Existence to nonlinear parabolic problems with unbounded weights

TL;DR

This paper investigates the existence of solutions to nonlinear weighted parabolic equations with unbounded weights, establishing conditions under which global weak solutions exist based on Hardy-type inequalities.

Contribution

It introduces new existence results for nonlinear parabolic problems with unbounded weights, extending the theory to broader classes of weights satisfying Hardy inequalities.

Findings

Existence of global weak solutions under certain conditions.

Applicable to a general class of unbounded weights.

Provides a threshold for the parameter λ based on Hardy inequality constants.

Abstract

We consider the weighted parabolic problem of the type \begin{equation*} \begin{split} \left\{\begin{array}{ll} u_t-\mathrm{div}(\omega_2(x)|\nabla u|^{p-2} \nabla u )= \lambda \omega_1(x) |u|^{p-2}u,& x\in\Omega, u(x,0)=f(x),& x\in\Omega, u(x,t)=0,& x\in\partial\Omega,\ t>0, \end{array}\right. \end{split} \end{equation*} for quite a general class of possibly unbounded weights $ \omega_1,\omega_2$ satisfying the Hardy-type inequality. We prove existence of a global weak solution in the weighted Sobolev spaces provided that $\lambda$ is smaller than the optimal constant in the inequality.