y + ct\cos\theta) \quad (\text{A.10})
\end{aligned}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\mathcal{T}_3^B (M) (t, x, y) &= \int_0^{\frac{1}{c \sin \theta} x - \frac{a_1}{c \sin \theta}} -Q(M) \left(s + t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta}, \right. \\
&\qquad \left. sc \sin \theta + a_1, -sc \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right) ds + \\
&\qquad N_3^{-} \left(t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta}, \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right)
\end{aligned}
\tag{A.11}
\end{equation}
$$
$$
\begin{equation}
\begin{aligned}
\mathcal{T}_3^C (M) (t, x, y) = & \int_0^{-\frac{1}{c \cos \theta} y + \frac{b_2}{c \cos \theta}} -Q(M) \left(s+t+\frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta}, \right. \\
& \left. \quad sc \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2, -sc \cos \theta + b_2 \right) ds + \right. \\
& \left. \qquad N_3^{++} \left(t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta}, x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2 \right) \right) \quad (A.11)
\end{aligned}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\mathcal{T}_4 (M) (t, x, y) = \mathcal{T}_4^A (M) (t, x, y) \cdot \mathbb{I} & \begin{cases} x + ct \cos \theta \le b_1 \\ y + ct \sin \theta \le b_2 \end{cases} (t, x, y) + \\
& \mathcal{T}_4^B (M) (t, x, y) \cdot \mathbb{I} \begin{cases} x + ct \cos \theta \ge b_1 \\ x \sin \theta - y \cos \theta \ge b_1 \sin \theta - b_2 \cos \theta \end{cases} (t, x, y) + \\
& \mathcal{T}_4^C (M) (t, x, y) \cdot \mathbb{I} \begin{cases} y + ct \sin \theta \ge b_2 \\ x \sin \theta - y \cos \theta \le b_1 \sin \theta - b_2 \cos \theta \end{cases} (t, x, y) \quad (A.12)
\end{aligned}
\end{equation}
$$

where

$$
\begin{equation}
\begin{aligned}
\mathcal{T}_4^A (M) (t, x, y) = & \int_0^t Q(M) (s, x - c(s-t)\cos\theta, y - c(s-t)\sin\theta) \, ds + \\
& N_4^0 (x + ct \cos\theta, y + ct \sin\theta) \quad (A.13)
\end{aligned}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\mathcal{T}_4^B (M) (t, x, y) = & \left( \int_0^{-\frac{1}{c \cos \theta} x + \frac{b_1}{c \cos \theta}} Q(M) \left( s + t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, \right. \right. \\
& \left. \left. \quad -sc \cos \theta + b_1, -sc \sin \theta - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} b_1 \right) ds + \right. \\
& \left. \qquad N_4^+ \left( t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, -\frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} b_1 \right) \right) \quad (A.14)
\end{aligned}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\mathcal{T}_4^C (M) (t, x, y) = & \int_0^{-\frac{1}{c \sin \theta} y + \frac{b_2}{c \sin \theta}} Q(M) \left(s + t + \frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, \right. \\
& \left. \quad -sc \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2, -sc \sin \theta + b_2 \right) ds + \\
& N_4^{++} \left(t + \frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2 \right).
\end{aligned}
\tag{A.15}
\end{equation}
$$

B Derivatives $\frac{\partial \mathcal{T}_i(M)}{\partial t}$, $\frac{\partial \mathcal{T}_i(M)}{\partial x}$, $\frac{\partial \mathcal{T}_i(M)}{\partial y}$, ($i=1,2,3,4$)

$\frac{\partial \mathcal{T}_1 (M)}{\partial t}$, $\frac{\partial \mathcal{T}_1 (M)}{\partial x}$, $\frac{\partial \mathcal{T}_1 (M)}{\partial y}$ are defined in $\mathcal{P}$ except perhaps on the planes given respectively by the equations $x - ct \cos \theta = a_1$; $y - ct \sin \theta = a_2$; $x \sin \theta -
$y \cos \theta = a_1 \sin \theta - a_2 \cos \theta$ and

$$ 
\begin{aligned}
& \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_1 (M) (t, x, y) = \\
& \qquad \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_1^A (M) (t, x, y) \cdot \mathbb{I} \left\{ \begin{array}{l} x - ct \cos \theta > a_1 \\ y - ct \sin \theta > a_2 \end{array} \right. (t, x, y) + \\
& \qquad \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_1^B (M) (t, x, y) \cdot \\
& \qquad \qquad \mathbb{I} \left\{ \begin{array}{l} x - ct \cos \theta < a_1 \\ x \sin \theta - y \cos \theta < a_1 \sin \theta - a_2 \cos \theta \end{array} \right. (t, x, y) + \\
& \qquad \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_1^C (M) (t, x, y) \cdot \mathbb{I} \left\{ \begin{array}{l} y - ct \sin \theta < a_2 \\ x \sin \theta - y \cos \theta > a_1 \sin \theta - a_2 \cos \theta \end{array} \right. (t, x, y)
\end{aligned}
\quad (B.1) $$

where

$$ 
\begin{aligned}
\frac{\partial \mathcal{T}_1^A (M)}{\partial t} (t, x, y) ={}& Q (M) (t, x, y) + \int_0^t \left[ -c \cos \theta \frac{\partial Q (M)}{\partial x} \right. \\
& \left. \qquad (s, x + c(s-t) \cos \theta, y + c(s-t) \sin \theta) - c \sin \theta \frac{\partial Q (M)}{\partial y} \right. \\
& \left. \qquad \left( s, x + c(s-t) \cos \theta, y + c(s-t) \sin \theta \right) \right] ds \\
& -c \cos \theta \frac{\partial N_1^0}{\partial x} (x - ct \cos \theta, y - ct \sin \theta) -c \sin \theta \frac{\partial N_1^0}{\partial y} (x - ct \cos \theta, y - ct \sin \theta)
\end{aligned}
\quad (B.2) $$

$$ 
\begin{aligned}
\frac{\partial \mathcal{T}_1^B (M)}{\partial t} (t, x, y) ={}& \int_0^{\frac{1}{c \cos \theta} x - \frac{a_1}{c \cos \theta}} \frac{\partial Q (M)}{\partial t} \left( s + t - \frac{1}{c \cos \theta} x + \frac{a_1}{c \cos \theta}, \right. \\
& \left. sc \cos \theta + a_1, sc \sin \theta - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} a_1 \right) ds + \\
& \frac{\partial N_1^-}{\partial t} \left( t - \frac{1}{c \cos \theta} x + \frac{a_1}{c \cos \theta}, -\frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} a_1 \right) (B.3)
\end{aligned}
$$

$$ 
\begin{aligned}
\frac{\partial \mathcal{T}_1^C (M)}{\partial t} (t, x, y) ={}& \int_0^{\frac{1}{c \sin \theta} y - \frac{a_2}{c \sin \theta}} \frac{\partial Q (M)}{\partial t} \left( s + t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta}, \right. \\
& \left. sc \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} a_2, sc \sin \theta + a_2 \right) ds + \\
& \frac{\partial N_1^{--}}{\partial t} \left( t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta}, x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} a_2 \right) (B.4)
\end{aligned}
$$
$$
\frac{\partial \mathcal{T}_1^A (M)}{\partial x} (t, x, y) = \int_0^t \left[ \frac{\partial Q (M)}{\partial x} \left( s, x + c (s-t) \cos \theta, y + c (s-t) \sin \theta \right) ds + \frac{\partial N_1^0}{\partial x} \left( x - ct \cos \theta, y - ct \sin \theta \right) \right] \tag{B.5}
$$

$$
\begin{equation}
\begin{split}
\frac{\partial \mathcal{T}_1^B (M)}{\partial x} (t, x, y) = {}& \frac{1}{c \cos \theta} Q (M) (t, x, y) + \\
& \int_0^{\frac{1}{c \cos \theta} x - \frac{a_1}{c \cos \theta}} \left( -\frac{1}{c \cos \theta} \frac{\partial Q (M)}{\partial t} \left( s + t - \frac{1}{c \cos \theta} x + \frac{a_1}{c \cos \theta}, \right. \right. \\
& \qquad \left. sc \cos \theta + a_1, sc \sin \theta - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} a_1 \right) \\
& - \frac{\sin \theta}{\cos \theta} \frac{\partial Q (M)}{\partial y} \left( s + t - \frac{1}{c \cos \theta} x + \frac{a_1}{c \cos \theta}, sc \cos \theta + a_1, \right. \\
& \qquad \left. sc \sin \theta - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} a_1 \right) \right) ds + \\
& - \frac{1}{c \cos \theta} \frac{\partial N_1^-}{\partial t} \left( t - \frac{1}{c \cos \theta} x + \frac{a_1}{c \cos \theta}, -\frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} a_1 \right) \\
& - \frac{\sin \theta}{\cos \theta} \frac{\partial N_1^-}{\partial t} \left( t - \frac{1}{c \cos \theta} x + \frac{a_1}{c \cos \theta}, -\frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} a_1 \right)
\end{split}
\tag{B.6}
\end{equation}
$$

$$
\begin{equation}
\begin{split}
\frac{\partial \mathcal{T}_1^C (M)}{\partial x} (t, x, y) = & \int_0^{\frac{1}{c \sin \theta} y - \frac{a_2}{c \sin \theta}} \frac{\partial Q (M)}{\partial y} \left( s + t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta}, \right. \\
& \left. sc \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} a_2, sc \sin \theta + a_2 \right) ds + \\
& \frac{\partial N_1^{--}}{\partial x} \left( t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta}, x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} a_2 \right)
\end{split}
\tag{B.7}
\end{equation}
$$

$$
\frac{\partial \mathcal{T}_1^A (M)}{\partial y} (t, x, y) = \int_0^t \left[ \frac{\partial Q (M)}{\partial y} \left( s, x + c (s-t) \cos \theta, y + c (s-t) \sin \theta \right) ds + \frac{\partial N_1^0}{\partial y} \left( x - ct \cos \theta, y - ct \sin \theta \right) \right] \tag{B.8}
$$

$$
\begin{equation}
\begin{split}
\frac{\partial \mathcal{T}_1^B (M)}{\partial y} (t, x, y) = & \int_0^{\frac{1}{c \cos \theta} x - \frac{a_1}{c \cos \theta}} \frac{\partial Q (M)}{\partial y} \left( s + t - \frac{1}{c \cos \theta} x + \frac{a_1}{c \cos \theta}, \right. \\
& \left. sc \cos \theta + a_1, sc \sin \theta - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} a_1 \right) ds + \\
& \frac{\partial N_1^-}{\partial y} \left( t - \frac{1}{c \cos \theta} x + \frac{a_1}{c \cos \theta}, -\frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} a_1 \right)
\end{split}
\tag{B.9}
\end{equation}
$$
$$
\begin{align*}
\frac{\partial \mathcal{T}_1^C (M)}{\partial y} (t, x, y) = & \\
& \frac{1}{c \sin \theta} Q (M) (t, x, y) + \int_0^{\frac{1}{c \sin \theta} y - \frac{a_1}{c \sin \theta}} \left( -\frac{1}{c \sin \theta} \frac{\partial Q (M)}{\partial t} \left( s + t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta}, \right. \right. \\
& \left. \left. \text{sc} \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} a_2, \text{sc} \sin \theta + a_2 \right) \right. \\
& - \frac{\cos \theta}{\sin \theta} \frac{\partial Q (M)}{\partial y} \left( s + t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta}, \text{sc} \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} a_2, \text{sc} \sin \theta + a_2 \right) \right) ds + \\
& - \frac{1}{c \sin \theta} \frac{\partial N_1^{--}}{\partial t} \left( t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta}, x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} a_2 \right) \\
& - \frac{\cos \theta}{\sin \theta} \frac{\partial N_1^{--}}{\partial x} \left( t - \frac{1}{c \sin \theta} y + \frac{a_2}{c \sin \theta}, x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} a_2 \right) \tag{B.10}
\end{align*}
$$

**** $\frac{\partial \mathcal{T}_2 (M)}{\partial t}$, $\frac{\partial \mathcal{T}_2 (M)}{\partial x}$, $\frac{\partial \mathcal{T}_2 (M)}{\partial y}$ are defined in $\mathcal{P}$ except perhaps on the planes given respectively by the equations $x + ct \sin \theta = b_1$; $y - ct \cos \theta = a_2$; $x \cos \theta + y \sin \theta = b_1 \cos \theta + a_2 \sin \theta$ and

$$
\begin{align*}
\left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_2 (M) (t, x, y) = & \\
& \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_2^A (M) (t, x, y) \cdot \mathbb{I} \begin{cases} x + ct \sin \theta < b_1 & (t, x, y) + \\ y - ct \cos \theta > a_2 & \end{cases} \\
& \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_2^B (M) (t, x, y) \cdot \mathbb{I} \begin{cases} x + ct \sin \theta > b_1 & (t, x, y) + \\ x \cos \theta + y \sin \theta > b_1 \cos \theta + a_2 \sin \theta & \end{cases} \\
& \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_2^C (M) (t, x, y) \cdot \mathbb{I} \begin{cases} y - ct \cos \theta < a_2 & (t, x, y) + \\ x \cos \theta + y \sin \theta < b_1 \cos \theta + a_2 \sin \theta & \end{cases} \tag{B.11}
\end{align*}
$$

where

$$
\begin{align}
\frac{\partial \mathcal{T}_2^A (M)}{\partial t} (t, x, y) = & -Q (M) (t, x, y) + \int_0^t \left[ -c \sin \theta \frac{\partial Q (M)}{\partial x} \right. \nonumber \\
& \left. \quad (s, x - c(s-t) \sin \theta, y + c(s-t) \cos \theta) + c \cos \theta \frac{\partial Q (M)}{\partial y} \right. \nonumber \\
& \left. \quad (s, x - c(s-t) \sin \theta, y + c(s-t) \cos \theta) \right] ds \nonumber \\
& + c \sin \theta \frac{\partial N_2^0}{\partial x} (x + ct \sin \theta, y - ct \cos \theta) - c \cos \theta \frac{\partial N_2^0}{\partial y} (x + ct \sin \theta, y - ct \cos \theta) \tag{B.12}
\end{align}
$$
$$
\begin{align}
\frac{\partial \mathcal{T}_2^B (M)}{\partial t} (t, x, y) &= \int_0^{-\frac{1}{c \sin \theta} x + \frac{b_1}{c \sin \theta}} -\frac{\partial Q (M)}{\partial t} \left( s + t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, \right. \nonumber \\
&\qquad \left. - s c \sin \theta + b_1, s c \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) ds + \nonumber \\
&\qquad \frac{\partial N_2^+}{\partial t} \left( t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) \tag{B.13}
\end{align}
$$

$$
\begin{align}
\frac{\partial \mathcal{T}_2^C (M)}{\partial t} (t, x, y) &= \int_0^{\frac{1}{c \cos \theta} y - \frac{a_2}{c \cos \theta}} -\frac{\partial Q (M)}{\partial t} \left( s + t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, \right. \nonumber \\
&\qquad \left. - s c \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2, s c \cos \theta + a_2 \right) ds + \nonumber \\
&\qquad \frac{\partial N_2^{--}}{\partial t} \left( t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2 \right) \tag{B.14}
\end{align}
$$

$$
\begin{align}
\frac{\partial \mathcal{T}_2^A (M)}{\partial x} (t, x, y) = & \int_0^t \left[ -\frac{\partial Q (M)}{\partial x} \left( s, x - c (s-t) \sin \theta, y + c (s-t) \cos \theta \right) ds + \right. \nonumber \\
& \left. \frac{\partial N_2^0}{\partial x} (x + ct \sin \theta, y - ct \cos \theta) \right] \tag{B.15}
\end{align}
$$

$$
\begin{align}
\frac{\partial \mathcal{T}_2^B (M)}{\partial x} (t, x, y) ={}& \frac{1}{c \sin \theta} Q (M) (t, x, y) + \nonumber \\
& \int_0^{-\frac{1}{c \sin \theta} x + \frac{b_1}{c \sin \theta}} \left( -\frac{1}{c \sin \theta} \frac{\partial Q (M)}{\partial t} \left( s + t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, \right. \right. \nonumber \\
& \qquad \left. - s c \sin \theta + b_1, s c \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) \right. \nonumber \\
& \qquad \left. - \frac{\cos \theta}{\sin \theta} \frac{\partial Q (M)}{\partial y} \left( s + t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, -s c \sin \theta + b_1, \right. \right. \nonumber \\
& \qquad \left. \left. s c \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) \right) ds + \nonumber \\
& + \frac{1}{c \sin \theta} \frac{\partial N_2^+}{\partial t} \left( t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) \nonumber \\
& + \frac{\cos \theta}{\sin \theta} \frac{\partial N_2^+}{\partial t} \left( t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) \tag{B.16}
\end{align}
$$

$$
\begin{align}
\frac{\partial \mathcal{T}_2^C (M)}{\partial x} (t, x, y) &= \int_0^{\frac{1}{c \cos \theta} y - \frac{a_2}{c \cos \theta}} -\frac{\partial Q (M)}{\partial x} \left( s + t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, \right. \nonumber \\
&\qquad \left. - s c \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2, s c \cos \theta + a_2 \right) ds + \nonumber \\
&\qquad \frac{\partial N_2^{--}}{\partial x} \left( t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2 \right) \tag{B.17}
\end{align}
$$
$$
\frac{\partial \mathcal{T}_2^A (M)}{\partial y} (t, x, y) =
\int_0^t -\frac{\partial Q (M)}{\partial y} \left( s, x - c (s-t) \sin \theta, y + c (s-t) \cos \theta \right) ds +
\frac{\partial N_2^0}{\partial y} \left( x + ct \sin \theta, y - ct \cos \theta \right) \quad (B.18)
$$

$$
\frac{\partial \mathcal{T}_2^B (M)}{\partial y} (t, x, y) =
\int_0^{-\frac{1}{c \sin \theta} x + \frac{b_1}{c \sin \theta}} -\frac{\partial Q (M)}{\partial y} \left( s + t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, \right.
\left. - s c \sin \theta + b_1, s c \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) ds +
\frac{\partial N_2^+}{\partial y} \left( t + \frac{1}{c \sin \theta} x - \frac{b_1}{c \sin \theta}, \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} b_1 \right) \quad (B.19)
$$

$$
\frac{\partial \mathcal{T}_2^C (M)}{\partial y} (t, x, y) =
-\frac{1}{c \cos \theta} Q (M) (t, x, y) + \int_0^{\frac{1}{c \cos \theta} y - \frac{a_2}{c \cos \theta}} \left( \frac{1}{c \cos \theta} \frac{\partial Q (M)}{\partial t} \left( s + t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, \right. \right.
\left. - s c \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2, s c \cos \theta + a_2 \right) \right.
- \frac{\sin \theta}{\cos \theta} \frac{\partial Q (M)}{\partial x} \left( s + t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, -s c \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2, s c \cos \theta + a_2 \right) \right) ds +
- \frac{1}{c \cos \theta} \frac{\partial N_2^{--}}{\partial t} \left( t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2 \right)
+ \frac{\sin \theta}{\cos \theta} \frac{\partial N_2^{--}}{\partial x} \left( t - \frac{1}{c \cos \theta} y + \frac{a_2}{c \cos \theta}, x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} a_2 \right) \quad (B.20)
$$

**** $\frac{\partial \mathcal{T}_3 (M)}{\partial t}$, $\frac{\partial \mathcal{T}_3 (M)}{\partial x}$, $\frac{\partial \mathcal{T}_3 (M)}{\partial y}$ are defined in $\mathcal{P}$ except perhaps on the planes given respectively by the equations $x - ct \sin \theta = a_1$; $y + ct \cos \theta =$
$b_2; x \cos \theta + y \sin \theta = a_1 \cos \theta + b_2 \sin \theta$ and

$$
\begin{equation}
\begin{aligned}
& \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_3 (M) (t, x, y) = \\
& \qquad \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_3^A (M) (t, x, y) \cdot \mathbb{I} \begin{cases} x - ct \sin \theta > a_1 \\ y + ct \cos \theta < b_2 \end{cases} (t, x, y) + \\
& \qquad \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_3^B (M) (t, x, y) \cdot \mathbb{I} \begin{cases} x - ct \sin \theta < a_1 \\ x \cos \theta + y \sin \theta < a_1 \cos \theta + b_2 \sin \theta \end{cases} (t, x, y) + \\
& \qquad \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_3^C (M) (t, x, y) \cdot \mathbb{I} \begin{cases} y + ct \cos \theta > b_2 \\ x \cos \theta + y \sin \theta > a_1 \cos \theta + b_2 \sin \theta \end{cases} (t, x, y)
\end{aligned}
\tag{B.21}
\end{equation}
$$

where

$$
\begin{equation}
\begin{aligned}
\frac{\partial \mathcal{T}_3^A (M)}{\partial t} (t, x, y) &= -Q (M) (t, x, y) + \int_0^t \left[ c \sin \theta \frac{\partial Q (M)}{\partial x} \right. \\
& \qquad \left. (s, x + c(s-t) \sin \theta, y - c(s-t) \cos \theta) - c \cos \theta \frac{\partial Q (M)}{\partial y} \right. \\
& \qquad \left. (s, x + c(s-t) \sin \theta, y - c(s-t) \cos \theta) \right] ds \\
& \qquad -c \sin \theta \frac{\partial N_3^0}{\partial x} (x - ct \sin \theta, y + ct \cos \theta) + c \cos \theta \frac{\partial N_3^0}{\partial y} (x - ct \sin \theta, y + ct \cos \theta)
\end{aligned}
\tag{B.22}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\frac{\partial \mathcal{T}_3^B (M)}{\partial t} (t, x, y) &= \int_0^{\frac{1}{c \sin \theta} x - \frac{a_1}{c \sin \theta}} -\frac{\partial Q (M)}{\partial t} \left( s + t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta} \right. \\
& \qquad \left. sc \sin \theta + a_1, -sc \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right) ds + \\
& \qquad \frac{\partial N_3^-}{\partial t} \left( t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta}, \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right)
\end{aligned}
\tag{B.23}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\frac{\partial \mathcal{T}_3^C (M)}{\partial t} (t, x, y) &= \int_0^{-\frac{1}{c \cos \theta} y + \frac{b_2}{c \cos \theta}} -\frac{\partial Q (M)}{\partial t} \left( s + t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta} \right. \\
& \qquad \left. sc \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2, -sc \cos \theta + b_2 \right) ds + \\
& \qquad \frac{\partial N_3^{++}}{\partial t} \left( t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta}, x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2 \right)
\end{aligned}
\tag{B.24}
\end{equation}
$$
$$ \frac{\partial \mathcal{T}_3^A (M)}{\partial x} (t, x, y) = \int_0^t \left[ -\frac{\partial Q (M)}{\partial x} \left( s, x + c(s-t)\sin\theta, y - c(s-t)\cos\theta \right) ds + \frac{\partial N_3^0}{\partial x} (x - ct\sin\theta, y + ct\cos\theta) \right] \quad (B.25) $$

$$ \frac{\partial \mathcal{T}_3^B (M)}{\partial x} (t, x, y) = -\frac{1}{c \sin \theta} Q (M) (t, x, y) + \int_0^{\frac{1}{c \sin \theta} x - \frac{a_1}{c \sin \theta}} \left( \frac{1}{c \sin \theta} \frac{\partial Q (M)}{\partial t} \left( s + t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta}, \right. \right. \\ \left. sc \sin \theta + a_1, -sc \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right) \\ - \frac{\cos \theta}{\sin \theta} \frac{\partial Q (M)}{\partial y} \left( s + t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta}, sc \sin \theta + a_1, \right. \\ \left. - sc \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right) \right) ds + \\ - \frac{1}{c \sin \theta} \frac{\partial N_3^-}{\partial t} \left( t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta}, \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right) \\ \left. + \frac{\cos \theta}{\sin \theta} \frac{\partial N_3^-}{\partial t} \left( t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta}, \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right) \right) \quad (B.26) $$

$$ \frac{\partial \mathcal{T}_3^C (M)}{\partial x} (t, x, y) = \int_0^{-\frac{1}{c \cos \theta} y + \frac{b_2}{c \cos \theta}} -\frac{\partial Q (M)}{\partial x} \left( s + t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta}, \right. \\ \left. sc \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2, -sc \cos \theta + b_2 \right) ds + \\ \frac{\partial N_3^{++}}{\partial x} \left( t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta}, x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2 \right) \quad (B.27) $$

$$ \frac{\partial \mathcal{T}_3^A (M)}{\partial y} (t, x, y) = \int_0^t \left[ -\frac{\partial Q (M)}{\partial y} \left( s, x + c(s-t)\sin\theta, y - c(s-t)\cos\theta \right) ds + \frac{\partial N_3^0}{\partial y} (x - ct\sin\theta, y + ct\cos\theta) \right] \quad (B.28) $$
$$
\begin{align*}
\frac{\partial \mathcal{T}_3^B (M)}{\partial y} (t, x, y) = & \int_0^{\frac{1}{c \sin \theta} x - \frac{a_1}{c \sin \theta}} - \frac{\partial Q (M)}{\partial y} \left( s + t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta}, \right. \\
& \left. \left( sc \sin \theta + a_1, -sc \cos \theta + \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right) \right) ds + \right. \\
& \left. + \frac{\partial N_3^-}{\partial y} \left( t - \frac{1}{c \sin \theta} x + \frac{a_1}{c \sin \theta}, \frac{\cos \theta}{\sin \theta} x + y - \frac{\cos \theta}{\sin \theta} a_1 \right) \right) \tag{B.29}
\end{align*}
$$

$$
\begin{align*}
\frac{\partial \mathcal{T}_3^C (M)}{\partial y} (t, x, y) ={}& \frac{1}{c \cos \theta} Q (M) (t, x, y) + \\
& \int_0^{-\frac{1}{c \cos \theta} y + \frac{b_2}{c \cos \theta}} \left( -\frac{1}{c \cos \theta} \frac{\partial Q (M)}{\partial t} \left( s + t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta} \right. \right. \\
& \left. \left. \qquad sc \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2, -sc \cos \theta + b_2 \right) \right. \\
& \left. - \frac{\sin \theta}{\cos \theta} \frac{\partial Q (M)}{\partial x} \left( s + t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta}, sc \sin \theta + x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2, \right. \right. \\
& \left. \left. - sc \cos \theta + b_2 \right) \right) ds + \frac{1}{c \cos \theta} \frac{\partial N_3^{++}}{\partial t} \left( t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta}, x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2 \right) \\
& + \frac{\sin \theta}{\cos \theta} \frac{\partial N_3^{++}}{\partial x} \left( t + \frac{1}{c \cos \theta} y - \frac{b_2}{c \cos \theta}, x + \frac{\sin \theta}{\cos \theta} y - \frac{\sin \theta}{\cos \theta} b_2 \right) \tag{B.30}
\end{align*}
$$

$\mathcal{T}_4^B$ are defined in $\mathcal{P}$ except perhaps on
the planes given respectively by the equations $x + ct \cos \theta = b_1$; $y + ct \sin \theta = b_2$; $x \sin \theta - y \cos \theta = b_1 \sin \theta - b_2 \cos \theta$ and

$$
\begin{align*}
\left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_4 (M) (t, x, y) = & \\
\left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_4^A (M) (t, x, y) \cdot \mathbb{I} & \begin{cases} x + ct \cos \theta < b_1 & (t, x, y) + \\ y + ct \sin \theta < b_2 & \end{cases} \\
& \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_4^B (M) (t, x, y) \cdot \mathbb{I} & \begin{cases} x + ct \cos \theta > b_1 & (t, x, y) + \\ x \sin \theta - y \cos \theta > b_1 \sin \theta - b_2 \cos \theta & \end{cases} \\
& \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right) \mathcal{T}_4^C (M) (t, x, y) \cdot \mathbb{I} & \begin{cases} y + ct \sin \theta > b_2 & (t, x, y) + \\ x \sin \theta - y \cos \theta < b_1 \sin \theta - b_2 \cos \theta & \end{cases} \tag{B.31}
\end{align*}
$$
where

$$
\begin{equation}
\begin{aligned}
\frac{\partial \mathcal{T}_4^A (M)}{\partial t} (t, x, y) ={}& Q(M) (t, x, y) + \int_0^t \left[ c \cos \theta \frac{\partial Q(M)}{\partial x} \right. \\
& \left. \qquad (s, x - c(s-t) \cos \theta, y - c(s-t) \sin \theta) + c \sin \theta \frac{\partial Q(M)}{\partial y} \right. \\
& \left. \qquad \left. (s, x - c(s-t) \cos \theta, y - c(s-t) \sin \theta) \right] ds \right. \\
& \left. + c \cos \theta \frac{\partial N_4^0}{\partial x} (x + ct \cos \theta, y + ct \sin \theta) + c \sin \theta \frac{\partial N_4^0}{\partial y} (x + ct \cos \theta, y + ct \sin \theta) \right]
\end{aligned}
\tag{B.32}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\frac{\partial \mathcal{T}_4^B (M)}{\partial t} (t, x, y) = & \int_0^{-\frac{1}{c \cos \theta} x + \frac{b_1}{c \cos \theta}} \frac{\partial Q(M)}{\partial t} \left( s + t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta} \right. \\
& \left. - s c \cos \theta + b_1, - s c \sin \theta - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} b_1 \right) ds + \\
& \frac{\partial N_4^+}{\partial t} \left( t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} b_1 \right)
\end{aligned}
\tag{B.33}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\frac{\partial \mathcal{T}_4^C (M)}{\partial t} (t, x, y) = & \int_0^{-\frac{1}{c \sin \theta} y + \frac{b_2}{c \sin \theta}} \frac{\partial Q(M)}{\partial t} \left( s + t + \frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta} \right. \\
& \left. - s c \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2, - s c \sin \theta + b_2 \right) ds + \\
& \frac{\partial N_4^{++}}{\partial t} \left( t + \frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2 \right)
\end{aligned}
\tag{B.34}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\frac{\partial \mathcal{T}_4^A (M)}{\partial x} (t, x, y) = & \int_0^t \left[ \frac{\partial Q(M)}{\partial x} (s, x - c(s-t) \cos \theta, y - c(s-t) \sin \theta) \right. ds + \\
& \left. \frac{\partial N_4^0}{\partial x} (x + ct \cos \theta, y + ct \sin \theta) \right]
\end{aligned}
\tag{B.35}
\end{equation}
$$
$$
\begin{align*}
\frac{\partial \mathcal{T}_4^B (M)}{\partial x} (t, x, y) = & -\frac{1}{c \cos \theta} Q (M) (t, x, y) + \\
& \int_0^{-\frac{1}{c \cos \theta} x + \frac{b_1}{c \cos \theta}} \left( \frac{1}{c \cos \theta} \frac{\partial Q (M)}{\partial t} \right) \left( s + t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, \right. \\
& \qquad \left. - s c \cos \theta + b_1, - s c \sin \theta - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} b_1 \right) \\
& - \frac{\sin \theta}{\cos \theta} \frac{\partial Q (M)}{\partial y} \left( s + t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, - s c \cos \theta + b_1, \right. \\
& \qquad \left. - s c \sin \theta - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} b_1 \right) ds + \\
& \frac{1}{c \cos \theta} \frac{\partial N_4^+}{\partial t} \left( t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} b_1 \right) \\
& - \frac{\sin \theta}{\cos \theta} \frac{\partial N_4^+}{\partial y} \left( t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} b_1 \right) \tag{B.36}
\end{align*}
$$

$$
\begin{align*}
\frac{\partial \mathcal{T}_4^C (M)}{\partial x} (t, x, y) = & \int_0^{-\frac{1}{c \sin \theta} y + \frac{b_2}{c \sin \theta}} \frac{\partial Q (M)}{\partial x} \left( s + t + \frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, \right. \\
& \left. - s c \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2, - s c \sin \theta + b_2 \right) ds + \\
& \frac{\partial N_4^{++}}{\partial x} \left( t + \frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2 \right) \tag{B.37}
\end{align*}
$$

$$
\begin{align*}
\frac{\partial \mathcal{T}_4^A (M)}{\partial y} (t, x, y) = & \\
& \int_0^t \left[ \frac{\partial Q (M)}{\partial y} \left( s, x - c (s - t) \cos \theta, y - c (s - t) \sin \theta \right) ds + \right. \\
& \left. \frac{\partial N_4^0}{\partial y} \left( x + c t \cos \theta, y + c t \sin \theta \right) \right] \tag{B.38}
\end{align*}
$$

$$
\begin{align*}
\frac{\partial \mathcal{T}_4^B (M)}{\partial y} (t, x, y) = & \int_0^{-\frac{1}{c \cos \theta} x + \frac{b_1}{c \cos \theta}} \frac{\partial Q (M)}{\partial y} \left( s + t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, \right. \\
& \left. - s c \cos \theta + b_1, - s c \sin \theta - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} b_1 \right) ds + \\
& \frac{\partial N_4^+}{\partial y} \left( t + \frac{1}{c \cos \theta} x - \frac{b_1}{c \cos \theta}, - \frac{\sin \theta}{\cos \theta} x + y + \frac{\sin \theta}{\cos \theta} b_1 \right) \tag{B.39}
\end{align*}
$$
$$
\begin{align*}
\frac{\partial \mathcal{T}_4^C (M)}{\partial y} (t, x, y) = & -\frac{1}{c \sin \theta} Q(M)(t, x, y) + \int_0^{-\frac{1}{c \sin \theta} y + \frac{b_2}{c \sin \theta}} \left( \frac{1}{c \sin \theta} \frac{\partial Q(M)}{\partial t} \left( s+t+\frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, \right. \right. \\
& \left. \left. -sc \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2, -sc \sin \theta + b_2 \right) \right. \\
& - \frac{\cos \theta}{\sin \theta} \frac{\partial Q(M)}{\partial x} \left( s+t+\frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, -sc \cos \theta + x - \frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2, \right. \\
& \left. \left. -sc \sin \theta + b_2 \right) \right) ds + \\
& + \frac{1}{c \sin \theta} \frac{\partial N_4^{++}}{\partial t} \left( t+\frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, x-\frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2 \right) \\
& - \frac{\cos \theta}{\sin \theta} \frac{\partial N_4^{++}}{\partial x} \left( t+\frac{1}{c \sin \theta} y - \frac{b_2}{c \sin \theta}, x-\frac{\cos \theta}{\sin \theta} y + \frac{\cos \theta}{\sin \theta} b_2 \right) \quad (B.40)
\end{align*}
$$

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