Existence of solutions to degenerate parabolic problems with two weights via the Hardy inequality

TL;DR

This paper demonstrates the existence of weak solutions for certain degenerate parabolic equations using Hardy inequalities with two weights, extending the analysis to unbounded domains.

Contribution

It introduces a novel approach applying Hardy inequalities with two weights to establish solutions for degenerate parabolic problems on unbounded domains.

Findings

Existence of weak solutions proved under specified conditions.

Application of Hardy inequality with two weights to PDE analysis.

Extension to unbounded domains in degenerate parabolic problems.

Abstract

The paper concentrates on the application of the following Hardy inequality \begin{equation*} \int_\Omega \ |\xi(x)|^p \omega_{1 }(x)dx\le \int_\Omega |\nabla \xi(x)|^p\omega_{2 }(x)dx, \end{equation*} to the proof of existence of weak solutions to degenerate parabolic problems of the type \begin{equation*} \left\{\begin{array}{ll} u_t-div(\omega_2(x)|\nabla u|^{p-2} \nabla u )= \lambda W(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{equation*} on an open subset $\Omega\subseteq\mathbb{R}^n$, not necessarily bounded, where \[W(x)\leq \min\{m,\omega_1(x)\},\qquad m\in\mathbb{R}_+.\]