On the stability of stationary states in diffusion models

M. V. Polovinkina^a, I. P. Polovinkin<sup>bc

a</sup> Voronezh State University of Engineering Technologies
^b Voronezh State University
^c Belgorod State University

Abstract: We consider the initial-boundary value problem for the system of partial differential equations:
\begin{equation*} \frac{\partial u_s}{\partial t}=\vartheta_s \Delta u_s+F_s(u),\ s=1,\dots,m, \ x=(x_1,\dots,x_n)\in\Omega\subset\mathbb{R}^n, \end{equation*}

\begin{equation*} \left.\left(\mu_s u_s + \eta_s\frac{\partial u_s}{\partial\overrightarrow{\nu}}\right)\right|_{x\in\partial\Omega}=B_s(x),\ \mu_s^2+\eta_s^2>0, \ \mu_s\geq0,\ \eta_s \geq 0, \end{equation*}

\begin{equation*} u_s(x,0)=u_s^0(x), \ s=1,\dots,m, \end{equation*}
where $\Omega$ is a bounded domain with a piecewise smooth boundary $\partial\Omega$, $\overrightarrow{\nu}$ is a unit external normal vector to the boundary $\partial\Omega$ of the domain $\Omega$, $u=(u_1(x,t),\dots,u_m(x,t))$, $\vartheta_s\geq 0$, ${\rm diam}\Omega=d$, $B_s(x)\in C(\partial\Omega)$, $u_s^0(x)\in C(\overline{\Omega})$, $s=1,\dots,m,$ ${\overline{\Omega}=\Omega\cup\partial\Omega},$ $\Delta$ is the Laplace operator defined by the formula
$$ \Delta v= \sum\limits_{j=1}^n\,\frac{\partial^2 v}{\partial x_j^2}. $$
It is assumed that the functions $F_s$ are differentiable at a stationary point. Let $w=(w_1(x),\dots,w_m(x))$ be a stationary solution of the considered problem, that is, the solution of the boundary problem
\begin{equation*} \vartheta_s \Delta w_s+F_s(w)=0,\ s=1,\dots,m, \ x\in\Omega, \end{equation*}

\begin{equation*} \left.\left(\mu_s w_s + \eta_s\frac{\partial w_s}{\partial\overrightarrow{\nu}}\right)\right|_{x\in\partial\Omega}=B_s(x),\ s=1,\dots,m. \end{equation*}
We showed that a sufficient condition for the stability of a stationary solution is the negative definiteness of the quadratic form
\begin{equation*} \sum\limits_{s=1}^m\sum\limits_{k=1}^m\,A_{sk}\,z_kz_s\,, \end{equation*}
where
\begin{equation*} A_{sk}=\frac{1}{2}\left(\frac{\partial F_s}{\partial x_k}+\frac{\partial F_k}{\partial x_s}\right)-\delta_{ks}\vartheta_s/d^2. \end{equation*}
We note from a general point of view that adding diffusion terms to ordinary differential equations, for example, to logistic ones, can in some cases improve sufficient conditions for the stability of a stationary solution. We give examples of models in which the addition of diffusion terms to ordinary differential equations changes the stability conditions of a stationary solution.

Keywords: diffusion model, initial boundary value problem, stationary solution (state), stability, sufficient stability condition.

UDC: 519.765, 51-7, 517.9

MSC: 35B35, 35Q99