Initial boundary value problem with nonlocal boundary condition for a nonlinear parabolic equation with memory

Abstract

We consider a nonlinear parabolic equation with memory $u_{t}=\Delta u+au^{p}\int\limits_0^t u^{q}(x,\tau)\mathrm{d}\tau-bu^{m}$ for $(x,t)\in \Omega\times (0,+\infty)$ under nonlinear nonlocal boundary condition $\left.\frac{\partial u(x,t)}{\partial v}\right|_{\partial\Omega\times (0,+\infty)}= \int\limits_\Omega k(x,y,t)u^{l}(y,t)\mathrm{d}y$ and initial data $u(x,0)=u_{0}(x), x\in\Omega$, where $a,b,q,m,l$ are positive constants; $p\geq 0$; $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$; $v$ is unit outward normal on $\partial\Omega$. Nonnegative continuous function $k(x,y,t)$ is defined for $x\in \partial\Omega, y\in\bar{\Omega}, t\geq 0$, nonnegative function $u_{0}(x)\in C^{1}(\bar\Omega)$, while it satisfies the condition $\frac{\partial u_{0}(x)}{\partial v}=\int\limits_\Omega k(x,y,0)u_{0}^{l}(y)\mathrm{d} y$ for $x\in\partial\Omega$. In this paper we study classical solutions. We establish the existence of a local maximal solution of the original problem. We introduce definitions of a supersolution and a subsolution. It is shown that under some conditions a supersolution is not less than a subsolution. We find conditions for the positiveness of solutions. As a consequence of the positiveness of solutions and the comparison principle of solutions, we prove the uniqueness theorem.