数理解析研究所講究録
第 1881 巻 2014 年 38-42

38Spectrum comparison in a system of reaction-diffusion equations
with conservation property
龍谷大学・理工学部 森田 善久 (Yoshihisa Morita)
Department of Applied Mathematics and Informatics
Ryukoku University

Abstract

We are dealing with a system of reaction-diffusion equations with a conservation ofmass in a bounded domain with the Neumann boundary condition. The stationary problem of this system can be reduced to a scalar semilinear elliptic equation with a nonlocal
term. We report some result for the spectrum comparison between the linearized eigenvalue problem of an equilibrium solution for the system and the corresponding hnearized
problem for the scalar nonlocal equation.

1Introduction

We are concerned with the following two component system of reaction-diffusion equations:
with the smooth boundary
in a bounded domain$\Omega\subset \mathbb{R}^{N}$$\partial\Omega$$\{\begin{array}{l}u_{t}=d\triangle u-g(u+\gamma v)+kv,v_{t}=\triangle v+g(u+\gamma v)-kv\end{array}$$(x\in\Omega, t>0)$(1.1)

with the Neumann boundary conditions$\frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial\nu}=0 (x\in\partial\Omega)$

(1.2)

,

are positive constants and is a constant satisfying $0\leq\gamma\leq 1$ . Throughout
whereand $0<d<1$ . This system has a
the present article we assume that is of classconservation of mass by the property
$d,$

$k$

$\gamma$$g$$C^{1}$$\frac{d}{dt}\int_{\Omega}(u(x, t)+v(x, t))dx=0$

in a time interval on which the solution $(u(x, t), v(x, t))$ is defined. Thus we consider$(1.1)-(1.2)$ with a constraint$s:= \frac{1}{|\Omega|}\int_{\Omega}(u(x, t)+v(x, t))dx (t\geq 0)$.

(1.3)

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We assume that the initial data of the solution $(u(x, t), v(x, t))$ are taken in an appropriate
function space in which the -semiflow is generated (for instance see [6]).
Moreover, this system possesses a Lyapunov function given by$C^{1}$$\mathcal{E}_{\gamma}(u, v) := \int_{\Omega}\{\frac{d}{2}|\nabla(u+\gamma v)|^{2}+(1-\prime d\gamma)G(u+\gamma v)$$+ \frac{dk}{2}(u+\gamma v)^{2}+\frac{k(1-\gamma)}{2(1-d)}(du+v)^{2}+\frac{\gamma}{2}|\nabla(du+v)|^{2}\}dx$

.

(1.4)

In fact, for the solution $(u(x, t), v(x, t))$$\frac{d}{dt}\mathcal{E}_{\gamma}(u(\cdot, t), v(\cdot, t))=-\frac{1-d^{2}\gamma}{1-d\gamma}\int_{\Omega}|(u+\gamma v)_{t}|^{2}dx$$- \frac{k(1-d\gamma)}{1-d}\int_{\Omega}|\nabla(du+v)|^{2}dx-\frac{\gamma(1-\gamma)}{1-d\gamma}\int_{\Omega}|(du+v)_{t}|^{2}dx\leq 0$

holds. This Lyapunov function is found in [9] and [7] for $\gamma=0$ and $\gamma\in(0,1]$ respectively.
By virtue of the Lyapunov function the omega-limit set of any bounded orbit in the phasespace consists of equilibrium solutions (see [4]).

2Spectrum comparison

Every equilibrium solution to $(1.1)-(1.2)$ with (1.3) is obtained by solving the stationary
problem$(x\in\Omega)$$\{\begin{array}{l}d\triangle u-g(u+\gamma v)+kv=0,\triangle v+g(u+\gamma v)-kv=0\end{array}$,$\frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial v}=0$$(x\in\partial\Omega)$

,

(2.1)

with the constraint$s= \overline{u+v} :=\frac{1}{|\Omega|}\int_{\Omega}(u(x)+v(x))dx$

By putting $w=u+\gamma v$ the equations are transformed int

.

(2.2)

$0$

$\{\begin{array}{l}d\triangle w-(1-d\gamma)g(w)+k(1-d\gamma)v=0,d\Delta w+(1-d\gamma)\triangle v=0.\end{array}$(2.3)

The second equation yields$dw+(1-d\gamma)v=c$ ,thus (2.3) turns a single equation for

$w$

as

$d\triangle w-(1-d\gamma)g(w)-kdw+kc=0,$

where$c$

is an unknown constant.

(2.4)

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We note that if $\gamma=0,$ $(2.4)$ becomes$u+\overline{v}=\overline{u}+\overline{v}+(d-1)\overline{u}=s-(1-d)\overline{u}=c,$because of $w=u$ . In the sequel when $\gamma=0$ , the system (2.3) is reduced to$d\Delta u-g(u)-kdu+k(s-(1-d)\overline{u})=0$.

(2.5)

On the other hand for $\gamma=1$ we obtain$d\Delta w-(1-d)(g(w)-c_{w})-kd(w-s)=0, c_{w}:=\overline{g(w)}$ .

(2.6)

We notice that (2.5) is the Euler-Lagrange equations of the energy functional$E_{0}(u):= \frac{d}{2}\Vert\nabla u\Vert^{2}+\int_{\Omega}F_{0}(u)dx+\frac{k}{2}(s-(1-d)\overline{u})^{2},$in$L^{2}(\Omega)$$F_{0}(u);= \int^{u}g(\xi)d\xi+\frac{kd}{2}u^{2},$$(2.7)$

while so is (2.6) of the functional$E_{1}(w):= \frac{d}{2}\Vert\nabla w\Vert^{2}+\int_{\Omega}F_{1}(w)dx,$$F_{1}(w):= \int^{w}(1-d)g(\xi)d\xi+\frac{kd}{2}w^{2}$

in $X:=\{w\in L^{2}(\Omega):\overline{w}=s\}$ respectively, where$\Vert\cdot\Vert$stands for the$L^{2}$,

(2.8)

norm.

Let $u=u^{*}(x)$ and $w=w^{*}(x)$ be solutions to (2.5) and (2.6). Then$(u^{*},v^{*})=(u^{*}(x), s-du^{*}(x)-(1-d)\overline{u^{*}})$(2.9)$(u^{*}, v^{*})= \frac{1}{1-d}(dw^{*}(x)-ds, s-dw^{*}(x))$(2.10)

andgive solutions to (2.1) for $\gamma=0$ and $\gamma=1$ respectively.
Now consider the hnearized operators at $u^{*}(x)$ and $w^{*}(x)$ for (2.5) and (2.6) as$\mathcal{L}_{0}(\varphi) :=-d\Delta\varphi+(g’(u^{*}(\cdot))+kd)\varphi+k(1-d)\overline{\varphi},$and$\mathcal{L}_{1}(\varphi):=-d\Delta\varphi+\{(1-d)g’(w^{*}(\cdot))+kd\}\varphi-(1-d)\overline{g’(w^{*})\varphi},$with the domains$\mathcal{D}(\mathcal{L}_{0})=\{\varphi\in H^{2}(\Omega):\partial\varphi/\partial v=0(x\in\partial\Omega)\},$

and$\mathcal{D}(\mathcal{L}_{1})=\{\varphi\in H^{2}(\Omega):\overline{\varphi}=0, \partial\varphi/\partial\nu=0(x\in\partial\Omega)\},$

41
respectively. Corresponding to (2.9) and (2.10), we have the linearized operators$\mathcal{A}_{C}(\begin{array}{l}\phi\psi\end{array}):=-(d\triangle\phi-g’(u^{*}(.\cdot))\phi+k\psi\triangle\psi+g’(u^{*}())\phi-k\psi)$,and$\mathcal{A}_{1}(\begin{array}{l}\phi\psi\end{array}):=-(\begin{array}{l}d\triangle\phi-g’(w^{*}(\cdot))(\phi+\psi)+k\psi\triangle\psi+g,(w^{*}(\cdot))(\phi+\psi)-k\psi\end{array})$

with the domain$\mathcal{D}(\mathcal{A}_{0})=\mathcal{D}(\mathcal{A}_{1})$$=\{(\phi, \psi)\in H^{2}(\Omega)\cross H^{2}(\Omega):\overline{\phi+\psi}=0, \partial\phi/\partial\nu=\partial\psi/\partial v=0(x\in\partial\Omega)\}.$

We note that the spectrum of the operator
for $j=1,2$ , consists of real eigenvalues.
It is also seen that every eigenvalue with non-positive real part ofor
is real (see[7] and [8] .
The number of all the negative eigenvalues with counting multiplicity for each linearized operator is called the Morse index, while the number of all the non-positive
eigenvalues with counting multiplicity for the operator is called the augmented Morseindex.
By [8] and [7] we have the following result:$\mathcal{L}_{j}$$\mathcal{A}_{C}$$\mathcal{A}_{1}$$)$Theorem 2.1 For each $j=0,1$ the Morse index and the augmented Morse index ofagree with those of$\mathcal{L}_{j}$$\mathcal{A}_{j}.$As an application of this theorem any stable equilibrium solution $(u^{*}(x), v^{*}(x))$ to(1.1) for $\gamma=0,1$ in the case that is a finite interval has a monotone profile such thatboth ofare monotone (constant or strictly monotone). Indeed,(resp.
$w^{*}(x))$ of (2.5) (resp. (2.6)) must be monotone if it is stable (see [5] or [10]). Considering(2.9) and (2.10), we get the monotonicity.
We remark that the spectral comparison idea appears in [1] where the Cahn-Hillard
equation and the phase-field system are treated. The authors of [1] consider the linearized
eigenvalue problems of those equations and compare them with certain scalar eigenvalue
problems. On the other hand in the present study we compare the systems with thescalar ones with some nonlocal terms. By considering the nonlocal problems the nicecorrespondence between the Morse indices is shown as stated in Theorem 2.1.
The main tool for the proof of Theorem 2.1 is the min-max principle for the eigenvalue
problem (see [2] and [3]). More precisely, in the case $\gamma=0$ , based on the research forthe phase-field system in [1], we transform the linearized operator
to a self-adjointoperator and apply the mini-max principle repeatedly to compare the eigenvalues ([8]).
On the other hand for the case $\gamma=1$ we are not able to find such a transformation
for. Nonetheless, by introducing an auxiliary eigenvalue problem with a parameter,
which allows a variational setting, we compare eigenvalues of with those of the auxiliary
problem. Then, apply a continuation argument between the auxiliary problem and that(for the details see [7]).
of$\Omega$$u^{*}(x),$ $v^{*}(x)$$u^{*}(x)$$\mathcal{A}_{0}$$\mathcal{A}_{1}$$\mathcal{L}_{1}$$\mathcal{A}_{1}$42Acknowledgements
This work was supported in part by the Grant-in-Aid for Scientific Research (B) No.
22340022 and Challenging Exploratory Research No.24654044, Japan Society for the Promotion of Science.

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