A parabolic problem involving $p(x)$-Laplacian, a power and a singular nonlinearity

TL;DR

This paper investigates the existence of solutions for nonlinear singular parabolic equations involving a variable exponent p(x)-Laplacian, considering different cases based on the source term and parameter ranges.

Contribution

It extends the analysis of p(x)-Laplacian equations to include singular nonlinearities and variable exponents, providing new existence results under various conditions.

Findings

Existence of non-negative weak solutions established.

Different parameter regimes analyzed for the source term.

Results applicable to bounded domains with Lipschitz boundaries.

Abstract

The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial u}{\partial t}-\Delta_{p(x)}u&=\lambda u^{q(x)-1} + u^{-\delta(x)}g+ f&&\text{in}~Q_T, u&= 0&&\text{on}~\Sigma_T, u(0,\cdot)&=u_0(\cdot)&&\text{in}~\Omega\nonumber. \end{align*} Here $Q_T=\Omega\times(0,T)$, $\Sigma_T=\partial\Omega\times(0,T)$, $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz continuous boundary $\partial\Omega$, $\lambda\in(0,\infty)$, $f\in L^1(Q_T)$, $g\in L^\infty(\Omega)$, $u_0\in L^r(\Omega)$ with $r\geq 2$, $\delta:\overline{\Omega}\rightarrow(0,\infty)$ is continuous, and $p,q\in C(\overline{\Omega})$ with $\underset{x\in\overline{\Omega}}{\max}~p(x)<N$, $q(\cdot)<p^*(\cdot)$. The article is distinguished into two cases according to the choice of $f$ with different range of parameters $p(\cdot)$, $q(\cdot)$.