On the weak solution for the nonlocal parabolic problem with p-Kirchhoff term via topological degree

Abstract

In this work, we study the existence of weak solution for the following nonlinear parabolic initial boundary value problem associated to the p-Kirchhoff-type equation, ?/ ?t ?M (?? (A(x, t,?u) + 1/ p |u|p) dx) div (a(x, t,?u) ? |?u|p?2?u) = f in Q. = ??(0, T) where ? ? Rn (N ? 2) is a bounded domain with Lipschitz boundar ??,M: R+ ?? R+ is the p-Kirchhoff-type function and a : Q ? RN ?? RN is a Carath?odory function. Under some appropriate assumptions, we obtain the existence of a weak solution for the problem above by using Berkovits and Mustonen topological degree theory, in the space Lp(0, T,W1,p 0 (?)).