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

Abstract

In this work, we study the existence of weak solution for the following nonlinear parabolic initial boundary value problem assosiated to the $p$-Kirchhoff-type equation,
$$
\displaystyle \frac{\partial u}{\partial t} - \mathcal{M}\Big( \int_{\Omega} \big(A(x,t,\nabla u)+\frac{1}{p}\vert u\vert^{p}\big)dx \Big)\mbox{div}\Big(a(x, t, \nabla u)-\vert\nabla u\vert^{p-2}\nabla u\Big)= f$$ in $\mathcal{Q}.
=\Omega\times(0,T)$ where $\Omega\subset \mathbb{R}^n \; ( N\geq 2)$ is a bounded domain with Lipschitz boundar $\partial \Omega$, $\mathcal{M}: \mathbb{R}^{+}\longrightarrow \mathbb{R}^{+}$ is the $p$-Kirchhoff-type function and $a:\mathcal{Q}\times \mathbb{R}^{N} \longrightarrow \mathbb{R}^{N}$ is a Carath\'eodory function. Under some appropriate assumptions, we obtain the existence of a weak solution for the problem above by using Berkovits and Mustonen?s topological degree theory, in the space $L^{p}(0,T,W^{1,p}_{0}(\Omega))$.