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

Abstract

In this paper we consider parabolic equation with nonlinear memory and absorption \begin{equation*} u_t= \Delta u + a \int_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_{\Omega}{k(x,y,t)u^l(y,t)}\,dy, \; x\in\partial\Omega, \; t > 0, \end{equation*} and nonnegative continuous initial datum. Here a , b , q , m , l are positive numbers, Ω is a bounded domain in R N for N ≥ 1 with smooth boundary ∂ Ω , k(x,y,t) is a nonnegative continuous function defined for x ∈ ∂ Ω , y ∈ Ω ‾ and t ≥ 0. We prove that every solution of the problem is global if max ⁡ ( q , l ) ≤ 1 or max ⁡ ( q , l ) > 1 and l < ( m + 1 ) / 2 , q ≤ m . For l > max ⁡ { 1 , ( p + 1 ) / 2 } and positive for small values of t function k(x,y,t) solutions blow up in finite time for large enough initial data. The obtained results improve previously established conditions for the existence and the absence of global solutions.