On a nonlinear parabolic problem: Stability properties of Ground States

TL;DR

This paper investigates the stability properties of ground state solutions to a nonlinear parabolic PDE with critical or supercritical nonlinearities, extending known results to broader classes of nonlinear potentials using a new unifying approach.

Contribution

It introduces a novel unifying method to analyze stability of ground states for a wider class of nonlinear potentials in nonlinear parabolic equations.

Findings

Extended stability results to larger classes of nonlinear potentials.

Established weak asymptotic stability of positive ground states.

Provided a new framework for analyzing nonlinear parabolic stability.

Abstract

We consider the Cauchy-problem for the following parabolic equation: \begin{equation*} \displaystyle u_t = \Delta u+ f(u,|x|), \end{equation*} where $x \in \mathbb{R}^n$, $n >2$, and $f=f(u,|x|)$ is either critical or supercritical with respect to the Joseph-Lundgren exponent. Using a new unifying approach we extend to a larger class of nonlinear potentials $f$, some known results concerning stability and weak asymptotic stability of positive Ground States.