Well-Posedness and Numerical Study for Solutions of a Parabolic Equation with Variable-Exponent Nonlinearities

Abstract

We consider the following nonlinear parabolic equation: $u_t-\mathrm{d}\mathrm{i}\mathrm{v}(|\nabla u{|}^{p(x)-\mathrm{2}}\nabla u)=f(x,t)$, where $f:\mathrm{\Omega }\times (\mathrm{0},T)\to \mathbb{R}$ and the exponent of nonlinearity $p(·)$ are given functions. By using a nonlinear operator theory, we prove the existence and uniqueness of weak solutions under suitable assumptions. We also give a two-dimensional numerical example to illustrate the decay of solutions.