GLOBAL AND BLOW–UP SOLUTIONS FOR A PARABOLIC EQUATION WITH NONLINEAR MEMORY UNDER NONLINEAR NONLOCAL BOUNDARY CONDITION

In this paper we consider parabolic equation with nonlinear memory and absorption \begin{equation*} u_t= \Delta u + a \int\limits_0^t u^q (x,\tau) d\tau - b u^m, x \in \Omega, t>0, \end{equation*} under nonlinear nonlocal boundary condition \begin{equation*} u(x,t) = \int\limits_{\Omega}{k(x,y,t)u^l(y,t)} dy, x\in\partial\Omega, t > 0, \end{equation*} and nonnegative continuous initial data. Here $a,$ $b,$ $q,$ $m,$ $l$ are positive numbers, $\Omega$ is a bounded domain in $\mathbb{R}^N,$ $N\geq1,$ with smooth boundary $\partial\Omega,$ $k(x,y,t)$ is a nonnegative continuous function defined for $x \in \partial \Omega$, $y \in \overline\Omega$ and $ t \ge 0.$ We prove that each solution of the problem is global if $\max (q,l) \leq 1$ or $\max (q,l) > 1$ and $ l < (m + 1)/2,$ $q \leq m.$ If $l>\max\{1, (p+1)/2\}$ and the function $k(x,y,t)$ is positive for small $t,$ the solutions blow up in finite time for large enough initial data. The obtained results improve previously established conditions for the existence and absence of global solutions.