A Strong Minimum principle and Large Time Asymptotics for viscosity solutions to a class of doubly nonlinear possibly degenerate parabolic equations

TL;DR

This paper establishes a strong minimum principle and analyzes the large time behavior of positive viscosity solutions for a class of doubly nonlinear parabolic equations, extending understanding of their qualitative properties.

Contribution

It introduces new results on the strong minimum principle and asymptotic behavior for a broad class of doubly nonlinear parabolic equations with homogeneous spatial operators.

Findings

Proved a strong minimum principle for viscosity solutions.

Derived large time asymptotics for solutions in bounded domains.

Extended classical results to degenerate and doubly nonlinear cases.

Abstract

We study a version of the strong minimum principle, and large time asymptotics of positive viscosity solutions to classes of doubly nonlinear parabolic equations of the form $$ H(Du,D^2u)-u^{k-1}u_t=0,\;\;k\geq 1,\quad\mbox{in $\Omega\times [0,T)$},$$ where $\Omega\subset \mathbb{R}^n$ is a bounded domain and $0<T\leq \infty$. The spatial operator $H$ is homogeneous with power $k$.