On a class of nonlocal porous medium equations of Kirchhoff type


Sert U.

TURKISH JOURNAL OF MATHEMATICS, vol.46, no.6, pp.2231-2249, 2022 (Peer-Reviewed Journal)

  • Publication Type: Article / Article
  • Volume: 46 Issue: 6
  • Publication Date: 2022
  • Journal Name: TURKISH JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded
  • Page Numbers: pp.2231-2249

Abstract

We study the Dirichlet problem for the degenerate parabolic equation of the Kirchhoff type
\[
u_{t}-a\left(\|u\|_{L^{p}(\Omega)}^{p}\right)\sum\limits_{i=1}^{n}D_{i}\left(
\left\vert u\right\vert ^{p-2}D_{i}u\right) +b\left(
x,t,u\right)=f\left( x,t\right) \quad \text{in $Q_T=\Omega \times (0,T)$},
\] where $p\geq2$, $T>0$, $\Omega \subset \mathbb{R}^{n}$, $n\geq 2$, is a smooth bounded domain. The coefficient $a(\cdot)$ is real-valued function
defined on $\mathbb{R}_+$ and $b(\cdot,\cdot,\tau)$ is a measurable function with variable nonlinearity in $\tau$. We prove existence of weak solutions of the considered problem under appropriate and general conditions on $a$ and $b$. Sufficient conditions for uniqueness are found and in the case $f\equiv0$ the decay rates for $\|u\|_{L^2(\Omega)}$ are obtained