# Existence results for nonlinear mono-energetic singular transport equations: $L^p$-spaces

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal

► To cite this version:

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $L^p$-spaces. Discrete and Continuous Dynamical Systems - Series S, 2022, 15 (1), pp.179-195. 10.3934/dcdss.2021028. hal-04910620

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# Existence results for nonlinear mono-energetic singular transport equations: $L^p$-spaces

Khalid Latrach<sup>a</sup> Hssaine Oummi<sup>b</sup> and Ahmed Zeghal<sup>b, *</sup>

<sup>a</sup>Université Clermont Auvergne

CNRS, LMBP

F-63000 Clermont-Ferrand, France

E-mail: khalid.latrach@uca.fr

<sup>b</sup>Laboratoire de Mathématiques et Applications

Université Sultan Moulay Slimane, Faculté des Sciences et Techniques

B.P. 523, 23000 Beni-Mellal, Morocco

E-mail: a.zeghal@fstbm.ac.ma

## Abstract

We establish some results regarding the existence of solutions to a nonlinear mono-energetic singular transport equation in slab geometry on $L^p$-spaces with $p \in (1, +\infty)$. Both the cases where the boundary conditions are specular reflections and periodic are discussed.

**Keywords:** nonlinear transport equation; fixed point theorems; existence results.

**AMS subject classifications:** 35F20; 45K05; 47H10

## 1 Introduction

In this section, we are concerned with existence of solutions to the nonlinear boundary value problem

$$ \xi \frac{\partial \psi}{\partial x}(x, \xi) + \sigma(\xi)\psi(x, \xi) + \lambda \psi(x, \xi) = \int_{-1}^{1} \kappa(x, \xi, \xi') f(x, \xi', \psi(x, \xi')) d\xi', \quad (1) $$

supplemented with specular reflecting boundary conditions i.e.

$$ \left\{ \begin{array}{ll} \psi(a, \xi) = \psi(a, -\xi), & \xi \in (-1, 0) \\ \psi(-a, \xi) = \psi(-a, -\xi), & \xi \in (0, 1) \end{array} \right. \quad (2) $$

on $L_p$ spaces with $1 < p < \infty$ where $x \in (-a, a)$ for a parameter $0 < a < \infty$, $\xi \in (-1, 1)$. It describes the transport of particles (neutrons, photons, molecules of gas, etc.) in a slab with thickness $2a$. The function $\psi(x, \xi)$ represents the number density of gas particles having the position $x$ and the direction cosine of propagation $\xi$. (The variable $\xi$ may be thought of as the cosine of

<footnote>*Corresponding author</footnote>
the angle between the velocity of particles and the x-direction). The functions $\sigma(\cdot)$ and $\kappa(\cdot, \cdot, \cdot)$ are measurable functions called, respectively, the collision frequency and the scattering kernel. The function $f(\cdot, \cdot, \cdot)$ is a measurable nonlinear function of the number density of gas particles $\psi$ and $\lambda$ is a complex number. The functions $\sigma(\cdot)$ and $\kappa(\cdot, \cdot, \cdot)$ are assumed to satisfy the following conditions:

(i) there exist a closed subset $O \subseteq (-1, 1)$ with zero Lebesgue measure and a constant $\sigma_0 > 0$ such that $\sigma(\cdot) \in L_{\text{loc}}^{\infty}((-1, 1)\setminus O)$, $\sigma(\xi) > \sigma_0$ a.e. on $(-1, 1)$ and $\sigma(\xi) = \sigma(-\xi) \quad \forall \xi \in (-1, 1)$;

(ii) the collision operator $K\psi(x, \xi) := \int_{-1}^{1} \kappa(x, \xi, \xi')\psi(x, \xi')d\xi'$ viewed as an operator from $L^p([-1, 1]; \sigma(\xi)d\xi)$ into $L^p([-1, 1]; d\xi)$ is assumed to be compact and positive (in the lattice sense).

These two assumptions were first made by M. Chabi and M. Mokhtar-Kharroubi [18] who have discussed the well posedness of the Cauchy problem governed by the linear operator $T_H + K$ in bounded geometry in the case of nonrentry boundary conditions, i.e. $H = 0$ (cf. Section 2 for notations). We note that the first work in this direction concerning linear multi-dimensional singular transport equation in bounded geometry with nonrentry boundary conditions on $L^1$-spaces, motivated by free gas models (see [19, 21]), was done by M. Chabi in his Phd thesis [7] (further developments were given in [16]). For the well posedness and the time asymptotic behavior of the solution to the Cauchy problem governed by the linear one-dimensional mono-energetic singular transport equation with reflecting boundary conditions (2), we refer to [8, 9]. Note that the assumption about $\sigma(\cdot)$ means that the singularities of the collision frequency are contained in a set of zero Lebesgue measure. In fact, unbounded and nonnegative collision frequencies act as strong absorptions which allow the unboundedness of the collision operator $K$.

Note that the nonlinear boundary value problem (1)-(2) was discussed for reflecting and periodic boundary conditions in [14] on $L^1$-spaces. The essential argument of proofs of the results established in [14] is a fixed point theorem for continuous weakly compact operators [13, Theorem 4.3.1] (see also [15]). The goal of this paper is to complete the results obtained in $L^1$-spaces and to extend them to $L^p$-spaces where $\sigma(\cdot)$ and $\kappa(\cdot, \cdot, \cdot)$ satisfy the conditions (i) and (ii) and the map $f(\cdot, \cdot, \cdot)$ is a Carathéodory function on $[-a, a] \times [-1, 1] \times \mathbb{C}$ which defines a continuous Nemytskii (or superposition) operator which maps bounded sets into bounded sets.

The outline of this work is as follows. In Section 2, we introduce the functional setting of the problem and we establish some preliminary results required in the rest of the paper. In Section 3, we present and prove our existence results in the case of specular reflecting boundary conditions. In Section 4, we discuss the Problem (1) for periodic boundary conditions.

## 2 Notations and Preliminaries

In this Section we will collect notations and preliminary facts connected to our problem. Let

$$X_p = L^p(\Omega; dxd\xi)$$

where $\Omega = [-a, a] \times [-1, 1]$, $(a > 0)$, and $p \in (1, \infty)$. Define the following sets representing the incoming and the outgoing boundaries of the phase space $\Omega$:

$$\Omega^i = \Omega_1^i \cup \Omega_2^i = \{-a\} \times [0, 1] \cup \{a\} \times [-1, 0],$$
$$ \Omega^o = \Omega_1^o \cup \Omega_2^o = \{-a\} \times [-1, 0] \cup \{a\} \times [0, 1]. $$

Moreover, we introduce the following boundary spaces:

$$ \begin{aligned} L_p^i &:= L^p(\Omega^i; |\xi|d\xi) \sim L^p(\Omega_1^i; |\xi|d\xi) \oplus L^p(\Omega_2^i; |\xi|d\xi) \\ &= L_{1,p}^i \oplus L_{2,p}^i, \end{aligned} $$

endowed with the norm

$$ \|\psi^i, L_p^i\| = (\|\psi_1^i, L_{1,p}^i\|^p + \|\psi_2^i, L_{2,p}^i\|^p)^{\frac{1}{p}} = \left[ \int_0^1 |\psi(-a, \xi)|^p |\xi| d\xi + \int_{-1}^0 |\psi(a, \xi)|^p |\xi| d\xi \right]^{\frac{1}{p}} $$

and

$$ \begin{aligned} L_p^o &:= L^p(\Omega^o; |\xi|d\xi) \sim L^p(\Omega_1^o; |\xi|d\xi) \oplus L^p(\Omega_2^o; |\xi|d\xi) \\ &:= L_{1,p}^o \oplus L_{2,p}^o, \end{aligned} $$

endowed with the norm

$$ \|\psi^o, L_p^o\| = (\|\psi_1^o, L_{1,p}^o\|^p + \|\psi_2^o, L_{2,p}^o\|^p)^{\frac{1}{p}} = \left[ \int_{-1}^0 |\psi(-a, \xi)|^p |\xi| d\xi + \int_0^1 |\psi(a, \xi)|^p |\xi| d\xi \right]^{\frac{1}{p}}, $$

where $\sim$ means the natural identification of these spaces.

The boundary conditions (2) may be written abstractly as an operator $H$ relating the incoming and the outgoing fluxes, namely

$$ \left\{ \begin{array}{l} H : L_{1,p}^o \oplus L_{2,p}^o \to L_{1,p}^i \oplus L_{2,p}^i \\ H \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} := \begin{pmatrix} H_{11} & 0 \\ 0 & H_{22} \end{pmatrix} \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \end{array} \right. $$

where

$$ \left\{ \begin{array}{l} H_{11} : L_{1,p}^o \to L_{1,p}^i \\ \psi(-a, \xi) \to \psi(-a, -\xi), \quad -1 < \xi < 0; \end{array} \right. $$

$$ \left\{ \begin{array}{l} H_{22} : L_{2,p}^o \to L_{2,p}^i \\ \psi(a, \xi) \to \psi(a, -\xi), \quad 0 < \xi < 1. \end{array} \right. $$

**Remark 2.1** Clearly $H$ is invertible. On the other hand, if $\begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \in L_p^o$, then

$$ \begin{aligned} \|Hu, L_p^i\| &= \|H_{11}u_1, L_{1,p}^i\|^p + \|H_{22}u_2, L_{2,p}^i\|^p \\ &= \int_0^1 |u_1(-a, -\xi)|^p |\xi| d\xi + \int_{-1}^0 |u_2(a, -\xi)|^p |\xi| d\xi \\ &= \|u_1, L_{1,p}^o\|^p + \|u_2, L_{2,p}^o\|^p = \|u, L_p^o\|^p. \end{aligned} $$

Thus, $H$ is a perfect reflecting boundary operator.

<figure><img src="image_2.png" /></figure>
Throughout this paper, the collision frequency $\sigma(\cdot)$ will be assumed to satisfies assumption (A1):

(A1) there exist a closed subset $\mathcal{O} \subset [-1, 1]$ with zero Lebesgue measure and a constant $\sigma_0 > 0$ such that $\sigma(\cdot) \in L_{\text{loc}}^{\infty}([-1, 1]\mathcal{O})$, $\sigma(\xi) > \sigma_0$ a.e. on $[-1, 1]$ and

$$ \sigma(\xi) = \sigma(-\xi) \quad \forall \xi \in [-1, 1]. $$

We need also the following functional spaces

$$ \mathcal{X}_p^{\sigma} := L^p([-a, a] \times [-1, 1]; \sigma(\xi)dxd\xi), $$

$$ \mathcal{L}^{p,\sigma} := L^p([-1, 1]; \sigma(\xi)d\xi), $$

$$ \mathcal{L}^p := L^p([-1, 1]; d\xi). $$

Next, we introduce be the following partial Sobolev space

$$ W_p := \{ \psi \in \mathcal{X}_p^{\sigma} \text{ such that } \xi \frac{\partial \psi}{\partial x} \in \mathcal{X}_p \}. $$

It should be noticed that elementary calculations using the estimate $\sigma(\xi) > \sigma_0$ (see (A1)) show that

$$ \mathcal{X}_p^{\sigma} \subset \mathcal{X}_p $$

and therefore

$$ W_p \subset \{ \psi \in \mathcal{X}_p \text{ such that } \xi \frac{\partial \psi}{\partial x} \in \mathcal{X}_p \}. $$

Accordingly, any function $\psi \in W_p$ possesses traces on $\Omega^i$ and $\Omega^o$ belonging to the spaces $L_p^i$ and $L_p^o$ (see, for instance, [5, 6] or [4]). They are denoted, respectively, by $\psi^i := \psi|_{\Omega^i}$ and $\psi^o := \psi|_{\Omega^o}$ and represent the incoming and the outgoing fluxes.

Now we are in position to define the streaming operator $T_H$

$$ \left\{ \begin{array}{l} T_H : D(T_H) \subseteq \mathcal{X}_p^{\sigma} \longrightarrow \mathcal{X}_p \\ \psi \longrightarrow T_H \psi(x, \xi) = -\xi \frac{\partial \psi}{\partial x}(x, \xi) - \sigma(\xi) \psi(x, \xi) \\ D(T_H) = \{ \psi \in W_p \text{ such that } \psi^i = H(\psi^o) \}, \end{array} \right. $$

where $\psi^o = (\psi_1^o, \psi_2^o)^T$ and $\psi^i = (\psi_1^i, \psi_2^i)^T$ with $\psi_1^o, \psi_2^o, \psi_1^i$ and $\psi_2^i$ are given by

$$ \left\{ \begin{array}{ll} \psi_1^i(\xi) = \psi(-a, \xi), & \xi \in (0, 1), \\ \psi_2^i(\xi) = \psi(a, \xi), & \xi \in (-1, 0), \\ \psi_1^o(\xi) = \psi(-a, \xi), & \xi \in (-1, 0), \\ \psi_2^o(\xi) = \psi(a, \xi), & \xi \in (0, 1). \end{array} \right. $$

Let $\varphi \in \mathcal{X}_p$ and consider the resolvent equation for the operator $T_H$

$$ (\lambda - T_H)(\psi) = \varphi \qquad (3) $$

where $\lambda$ is a complex number and the unknown $\psi$ must be sought in $D(T_H)$. Thus, for $\text{Re}\lambda > -\sigma_0$, the solution of (3) is formally given by
$$ \psi(x, \xi) = \begin{cases} \psi(-a, \xi) \exp\left(-\frac{(\lambda + \sigma(\xi))|a+x|}{|\xi|}\right) + \frac{1}{|\xi|} \int_{-a}^{x} \exp\left(-\frac{(\lambda + \sigma(\xi))|x-y|}{|\xi|}\right) \varphi(y, \xi) dy, & \text{for } \xi \in (0, 1) \\ \\ \psi(a, \xi) \exp\left(-\frac{(\lambda + \sigma(\xi))|a-x|}{|\xi|}\right) + \frac{1}{|\xi|} \int_{x}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))|x-y|}{|\xi|}\right) \varphi(y, \xi) dy, & \text{for } \xi \in (-1, 0). \end{cases} \quad (4) $$

For $x = a$, we get for $\xi \in (0, 1)$

$$ \psi(a, \xi) = \psi(-a, \xi) \exp\left(-2a\frac{(\lambda + \sigma(\xi))}{|\xi|}\right) + \frac{1}{|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))|a-x|}{|\xi|}\right) \varphi(x, \xi) dx, \quad (5) $$

and for $x = -a$, we get for $\xi \in (-1, 0)$

$$ \psi(-a, \xi) = \psi(a, \xi) \exp\left(-2a\frac{(\lambda + \sigma(\xi))}{|\xi|}\right) + \frac{1}{|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))|a+x|}{|\xi|}\right) \varphi(x, \xi) dx. \quad (6) $$

Set

$$ \mathbf{m}(\lambda, \sigma(\xi), \xi) = \exp\left(-2a\frac{(\lambda + \sigma(\xi))}{|\xi|}\right). $$

Using (4)-(6) and taking into account of the boundary conditions, simple calculations show that the solution of (3) writes

$$ \psi(x, \xi) = \frac{1}{|\xi|(1 - (\mathbf{m}(\lambda, \sigma(\xi), \xi))^2)} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(4a + x - y)}{|\xi|}\right) \varphi(y, \xi) dy \\ + \frac{1}{|\xi|(1 - (\mathbf{m}(\lambda, \sigma(\xi), \xi))^2)} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a + x + y)}{|\xi|}\right) \varphi(y, -\xi) dy \\ + \frac{1}{|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(x - y)}{|\xi|}\right) \varphi(y, \xi) dy \\ \text{for } \xi \in (0, 1), $$

and

$$ \psi(x, \xi) = \frac{1}{|\xi|(1 - (\mathbf{m}(\lambda, \sigma(\xi), \xi))^2)} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(4a - x + y)}{|\xi|}\right) \varphi(y, \xi) dy \\ + \frac{1}{|\xi|(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a - x - y)}{|\xi|}\right) \varphi(y, -\xi) dy \\ + \frac{1}{|\xi|} \int_{x}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(y - x)}{|\xi|}\right) \varphi(y, \xi) dy \\ \text{for } \xi \in (-1, 0). $$
Let $\lambda \in \mathbb{C}$ be such that $\mathrm{Re}\lambda > -\sigma_0$ and define the operators

$$
\left\{
\begin{array}{ll}
\mathcal{P}_{\lambda} : \mathbf{X}_p \to \mathbf{X}_p^{\sigma}, & \\
\mathcal{P}_{\lambda\varphi} := \chi_{(-1,0)}(\xi)\mathcal{P}_{\lambda}^{-}\varphi + \chi_{(0,1)}(\xi)\mathcal{P}_{\lambda}^{+}\varphi, & \text{where} \\
(\mathcal{P}_{\lambda}^{+}\varphi)(x, \xi) := \frac{1}{|\xi|(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(4a + x - y)}{|\xi|}\right) \varphi(y, \xi)dy, & \\
\mathcal{P}_{\lambda}^{-}\varphi(x, \xi) := \frac{1}{|\xi|(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(4a - x + y)}{|\xi|}\right) \varphi(y, \xi)dy, & \text{for } \xi \in (0, 1), \\
\mathcal{P}_{\lambda}^{-}\varphi(x, \xi) := \frac{1}{|\xi|(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(4a - x - y)}{|\xi|}\right) \varphi(y, \xi)dy, & \text{for } \xi \in (-1, 0),
\end{array}
\right.
$$

$$
\left\{
\begin{array}{ll}
S_{\lambda} : \mathbf{X}_p \to \mathbf{X}_p^{\sigma}, & \\
S_{\lambda\varphi} := \chi_{(-1,0)}(\xi)S_{\lambda}^{-}\varphi + \chi_{(0,1)}(\xi)S_{\lambda}^{+}\varphi, & \text{where} \\
(S_{\lambda}^{+}\varphi)(x, \xi) := \frac{1}{|\xi|(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a + x + y)}{|\xi|}\right) \varphi(y, -\xi)dy, & \\
(S_{\lambda}^{-}\varphi)(x, \xi) := \frac{1}{|\xi|(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a - x - y)}{|\xi|}\right) \varphi(y, -\xi)dy, & \text{for } \xi \in (0, 1), \\
(S_{\lambda}^{-}\varphi)(x, \xi) := \frac{1}{|\xi|(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a - x + y)}{|\xi|}\right) \varphi(y, -\xi)dy, & \text{for } \xi \in (-1, 0)
\end{array}
\right.
$$

and

$$
\left\{
\begin{array}{ll}
Q_{\lambda} : \mathbf{X}_p \to \mathbf{X}_p^{\sigma}, & \\
Q_{\lambda\varphi} := \chi_{(-1,0)}(\xi)Q_{\lambda}^{-}\varphi + \chi_{(0,1)}(\xi)Q_{\lambda}^{+}\varphi, & \text{where} \\
(Q_{\lambda}^{+}\varphi)(x, \xi) := \frac{1}{|\xi|} \int_{-a}^{x} \exp\left(-\frac{(\lambda + \sigma(\xi))|x - y|}{|\xi|}\right) \varphi(y, \xi)dy, & \xi > 0, \\
(Q_{\lambda}^{-}\varphi)(x, \xi) := \frac{1}{|\xi|} \int_{x}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))|x - y|}{|\xi|}\right) \varphi(y, \xi)dy, & \xi < 0,
\end{array}
\right.
$$

where $\chi_{(-1,0)}$, $\chi_{(0,1)}$ denote, respectively, the characteristic functions of the intervals $(-1,0)$ and $(0,1)$.

The following lemma was established in [9].

**Lemma 2.1** Let $\lambda$ be a complex number such that $\mathrm{Re}\lambda + \sigma_0 > 0$ and assume that the condition (A1) holds true. Then the operators $\mathcal{P}_\lambda$, $\mathcal{S}_\lambda$ and $\mathcal{Q}_\lambda$ belong to $\mathcal{L}(\mathbf{X}_p, \mathbf{X}_p^\sigma)$ and satisfy

$$
\|\mathcal{P}_{\lambda}\|_{\mathcal{L}(\mathbf{X}_p, \mathbf{X}_p^{\sigma})} \le \frac{1}{1 - (\mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1))^2} \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)^{\frac{1}{p}}}{\mathrm{Re}\lambda + \sigma(\xi)},
$$

$$
\| \mathcal{S}_{\lambda} \|_{\mathcal{L}(\mathbf{X}_p, \mathbf{X}_p^{\sigma})} \le \frac{1}{1 - (\mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1))^2} \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)^{\frac{1}{p}}}{\mathrm{Re}\lambda + \sigma(\xi)},
$$

and

$$
\| \mathcal{Q}_{\lambda} \|_{\mathcal{L}(\mathbf{X}_p, \mathbf{X}_p^{\sigma})} \le \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)^{\frac{1}{p}}}{\mathrm{Re}\lambda + \sigma(\xi)}.
$$
As an immediate consequence of this lemma we have

**Corollary 2.1** Assume that the hypotheses of Lemma 2.1 hold true. Then $D(T_H) \subseteq \mathcal{X}_p^\sigma$ with continuous embedding. Further, for $\lambda \in \mathbb{C}$ such that $\mathrm{Re}\lambda > -\sigma_0$, the inverse of the operator $(\lambda - T_H)$ is given by

$$ (\lambda - T_H)^{-1} = \mathcal{P}_\lambda + \mathcal{S}_\lambda + \mathcal{Q}_\lambda \qquad (7) $$

and satisfies

$$ \|(\lambda - T_H)^{-1}\|_{\mathcal{L}(\mathcal{X}_p, \mathcal{X}_p^\sigma)} \le \frac{3\sigma_0^{-\frac{1}{q}}}{1 - (\mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1))^2}, \qquad (8) $$

where $q = \frac{p}{p-1}$ is the conjugate of p.

The estimate (8) follows from the fact that $\sigma(\cdot)$ is assumed to be unbounded because in this case we have

$$ \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)}{\mathrm{Re}\lambda + \sigma(\xi)} = 1 \quad \text{regardless of } \lambda. $$

**Remark 2.2** Note that, for any real number $\lambda$ such that $\lambda > -\sigma_0$, the operators $\mathcal{P}_\lambda$, $\mathcal{S}_\lambda$ and $\mathcal{Q}_\lambda$ are positive (in the lattice sense) on $\mathcal{X}_p$, that is, they map the positive cone of $\mathcal{X}_p$ into the positive cone of $\mathcal{X}_p^\sigma$. Therefore, for such reals, say $\lambda$, the operator $(\lambda - T_H)^{-1}$ is also positive on $\mathcal{X}_p$ (use (7)). $\square$

It should be noticed that according to Corollary 2.1 the embedding $D(T_H) \hookrightarrow \mathcal{X}_p^\sigma$ is continuous, we can suppose that the collision operator defined by

$$ K\psi(x, \xi) = \int_{-1}^{1} \kappa(x, \xi, \xi')\psi(x, \xi')d\xi', $$

is a bounded linear operator from $\mathcal{X}_p^\sigma$ into $\mathcal{X}_p$ ($K \in \mathcal{L}(\mathcal{X}_p^\sigma, \mathcal{X}_p)$).

We recall that (this comment is adapted to the one-dimensional setting), if $[-a, a]$ denotes the set of particle positions, the physical collision operators used in linear kinetic theory are bounded and in the form

$$ K\psi(x, \xi) = \int_{-1}^{1} \kappa(x, \xi, \xi')\psi(x, \xi')d\xi' \qquad (9) $$

where the kernel $\kappa(\cdot, \cdot, \cdot)$ is assumed to be a measurable nonnegative real valued function from $[-a, a] \times [-1, 1] \times [-1, 1]$ into $\mathbb{R}$. These operators possess some compactness properties with respect to the velocities (in one-dimensional case, the variable $\xi$ plays the role of the velocity variable). To highlight these compactness properties, M. Mokhtar-Kharroubi [17, 18] introduced the class of regular collision operators in a general $L^p$-spaces setting ($1 < p < +\infty$). Basically, a linear operator $K \in \mathcal{L}(L^p(\Omega; \sigma(\xi)dxd\xi), L^p(\Omega; dxd\xi))$ is regular if it is local with respect to $x \in [-a, a]$ and compact with respect to $\xi \in [-1, 1]$.
It is clear that the operator K is local in x, so it may be regarded as an operator from L<sup>p,σ</sup> into L<sup>p</sup> depending on a parameter x ∈ [-a, a]. We assume that the map K(·) from [-a, a] into L<sup>(L<sup>p,σ</sup>, L<sup>p</sup>)</sup> is strongly measurable, i.e. for each u ∈ L<sup>p,σ</sup>, the function

$$x \in [-a, a] \mapsto K(x)u \in L^p$$

is measurable and the map x ∈ [-a, a] ↦→ ||K(x)||<sub>L(L<sup>p,σ</sup>, L<sup>p</sup>)</sub> is essentially bounded. Here L(L<sup>p,σ</sup>, L<sup>p</sup>) stands for the set of all bounded linear operators from L<sup>p,σ</sup> into L<sup>p</sup>.

**Definition 2.1** Let X and Y be two normed spaces and denote by $\mathbb{B}_1$ the closed unit ball of X. A set $C$ of $\mathcal{L}(X, Y)$ is said to be collectively compact if, and only if, the set $C(\mathbb{B}_1) = \{C(x), C \in \mathcal{C}, x \in \mathbb{B}_1\}$ is relatively compact in Y.

**Definition 2.2** [18] We say that $K \in \mathcal{L}(X_p^\sigma, X_p)$ is a regular operator if,

a) $\{K(x) : x \in [-a, a]\}$ is a set of collectively compact operators from $L^{p,\sigma}$ into $L^p$, i.e.

$$\{K(x)u : x \in [-a, a], \|u\|_{L^{p,\sigma}} \le 1\} \text{ is relatively compact in } L^p.$$

b) For every $u' \in L^q$,

$$\{K'(x)u' : x \in [-a, a], \|u'\|_{L^q} \le 1\} \text{ is relatively compact in } L^{q,\sigma}$$

where $K'(x)$ denotes the dual operator of $K(x)$ and $q$ is the conjugate exponent of $p$.

Next, we recall the following useful approximation property of regular collision operators established in [18] (see also [9]).

**Lemma 2.2** The class of regular operators in $\mathcal{L}(X_p^\sigma, X_p)$ is the closure in the operator norm of the class of collision operators with kernels of the form

$$\kappa(x, \xi, \xi') = \sum_{i \in I} \delta_i(x) \kappa_i(\xi) \eta_i(\xi')$$

where $\delta_i(\cdot) \in L^\infty((-a, a); dx)$, $\kappa_i(\cdot) \in L^p$ and $\sigma(\cdot)^{-1/p} \eta_i(\cdot) \in L^q$ ($I$ is finite and $q$ denotes the conjugate of $p$).

**Lemma 2.3** Let $\lambda$ be a complex number satisfying Re$\lambda > -\sigma_0$ and let K be a regular collision operator. Then the operators $P_\lambda K$, $S_\lambda K$ and $Q_\lambda K$ are compact.

**Proof.** According to Lemma 2.2, there exists a sequence of finite rank operators on $\mathcal{L}(X_p^\sigma, X_p)$ which converges, in the operator norm, to K. So, it suffices to establish the result for an operator with kernel of the form

$$\kappa(x, \xi, \xi') = \delta(x) \kappa(\xi) \eta(\xi')$$
where $\delta(\cdot) \in L^\infty((-a, a); dx)$, $\kappa(\cdot) \in L^p$, $\sigma(\cdot)^{-\frac{1}{p}}\eta(\cdot) \in L^q$ with $\frac{1}{p} + \frac{1}{q} = 1$. To do so, let $\psi \in X_p^\sigma$. Since

$$
\begin{align*}
\mathcal{P}_\lambda K \psi(x, \xi) &= \chi_{(-1,0)}(\xi) \mathcal{P}_\lambda^- K \psi(x, \xi) + \chi_{(0,1)}(\xi) \mathcal{P}_\lambda^+ K \psi(x, \xi) \\
&= \frac{1}{|\xi|(1 - \mathbf{m}(\lambda, \sigma(\xi), \xi)^2)} \left[ \chi_{(-1,0)}(\xi) \int_{-a}^{a} e^{-\frac{(\lambda+\sigma(\xi))(4a+x-y)}{|\xi|}} K\varphi(y, \xi)dy \right. \\
&\qquad \left. + \chi_{(0,1)}(\xi) \int_{-a}^{a} e^{-\frac{(\lambda+\sigma(\xi))(4a-x+y)}{|\xi|}} K\varphi(y, \xi)dy \right] \\
&= \frac{1}{|\xi|(1 - \mathbf{m}(\lambda, \sigma(\xi), \xi)^2)} \left[ \chi_{(-1,0)}(\xi) \int_{-a}^{a} e^{-\frac{(\lambda+\sigma(\xi))(4a+x-y)}{|\xi|}} \delta(y)\kappa(\xi) \int_{-1}^{1} \eta(\xi')\varphi(y, \xi')dyd\xi' \right. \\
&\qquad \left. + \chi_{(0,1)}(\xi) \int_{-a}^{a} e^{-\frac{(\lambda+\sigma(\xi))(4a-x+y)}{|\xi|}} \delta(y)\kappa(\xi) \int_{-1}^{1} \eta(\xi')\varphi(y, \xi')dyd\xi' \right],
\end{align*}
$$

one can write the operator $\mathcal{P}_\lambda K$ in the form $\mathcal{P}_\lambda K = \Lambda_\lambda R$, where $R$ is the following bounded operator

$$
\left\{
\begin{array}{l}
R : X_p^\sigma \to L^p((-a, a); dx) \\
\psi \to R\psi := \int_{-1}^{1} \eta(\xi)\psi(x, \xi)d\xi
\end{array}
\right.
$$

while $\Lambda_\lambda$ stands of the integral operator (see [11, p.264])

$$
\left\{
\begin{array}{l}
\Lambda_{\lambda} : L^p((-a, a); dx) \to X_p^{\sigma}, \Lambda_{\lambda} \varphi := \chi_{(-1,0)}(\xi) \Lambda_{\lambda}^{-} \varphi + \chi_{(0,1)}(\xi) \Lambda_{\lambda}^{+} \varphi \quad \text{with} \\
\qquad (\Lambda_{\lambda}^{+} \varphi)(x, \xi) := \frac{\kappa(\xi)}{|\xi|(1 - \mathbf{m}(\lambda, \sigma(\xi), \xi)^2)} \int_{-a}^{a} \delta(y) \exp\left(-\frac{(\lambda + \sigma(\xi))(4a - x + y)}{|\xi|}\right) \varphi(y) dy, \\
\qquad \text{for } \xi \in (0, 1), \\
\\
\qquad (\Lambda_{\lambda}^{-} \varphi)(x, \xi) := \frac{\kappa(\xi)}{|\xi|(1 - \mathbf{m}(\lambda, \sigma(\xi), \xi)^2)} \int_{-a}^{a} \delta(y) \exp\left(-\frac{(\lambda + \sigma(\xi))(4a + x - y)}{|\xi|}\right) \varphi(y) dy, \\
\qquad \text{for } \xi \in (-1, 0).
\end{array}
\right.
$$

In order to prove the compactness of $\mathcal{P}_\lambda K$, it suffices to check that the integral operators $\Lambda_\lambda^+$ and $\Lambda_\lambda^-$ are compact. Indeed, using the Hölder inequality, we get

$$
\begin{align*}
| (\Lambda_{\lambda}^{+} \varphi)(x, \xi) | &\le \frac{\|\delta\|_{\infty} |\kappa(\xi)|}{|\xi|(1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1)^2)} \left( \int_{-a}^{a} e^{-\frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|} (4a - x + y)} dy \right)^{\frac{1}{q}} \\
&\qquad \left( \int_{-a}^{a} e^{-\frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|} (4a - x + y)} |\varphi(y)|^p dy \right)^{\frac{1}{p}} \\
&\le \frac{\|\delta\|_{\infty} |\kappa(\xi)|}{1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1)^2} \frac{|\xi|^{\frac{1}{q}}}{|\xi|(\mathrm{Re}\lambda + \sigma(\xi))^{\frac{1}{q}}} \left( \int_{-a}^{a} e^{-\frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|} (4a - x + y)} |\varphi(y)|^p dy \right)^{\frac{1}{p}}.
\end{align*}
$$

Hence

$$
| (\Lambda_{\lambda}^{+} \varphi)(x, \xi) |^p \le \frac{\|\delta\|_{\infty}^p |\kappa(\xi)|^p}{(1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1))^p} \frac{|\xi|^{p(\frac{1}{q}-1)}}{(\mathrm{Re}\lambda + \sigma(\xi))^q} \int_{-a}^{a} e^{-\frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|} (4a - x + y)} |\varphi(y)|^p dy.
$$
Integrating $|\Lambda_{\lambda}^{+} \varphi|^{p}$, we get

$$ \begin{aligned} \int_{-a}^{a} |(\Lambda_{\lambda}^{+} \varphi)(x, \xi)|^{p} dx & \le \frac{\|\delta\|_{\infty}^{p} |\kappa(\xi)|^{p}}{(1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_{0}, 1)^{2})^{p}} \frac{1}{|\xi|(\mathrm{Re}\lambda + \sigma(\xi))^{\frac{p}{q}}} \frac{|\xi|}{\mathrm{Re}\lambda + \sigma(\xi)} \int_{-a}^{a} |\varphi(y)|^{p} dy \\ & = \frac{\|\delta\|_{\infty}^{p} \|\varphi\|_{L^{p}(-a,a)}^{p}}{(1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_{0}, 1)^{2})^{p}} \frac{|\kappa(\xi)|^{p}}{(\mathrm{Re}\lambda + \sigma(\xi))^{p}} \end{aligned} $$

and therefore,

$$ \int_{0}^{1} \int_{-a}^{a} |(\Lambda_{\lambda}^{+} \varphi)(x, \xi)|^{p} \sigma(\xi) dx d\xi \le \frac{\|\delta\|_{\infty}^{p} \|\varphi\|_{L^{p}(-a,a)}^{p}}{(1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_{0}, 1)^{2})^{p}} \sup_{\xi \in (-1, 1)} \frac{\sigma(\xi)}{(\mathrm{Re}\lambda + \sigma(\xi))^{p}} \int_{0}^{1} |\kappa(\xi)|^{p} d\xi. $$

Similarly we have

$$ \int_{-1}^{0} \int_{-a}^{a} |(\Lambda_{\lambda}^{+} \varphi)(x, \xi)|^{p} \sigma(\xi) dx d\xi \le \frac{\|\delta\|_{\infty}^{p} \|\varphi\|_{L^{p}(-a,a)}^{p}}{(1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_{0}, 1)^{2})^{p}} \sup_{\xi \in (-1, 0)} \frac{\sigma(\xi)}{(\mathrm{Re}\lambda + \sigma(\xi))^{p}} \int_{-1}^{0} |\kappa(\xi)|^{p} d\xi. $$

Consequently (use (A1))

$$ \begin{aligned} \|\Lambda_{\lambda} \varphi\|_{X_{p}^{\sigma}} &= \left[ \int_{0}^{1} \int_{-a}^{a} |(\Lambda_{\lambda}^{+} \varphi)(x, \xi)|^{p} \sigma(\xi) dx d\xi + \int_{-1}^{0} \int_{-a}^{a} |(\Lambda_{\lambda}^{+} \varphi)(x, \xi)|^{p} \sigma(\xi) dx d\xi \right]^{\frac{1}{p}} \\ &\le \frac{\|\delta\|_{\infty} \|\varphi\|_{L^{p}(-a,a)}}{1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_{0}, 1)^{2}} \sup_{\xi \in (-1, 1)} \frac{\sigma(\xi)^{\frac{1}{p}}}{\mathrm{Re}\lambda + \sigma(\xi)} \|\kappa\|_{L^{p}}. \end{aligned} $$

Therefore

$$ \|\Lambda_{\lambda}\|_{X_{p}^{\sigma}} \le \frac{\|\delta\|_{\infty}}{1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_{0}, 1)^{2}} \sup_{\xi \in (-1, 1)} \frac{\sigma(\xi)^{\frac{1}{p}}}{\mathrm{Re}\lambda + \sigma(\xi)} \|\kappa\|_{L^{p}}. $$

The last inequality shows that $\Lambda_{\lambda}$ depends continuously on $\kappa \in L^{p}$. Now, since $\Lambda_{\lambda}$ is an integral operator, and the set of bounded functions which vanish in a neighborhood of $\xi = 0$ is dense in $L^{p}$, it follows that $\Lambda_{\lambda}$ is a limit, in the uniform topology, of integral operators with bounded kernels. Hence $\Lambda_{\lambda}$ is compact (see [11, Theorem 11.6, p. 275]). Furthermore, since $R \in \mathcal{L}(X_{p}^{\sigma}; L^{p}((-a, a); dx))$, then the operator $\mathcal{P}_{\lambda}K$ is compact.

The operator $\mathcal{S}_{\lambda}$ possesses the same structure as $\mathcal{P}_{\lambda}$, similar reasoning as above leads to the compactness of $\mathcal{S}_{\lambda}K$.

In order to prove that the operator $\mathcal{Q}_{\lambda}K$ is compact, we introduce the following integral operator

$$ \left\{ \begin{array}{ll} \Pi_{\lambda}: L^{p}((-a, a); dx) \to X_{p}^{\sigma}, \Pi_{\lambda} \varphi := \chi_{(-1,0)}(\xi) \Pi_{\lambda}^{-} \varphi + \chi_{(0,1)}(\xi) \Pi_{\lambda}^{+} \varphi & \text{with} \\ (\Pi_{\lambda}^{+} \varphi)(x, \xi) := \frac{\kappa(\xi)}{|\xi|} \int_{-a}^{x} \delta(y) \exp \left( - \frac{\lambda + \sigma(\xi)}{|\xi|} |x - y| \right) \varphi(y) dy, & \text{for } \xi \in (0, 1), \\ (\Pi_{\lambda}^{-} \varphi)(x, \xi) := \frac{\kappa(\xi)}{|\xi|} \int_{x}^{a} \delta(y) \exp \left( - \frac{\lambda + \sigma(\xi)}{|\xi|} |x - y| \right) \varphi(y) dy, & \text{for } \xi \in (-1, 0). \end{array} \right. $$
The operator $Q_\lambda K$ may be written in the form $Q_\lambda K = \Pi_\lambda R$. Since $R \in \mathcal{L}(X_p^\sigma; L^p((-a, a); dx))$, then it suffices to prove that the operator $\Pi_\lambda$ is compact. Indeed, for all $\varphi \in L^p(-a, a)$, $dx)$, we have

$$
\begin{align*}
|\langle \Pi_{\lambda}^{+} \varphi \rangle(x, \xi)| &\le \frac{|\kappa(\xi)|}{|\xi|} \int_{-a}^{x} |\delta(y)| e^{-\frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|} |x-y|} |\varphi(y)| dy \\
&\le \frac{\|\delta\|_{\infty} |\kappa(\xi)|}{|\xi|} \left( \int_{-a}^{x} e^{-\frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|} |x-y|} dy \right)^{\frac{1}{q}} \left( \int_{-a}^{x} e^{-\frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|} |x-y|} |\varphi(y)|^p dy \right)^{\frac{1}{p}} \\
&= \frac{\|\delta\|_{\infty} |k(\xi)|}{|\xi|} \frac{|\xi|^{\frac{1}{q}}}{(\mathrm{Re}\lambda + \sigma(\xi))^{\frac{1}{q}}} \left( \int_{-a}^{x} e^{-\frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|} |x-y|} |\varphi(y)|^p dy \right)^{\frac{1}{p}}
\end{align*}
$$

and therefore

$$
|\langle \Pi_{\lambda}^{+} \varphi \rangle(x, \xi)|^p \le \frac{|\kappa(\xi)|^p}{|\xi|^p} \frac{\|\delta\|_{\infty}^p |\xi|^{\frac{p}{q}}}{(\mathrm{Re}\lambda + \sigma(\xi))^{\frac{p}{q}}} \int_{-a}^{x} e^{-\frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|} |x-y|} |\varphi(y)|^p dy.
$$

Integrating $|(\Pi_{\lambda}^{+} \psi)(x, \xi)|^p$ with respect to $x$ and using Fubini's theorem, we get

$$
\begin{align*}
\int_{-a}^{a} |(\Pi_{\lambda}^{+} \varphi)(x, \xi)|^p dx
&\le \frac{|\kappa(\xi)|^p}{|\xi|^p} \frac{\|\delta\|_{\infty}^p |\xi|^{\frac{p}{q}}}{(\mathrm{Re}\lambda + \sigma(\xi))^{\frac{p}{q}}} \int_{-a}^{a} |\varphi(y)|^p dy \int_{y}^{a} e^{-\frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|} |x-y|} dx \\
&\le \frac{\|\varphi\|_{L^p(-a,a)}^p}{|\xi|} \frac{\|\delta\|_{\infty}^p}{(\mathrm{Re}\lambda + \sigma(\xi))^{\frac{p}{q}}} \frac{|\xi|}{\mathrm{Re}\lambda + \sigma(\xi)} |\kappa(\xi)|^p \\
&= \frac{\|\delta\|_{\infty}^p \|\varphi\|_{L^p(-a,a)}^p}{(\mathrm{Re}\lambda + \sigma(\xi))^p} |\kappa(\xi)|^p.
\end{align*}
$$

Therefore

$$
\int_0^1 \int_{-a}^a |(\Pi_\lambda^+ \varphi)(x, \xi)|^p \sigma(\xi) dx dv \le \|\delta\|_{\infty}^p \|\varphi\|_{L^p(-a,a)}^p \sup_{\xi \in (0,1)} \frac{\sigma(\xi)}{(\mathrm{Re}\lambda + \sigma(\xi))^p} \int_0^1 |\kappa(\xi)|^p d\xi.
$$

Similarly, we have

$$
\int_{-1}^{0} \int_{-a}^{a} |(\Pi_{\lambda}^{-} \varphi)(x, \xi)|^{p} \sigma(\xi) dx d\xi \leq \|\delta\|_{\infty}^{p} \|\varphi\|_{L^{p}(-a,a)}^{p} \sup_{\xi \in (-1,0)} \frac{\sigma(\xi)}{(\mathrm{Re}\lambda + \sigma(\xi))^{p}} \int_{-1}^{0} |\kappa(\xi)|^{p} d\xi.
$$

Hence we infer that,

$$
\|\Pi_{\lambda}\varphi\|_{X_p^\sigma} \le \|\delta\|_{\infty} \|\varphi\|_{L^p(-a,a)} \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)^{\frac{1}{p}}}{\mathrm{Re}\lambda + \sigma(\xi)} \|\kappa\|_{L^p}
$$

and therefore

$$
\|\Pi_{\lambda}\|_{X_p^\sigma} \le \|\delta\|_{\infty} \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)^{\frac{1}{p}}}{\mathrm{Re}\lambda + \sigma(\xi)} \|\kappa\|_{L^p}.
$$

The last inequality shows that $\Pi_\lambda$ depends continuously on $\kappa \in L^p$. Now, since $\Pi_\lambda$ is an integral operator, and the set of of bounded functions which vanish in a neighborhood of $\xi = 0$ is dense in $L^p(-1, 1)$, it follows that $\Pi_\lambda$ is a limit, in the uniform topology, of integral operators with bounded kernels. Hence $\Pi_\lambda$ is compact which completes the proof.

Q.E.D.

We shall now recall some fact about Nemytskii (or superposition) operators on $L^p$ spaces.
**Definition 2.3** A function $f : \Omega \times \mathbb{C} \to \mathbb{C}$ is said to satisfy the Carathéodory conditions on $\Omega \times \mathbb{C}$ if

(a) the function $t \mapsto f(t, u)$ is measurable on $\Omega$ for all $u \in \mathbb{C}$,

(b) the function $u \mapsto f(t, u)$ is continuous on $\mathbb{C}$ for almost all $t \in \Omega$.

□

If $f$ satisfies the Carathéodory conditions, we can define the operator $\mathcal{N}_f$ on the set of functions $\psi : \Omega \to \mathbb{C}$ by

$$ (\mathcal{N}_f\psi)(x, \xi) := f(x, \xi, \psi(x, \xi)) $$

for every $(x, \xi) \in \Omega$. The operator $\mathcal{N}_f$ is called the Nemytskii operator generated by $f$. In $L_p$-spaces the Nemytskii operator has been extensively investigated (see, for example, [1, 2, 12] and the references therein). The first results on continuity and boundedness of this operator in Lebesgue spaces over bounded domains $\Omega$ in Euclidean space are due to M.A. Krasnosel'skii (see for example [2, Theorem 3.1]).

The following result (established in a more general setting in [10] (see also [3])) provides necessary and sufficient conditions under which the Nemytskii operator maps continuously a weighted $L^p$-space into itself. (The statement of this result is adapted to our context.)

**Lemma 2.4** Assume that $f$ satisfies the Carathéodory conditions. Then $\mathcal{N}_f$ maps continuously $X_p^\sigma$ into itself, if and only if, the function $f$ satisfies the following growth condition

$$ |f(x, \xi, u)| \le \frac{\alpha(x, \xi)}{\sigma(\xi)^{\frac{1}{p}}} + \beta|u|, \quad (10) $$

for a.e. $(x, \xi) \in \Omega$, and every $u \in \mathbb{C}$ with a fixed nonnegative function $\alpha \in X_p$ and a fixed nonnegative constant $\beta$. Further, it maps bounded sets of $X_p^\sigma$ into bounded sets of $X_p^\sigma$.

The continuity of $\mathcal{N}_f$ was established in [10]. The fact that it maps bounded sets of $X_p^\sigma$ into bounded sets of $X_p^\sigma$ is an immediate consequence of the estimate (10).

### 3 Existence results

We shall introduce the following two assumptions

(A2) $f(\cdot, \cdot, \cdot)$ is a Carathéodory function satisfying condition (10),

(A3) $\beta \|K\|_{L(X_p^\sigma, X_p)} \le \frac{\sigma_0^{1/q}}{12}$,

where $\beta$ is the constant appearing in the condition (10).

For any $r > 0$, we denote by $\mathbb{B}_r^\sigma$ the set

$$ \mathbb{B}_r^\sigma = \{ \psi \in X_p^\sigma : \|\psi\|_{X_p^\sigma} \le r \}. $$
**Theorem 3.1** Assume that the hypotheses (A1)-(A3) are satisfied. Then, for all $\lambda \in \mathbb{C}$ and $r > 0$ satisfying

$$ \mathrm{Re}\lambda \geq \frac{\ln(2)}{4a} - \sigma_0 \quad \text{and} \quad r \geq 12\sigma_0^{-\frac{1}{q}} \|K\|_{\mathcal{L}(\mathcal{X}_p^\sigma, \mathcal{X}_p)} \|\alpha\|_{\mathcal{X}_p}, \quad (11) $$

the problem

$$ \left\{ \begin{array}{l} \xi \frac{\partial \psi}{\partial x}(x, \xi) + \sigma(\xi)\psi(x, \xi) + \lambda\psi(x, \xi) = \int_{-1}^{1} k(x, \xi, \xi')f(x, \xi', \psi(x, \xi'))d\xi' \\ \\ \psi^i = H(\psi^o), \end{array} \right. \quad (12) $$

has at least one solution on $\mathbb{B}_r^\sigma$.

**Proof.** Let $\lambda$ be a complex number such that $\mathrm{Re}\lambda > -\sigma_0$. It follows from Corollary 2.1 that the operator $\lambda - T_H$ is invertible and therefore Problem (12) may be written in the form

$$ \psi = \Pi(\lambda)\psi, \quad \psi^i = H(\psi^o), $$

where $\Pi(\lambda) = (\lambda - T_H)^{-1}K\mathcal{N}_f$.

Note first that the continuity of $\Pi(\lambda)$ on $\mathcal{X}_p^\sigma$ is obvious. Further, making use of Lemma 2.3 we conclude that $\Pi(\lambda)$ is compact on $\mathcal{X}_p^\sigma$. In particular, it is continuous and compact on $\mathbb{B}_r^\sigma$.

We claim that, for all $\lambda \in \mathbb{C}$ such that $\mathrm{Re}\lambda \ge \frac{\ln(2)}{4a} - \sigma_0$ and for all $r \ge 12\sigma_0^{-\frac{1}{q}} \|K\|_{\mathcal{L}(\mathcal{X}_p^\sigma, \mathcal{X}_p)} \|\alpha\|_{\mathcal{X}_p}$, the operator $\Pi(\lambda)$ maps $\mathbb{B}_r^\sigma$ into itself. Indeed, according to Lemma 2.4, there exist a constant $\beta > 0$ and a function $\alpha \in \mathcal{X}_p$ such that

$$ |f(x, \xi, \psi(x, \xi))| \le \frac{\alpha(x, \xi)}{\sigma(\xi)^{\frac{1}{p}}} + \beta|\psi(x, \xi)| \quad \text{a.e. } (x, \xi) \in \Omega \text{ and for all } \psi \in \mathcal{X}_p^\sigma. \quad (13) $$

Let $\psi \in \mathbb{B}_r^\sigma$, where $r \ge 12\sigma_0^{-\frac{1}{q}} \|K\|_{\mathcal{L}(\mathcal{X}_p^\sigma, \mathcal{X}_p)} \|\alpha\|_{\mathcal{X}_p}$. Using Corollary 2.1 and (13), we obtain

$$ \begin{aligned} \|\Pi(\lambda)\psi\|_{\mathcal{X}_p^\sigma} &= \|(\lambda - T_H)^{-1}K\mathcal{N}_f\psi\|_{\mathcal{X}_p^\sigma} \\ &\le \|(\lambda - T_H)^{-1}\|_{\mathcal{L}(\mathcal{X}_p, \mathcal{X}_p^\sigma)} \|K\|_{\mathcal{L}(\mathcal{X}_p^\sigma, \mathcal{X}_p)} \|\mathcal{N}_f(\psi)\|_{\mathcal{X}_p^\sigma} \\ &\le \frac{3\sigma_0^{-\frac{1}{q}} (\|\alpha\|_{\mathcal{X}_p} + \beta r) \|K\|_{\mathcal{L}(\mathcal{X}_p^\sigma, \mathcal{X}_p)}}{1 - (\mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1))^2}. \end{aligned} $$

Note that the map $t \mapsto \frac{1}{1-(\mathbf{m}(t,\sigma_0,1))^2}$ is strictly decreasing on $(-\sigma_0, +\infty)$ and for all $\lambda \in \mathbb{C}$ such that $\mathrm{Re}\lambda \ge \frac{\ln(2)}{4a} - \sigma_0$, we have $\frac{1}{1-(\mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1))^2} \le 2$. The last estimate yields that

$$ \begin{aligned} \|\Pi(\lambda)\psi\|_{\mathcal{X}_p^\sigma} &\le 6\sigma_0^{-\frac{1}{q}} (\|\alpha\|_{\mathcal{X}_p} + \beta r) \|K\|_{\mathcal{L}(\mathcal{X}_p^\sigma, \mathcal{X}_p)} \\ &\le 6\sigma_0^{-\frac{1}{q}} \|\alpha\|_{\mathcal{X}_p} \|K\|_{\mathcal{L}(\mathcal{X}_p^\sigma, \mathcal{X}_p)} + \frac{r}{2} \quad (\text{use (A3)}). \end{aligned} \quad (14) $$
Further, it follows from (11) that $6\sigma_0^{-\frac{1}{q}}\|\alpha\|_{X_p} \|K\|_{L(\mathcal{X}_p^\sigma, X_p)} \le \frac{r}{2}$. This estimate together with (14) implies that $\|\Pi(\lambda)\psi\|_{X_p^\sigma} \le r$ which proves our claim.

Finally, the use of the Schauder fixed point theorem (see, for example [13, p. 118] or [20, p. 15]) applied to operator $\Pi(\lambda)$ on the convex set $\mathbb{B}_r^\sigma$ shows that there exists $\psi \in \mathbb{B}_r^\sigma$ such that $\Pi(\lambda)\psi = \psi$. This completes the proof.

Q.E.D.

The following two results are devoted to the existence of positive solutions to the boundary value problem (12). Note that $X_p$ and $X_p^\sigma$ are Banach lattice spaces; their positive cone are denoted by $X_p^+$ and $X_p^{\sigma+}$, respectively.

Let $r > 0$. We define the set $\mathbb{B}_r^{\sigma,+}$ by

$$ \mathbb{B}_r^{\sigma,+} = \mathbb{B}_r^{\sigma} \cap X_p^{\sigma+}. $$

It is clear that according to Remark 2.2, for all real number $\lambda$ such that $\lambda > -\sigma_0$, the operator $(\lambda - T_H)^{-1}$ is a positive (in the lattice sense), that is $(\lambda - T_H)^{-1}(X_p^+) \subset X_p^{\sigma+}$. If further the hypotheses of Theorem 3.1 are satisfied, $\mathcal{N}_f(X_p^{\sigma+}) \subset X_p^{\sigma+}$ and $K$ is positive, then $(\lambda - T_H)^{-1}KN_f$ leaves the positive cone $X_p^{\sigma+}$ invariant. So, if we replace $\mathbb{B}_r^\sigma$ by $\mathbb{B}_r^{\sigma,+}$ and we argue as in the proof of Theorem 3.1, then we obtain the following result.

**Corollary 3.1** Assume that the hypotheses of Theorem 3.1 hold true and $\mathcal{N}_f(X_p^{\sigma+}) \subseteq X_p^{\sigma+}$. Then, for all reals $\lambda$ and $r$ satisfying (11), Problem (12) has at least one solution on $\mathbb{B}_r^{\sigma,+}$.

In the following we will focus our attention on the existence of nontrivial positive solutions.

**Corollary 3.2** Assume that the hypotheses of Corollary 3.1 are satisfied. If there is $\rho > 0$ and $0 \ne \psi_0 \in \mathbb{B}_r^{\sigma,+}$ such that

(a) $\psi_0 \notin N(K)$ where $N(K)$ denotes the null space of $K$,

(b) $(\mathcal{N}_f\psi)(x, \xi) \ge \rho\psi_0(x, \xi)$ for all $\psi \in \mathbb{B}_r^{\sigma,+}$.

Then, for all reals $\lambda$ and $r$ satisfying (11), there exists $\eta > 0$ such that the problem

$$ \begin{cases} \xi \frac{\partial \psi}{\partial x}(x, \xi) + \sigma(\xi)\psi(x, \xi) + \lambda\psi(x, \xi) = \eta \int_{-1}^{1} k(x, \xi, \xi')f(x, \xi', \psi(x, \xi'))d\xi', \\ \psi^i = H(\psi^o), \end{cases} $$

has at least one solution $\psi^* \in \mathbb{B}_r^{\sigma,+}$ satisfying $\|\psi^*\|_{X_p^\sigma} = r$.

**Proof.** Arguing as in the proof of Theorems 3.1 we infer that, for all real number $\lambda$ such that $\lambda > \bar{\lambda} := \frac{\ln 2}{4a} - \sigma_0$ and for all $r \ge 12\sigma_0^{-\frac{1}{q}}\|\alpha\|_{X_p} \|K\|_{L(\mathcal{X}_p^\sigma, X_p)}$, the operator $\Pi(\lambda)$ is continuous,
compact and maps $\mathbb{B}_r^\sigma$ into itself. On the other hand, for $\lambda \ge \bar{\lambda}$, using the positivity of $(\lambda - T_H)^{-1}$ and $K$ one sees that

$$ \Pi(\lambda)(\psi) \ge \rho(\lambda - T_H)^{-1} K(\psi_0) \ge 0 \text{ and } (\lambda - T_0)^{-1} K(\psi_0) \ne 0 \text{ for all } \psi \in \mathbb{B}_r^{\sigma,+}, $$

where $(\lambda - T_0)^{-1}$ is nothing else but the operator $Q_\lambda$. Accordingly,

$$ \|\Pi(\lambda)(\psi)\|_{X_p^\sigma} \ge \rho \|(\lambda - T_H)^{-1}(\psi_0)\|_{X_p^\sigma} > 0 \quad \text{for all } \psi \in \mathbb{B}_r^{\sigma,+} $$

and consequently

$$ \inf \{ \|\Pi(\lambda)(\psi)\|_{X_p^\sigma}, \psi \in \mathbb{B}_r^{\sigma,+} \} > 0. $$

Hence, for $\lambda \ge \bar{\lambda}$, one can define the operator

$$ R_{\lambda}(\psi) = r \frac{\Pi(\lambda)(\psi)}{\|\Pi(\lambda)(\psi)\|_{X_p^{\sigma}}} \quad \text{for all } \psi \in \mathbb{B}_r^{\sigma,+}. $$

Now, using the same arguments as in the proof of the Theorem 3.1, we see that $R_\lambda$ is continuous, compact and maps $\mathbb{B}_r^{\sigma,+}$ into itself. So, applying the Schauder fixed point theorem, one sees that $R(\lambda)$ has a fixed point $\psi^* \in \mathbb{B}_r^{\sigma,+}$ satisfying $\|\psi^*\|_{X_p^\sigma} = \mathbf{r}$. Putting $\eta = \frac{\mathbf{r}}{\|R_\lambda(\psi^*)\|_{X_p^\sigma}}$, we get $R_\lambda(\psi^*) = \eta^{-1}\psi^*$. Consequently, $\psi^* \in D(T_H) \cap \mathbb{B}_r^{\sigma,+}$ and

$$ \xi \frac{\partial \psi^*}{\partial x}(x, \xi) + \sigma(\xi) \psi^*(x, \xi) + \lambda \psi^*(x, \xi) = \eta \int_{-1}^{1} k(x, \xi, \xi') f(x, \xi', \psi^*(x, \xi')) d\xi' $$

which completes the proof.

Q.E.D.

# 4 Periodic boundary conditions

It should be noticed that, when dealing with perfect periodic boundary conditions, i.e.

$$ \left\{ \begin{array}{l} H : L_{1,p}^o \oplus L_{2,p}^o \to L_{1,p}^i \oplus L_{2,p}^i \\ H \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} := \begin{pmatrix} 0 & H_{12} \\ H_{21} & 0 \end{pmatrix} \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \end{array} \right. \qquad (15) $$

where

$$ \left\{ \begin{array}{l} H_{12} : L_{2,p}^o \to L_{1,p}^i \\ \psi(-a, \xi) \to \psi(a, \xi), \end{array} \right. \qquad \left\{ \begin{array}{l} H_{21} : L_{1,p}^o \to L_{2,p}^i \\ \psi(-a, \xi) \to \psi(a, \xi). \end{array} \right. $$

Using slight modifications, one sees that the results of Section 3 hold also true. Indeed, for $\mathrm{Re}\lambda > -\sigma_0$, similar arguments and calculations as in Section 2 give

$$ (\lambda - T_H)^{-1} = J_\lambda + Q_\lambda, $$
where $J_λ$ is the bounded operator defined by

$$
\left\{
\begin{array}{l}
J_{\lambda} : X_p \to X_p^{\sigma}, \quad J_{\lambda}\varphi := \chi_{(-1,0)}(\xi)J_{\lambda}^{-}\varphi + \chi_{(0,1)}(\xi)J_{\lambda}^{+}\varphi \quad \text{with} \\
(J_{\lambda}^{+}\varphi)(x, \xi) = \frac{1}{|\xi|(1 - \mathbf{m}(\lambda, \sigma(\xi), \xi))} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a + x - y)}{|\xi|}\right) \varphi(y, \xi)dy, \quad \xi > 0, \\
(J_{\lambda}^{-}\varphi)(x, \xi) = \frac{1}{|\xi|(1 - \mathbf{m}(\lambda, \sigma(\xi), \xi))} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a - x + y)}{|\xi|}\right) \varphi(y, \xi)dy, \quad \xi < 0.
\end{array}
\right.
$$

It is clear that $J_λ$ possesses the same structure as the operator $\mathcal{P}_λ$. So, we have

**Lemma 4.1** Let $\lambda$ be a complex number such that $\mathrm{Re}\lambda + \sigma_0 > 0$ and assume that the condition (A1) holds true. Then the operators $J_\lambda$, and $Q_\lambda$ belong to $\mathcal{L}(X_p, X_p^\sigma)$ and satisfy the estimates

$$
\| J_{\lambda} \|_{\mathcal{L}(X_p, X_p^{\sigma})} \le \frac{1}{1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1)} \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)^{\frac{1}{p}}}{\mathrm{Re}\lambda + \sigma(\xi)}
$$

and

$$
\| Q_{\lambda} \|_{\mathcal{L}(X_p, X_p^{\sigma})} \le \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)^{\frac{1}{p}}}{\mathrm{Re}\lambda + \sigma(\xi)}.
$$

Moreover, $D(T_H) \subseteq X_p^\sigma$ with continuous embedding and, for $\lambda \in \mathbb{C}$ such that $\mathrm{Re}\lambda > -\sigma_0$, the operator $(\lambda - T_H)$ is invertible and its inverse is given by

$$
(\lambda - T_H)^{-1} = J_{\lambda} + Q_{\lambda} \quad (16)
$$

and

$$
\|(\lambda - T_H)^{-1}\|_{\mathcal{L}(X_p, X_p^\sigma)} \le \frac{2\sigma_0^{\frac{-1}{q}}}{1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1)}. \quad (17)
$$

The proof is similar to that of Lemma 2.1 and Corollary 2.1, so it is dropped.

Let $\lambda$ be a complex number such that $\mathrm{Re}\lambda + \sigma_0 > 0$ and assume that the condition (A1) holds true. We know from Lemma 2.3 that, if the collision operator $K$ is regular, then the operator $Q_\lambda K$ is compact on $X_p^\sigma$. Moreover, since $J_\lambda$ has the same structure as the operator $\mathcal{P}_\lambda$, a similar reasoning as in the proof of Lemma 2.3 ensures that $J_\lambda K$ is also compact on $X_p^\sigma$. Hence, we have the following result

**Lemma 4.2** Let $\lambda$ be a complex number satisfying $\mathrm{Re}\lambda > -\sigma_0$ and assume that the condition (A1) holds true. If $K$ is a regular collision operator, then the operators $(\lambda - T_H)^{-1}K$ is compact on $X_p^\sigma$.

Let us now introduce the following assumption

$$
(A4) \quad \beta \|K\|_{\mathcal{L}(X_p^\sigma, X_p)} \le \frac{\sigma_0^{\frac{1}{q}}}{8},
$$
**Theorem 4.1** Assume that the conditions (A1), (A2) and (A4) are satisfied. Then, for all $\lambda \in \mathbb{C}$ and $r > 0$ satisfying

$$ \mathrm{Re}\lambda \geq \frac{\ln(2)}{2a} - \sigma_0 \quad \text{and} \quad r \geq 8\sigma_0^{\frac{-1}{q}} \|K\|_{\mathcal{L}(\mathcal{X}_p^{\sigma}, \mathcal{X}_p)} \|\alpha\|_{\mathcal{X}_p}, \qquad (18) $$

the problem

$$ \left\{ \begin{array}{l} \xi \frac{\partial \psi}{\partial x}(x, \xi) + \sigma(\xi)\psi(x, \xi) + \lambda \psi(x, \xi) = \int_{-1}^{1} \kappa(x, \xi, \xi') f(x, \xi', \psi(x, \xi')) d\xi' \\ \\ \psi^i = H(\psi^o), \end{array} \right. \qquad (19) $$

has at least one solution on $\mathbb{B}_r^{\sigma}$.

**Proof.** It is clear that, for all $\lambda \in \mathbb{C}$ such that $\mathrm{Re}\lambda \geq \frac{\ln(2)}{2a} - \sigma_0$, Problem (19) may be written as a fixed point problem (use Lemma 4.1), that is, $\psi = \Xi(\lambda)(\psi)$, $\psi^i = H(\psi^o)$ where $\Xi(\lambda) = (\lambda - T_H)^{-1}KN_f$. So, in order to prove that Problem (19) has a solution, it is enough to prove that the operator $\Xi(\lambda)$ has a fixed point in $\mathbb{B}_r^{\sigma}$. Note that according to Lemmas 2.4 and 4.1, the operator $\Xi(\lambda)$ is continuous on $\mathcal{X}_p^{\sigma}$ and maps bounded sets into bounded ones. Further, using Lemma 4.2, one sees that $\Xi(\lambda)$ is compact on $\mathcal{X}_p^{\sigma}$. In particular, it is continuous and compact on $\mathbb{B}_r^{\sigma}$. It is clear that the map $t \mapsto \frac{1}{1 - \mathbf{m}(t, \sigma_0, 1)}$ is strictly decreasing on $(-\sigma_0, +\infty)$ and, for all $\lambda \in \mathbb{C}$ such that $\mathrm{Re}\lambda > \frac{\ln 2}{2a} - \sigma_0$, we have

$$ \frac{1}{1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1)} \leq \frac{1}{1 - \mathbf{m}(\frac{\ln 2}{2a} - \sigma_0, \sigma_0, 1)} = 2. $$

So, for all $\lambda \in \mathbb{C}$ such that $\mathrm{Re}\lambda > \frac{\ln 2}{2a} - \sigma_0$, the operator $\lambda - T_H$ is invertible, so for such $\lambda$ and for all $\psi$ in $\mathbb{B}_r^{\sigma}$ we have

$$ \begin{aligned} \|\Xi(\lambda)(\psi)\|_{\mathcal{X}_p^{\sigma}} &\leq \|(\lambda - T_H)^{-1}\|_{\mathcal{L}(\mathcal{X}_p, \mathcal{X}_p^{\sigma})} \|K\|_{\mathcal{L}(\mathcal{X}_p^{\sigma}, \mathcal{X}_p)} \|\mathcal{N}_f(\psi)\|_{\mathcal{X}_p^{\sigma}} \\ &\leq \frac{2\sigma_0^{\frac{-1}{q}} (\|\alpha\|_{\mathcal{X}_p} + \beta r) \|K\|_{\mathcal{L}(\mathcal{X}_p^{\sigma}, \mathcal{X}_p)}}{1 - \mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1)} \\ &\leq 4\sigma_0^{\frac{-1}{q}} \|\alpha\|_{\mathcal{X}_p} \|K\|_{\mathcal{L}(\mathcal{X}_p^{\sigma}, \mathcal{X}_p)} + \frac{r}{2} \quad (\text{use (A4) and the estimate(18)}). \end{aligned} $$

So, for all $r \geq 8\sigma_0^{\frac{-1}{q}} \|\alpha\|_{\mathcal{X}_p} \|K\|_{\mathcal{L}(\mathcal{X}_p^{\sigma}, \mathcal{X}_p)}$, we have $\|\Xi(\lambda)(\psi)\|_{\mathcal{X}_p^{\sigma}} \leq r$ and therefore $\Xi(\lambda)$ maps $\mathbb{B}_r^{\sigma}$ into itself. Accordingly, the operator $\Xi(\lambda)$ satisfies all the conditions of the Schauder fixed point theorem on the convex $\mathbb{B}_r^{\sigma}$. Hence, there exists $\psi \in \mathbb{B}_r^{\sigma}$ such that $\psi = \Xi(\psi)$ which concludes the proof. Q.E.D.

**Concluding remark.**

1) It should be noticed that the discussion of existence of positive solutions or nontrivial positive solutions to Problem (19) may be carried out in the same way as in Corollaries 3.1 and 3.2.
2) In this paper we deal with both perfect reflecting and perfect periodic boundary conditions, i.e. $||Hu, L_p^i|| = ||u, L_p^o||$ and $||Hu, L_p^i|| = ||u, L_p^o||$. However, this hypothesis is not necessary, it was used in order to simplify the calculations (which are very tedious). Actually, our analysis works also for reflecting and periodic boundary conditions satisfying $||Hu, L_p^i|| = \alpha||u, L_p^o||$ and $||Hu, L_p^i|| = \alpha||u, L_p^o||$ where $\alpha \in (0, 1)$ denotes an accommodation coefficient. □

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