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

Soukaina Yacini, Chakir Allalou, Khalid Hilal

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 $L^{p}(0,T,W^{1,p}_{0}(\Omega))$.


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