Existence of solutions for degenerate parabolic equations with singular terms

TL;DR

This paper investigates the existence of solutions for a class of degenerate parabolic equations with singular gradient-dependent terms, extending understanding of such equations under specific boundary and initial conditions.

Contribution

It establishes the existence of solutions for a degenerate parabolic problem involving singular gradient terms, which was previously not well-understood.

Findings

Existence of solutions under given conditions

Extension to equations with singular gradient terms

Framework for analyzing similar degenerate problems

Abstract

In this paper we deal with parabolic problems whose simplest model is $$ \begin{cases} u'- \Delta_{p} u + B\frac{|\nabla u|^p}{u} = 0 & \text{in} (0,T) \times \Omega,\newline u(0,x)= u_0 (x) &\text{in}\ \Omega, \newline u(t,x)=0 &\text{on}\ (0,T) \times \partial\Omega, \end{cases} $$ where $T>0$, $N\geq 2$, $p>1$, $B > 0$, and $u_{0}$ is a positive function in $L^{\infty}(\Omega)$ bounded away from zero.