Global and blow-up solutions for a parabolic equation with nonlinear memory under nonlinear nonlocal boundary condition

Аннотация

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, \quad x \in \Omega,\quad 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, \quad x\in\partial\Omega, \quad 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.