MMN-4708
An existence result for parabolic equation of Kirchhoff type by topological degree method
Soukaina Yacini; Chakir Allalou; Khalid Hilal;Abstract
In the present papere, we will study the existence of at least one weak solution for the nonlinear parabolic initial boundary value problem associated to the following equation of Kirchoff type
$$ \frac{\partial u}{\partial t} - \mathcal{M}\big(\int_{\Omega}(\mathcal{B}(x,t,\nabla u)+ \frac{1}{\theta}\vert\nabla u\vert^{\theta})\;dx\big)\mbox{div}\big(b(x, t, \nabla u)-\vert\nabla u\vert^{\theta-2}\nabla u\big)= f(x,t) -g(x,t,u,\nabla u) $$
By using the Topological degree theory for operators of the type $ \mathtt{L}+\mathtt{S}+\mathtt{C}$, where $ \mathtt{L} $
%D(\mathcal{L})\subset \mathcal{X}\longrightarrow \mathcal{X}^*
is a maximal monotone map, $\mathtt{S} $ is bounded demicontinuous map of class $(S_{+})$ and $\mathtt{C}$ be compact and belongs to $\Gamma^{\tau}_{\sigma}$ (i.e there exist $\tau,\sigma\geq 0$ such that $\Vert \mathtt{C}x\Vert \leq \tau \Vert x\Vert +\sigma )
$.Our focus of study is centered on this problem in the space. $L^{\theta}(0,T,W^{1,\theta}(\Omega))$, wheer $\theta\geq 2$ and $\Omega$ is a bounded open domain in $\mathbb{R}^N$.
Vol. 26 (2025), No. 2, pp. 1089-1102