Quasilinear parabolic problem with $p(x)$-Laplacian: existence, uniqueness of weak solutions and stabilization

TL;DR

This paper investigates the existence, uniqueness, and long-term behavior of weak solutions to a quasilinear parabolic PDE involving the variable exponent p(x)-Laplacian, including stabilization properties.

Contribution

It establishes the existence and uniqueness of weak solutions for the p(x)-Laplacian parabolic problem and analyzes their stabilization behavior over time.

Findings

Proved existence and uniqueness of weak solutions.

Analyzed the stabilization and long-term behavior of solutions.

Abstract

We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-\Delta _{p(x)}u = f(x,u)$ in $ (0,T)\times\Omega$; $u = 0$ on $(0,T)\times\partial\Omega$; $u(0,x)=u_0(x)$ in $\Omega$; involving the $p(x)$-Laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.