On the viscosity solutions to a class of nonlinear degenerate parabolic differential equations

TL;DR

This paper establishes existence and uniqueness of positive solutions for a class of nonlinear degenerate parabolic equations with specific conditions on the operator and functions involved, extending to solutions without sign restrictions.

Contribution

It provides new existence and uniqueness results for solutions of nonlinear degenerate parabolic equations with general operators and conditions, including cases with no sign restrictions.

Findings

Proved existence and uniqueness of positive solutions.

Extended results to solutions without sign restrictions.

Analyzed equations with operators satisfying homogeneity conditions.

Abstract

In this work, we show existence and uniqueness of positive solutions of $H(Du, D^2u)+\chi(t)|Du|^\Gamma-f(u)u_t=$ in $\Omega\times(0, T)$ and $u=h$ on its parabolic boundary. The operator $H$ satisfies certain homogeneity conditions, $\Gamma>0$ and depends on the degree of homogeneity of $H$, $f>0$, increasing and meets a concavity condition. We also consider the case $f\equiv 1$ and prove existence of solutions without sign restrictions.