Existence for a Nonlocal Porous Medium Equations of Kirchhoff Type with Logarithmic Nonlinearity

Abstract

We study the Dirichlet problem for the nonlocal parabolic equation of the Kirchhoff type \[ u_{t}-a\left(\|u\|_{L^{p}(\Omega)}^{p}\right)\sum\limits_{i=1}^{n}D_{i}\left( \left\vert u\right\vert ^{p-2}D_{i}u\right) +b(x,t) \left\vert u \right\vert ^{\alpha \left( x,t\right) -2}u\log|u|=f\left( x,t\right) \quad \text{in $Q_T=\Omega \times (0,T)$}, \] 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 $\mathbb{R}_+$. It is shown that the problem has a weak solution under appropriate and general conditions on $a(\cdot)$, $\alpha(\cdot,\cdot)$ and $b(\cdot)$.