Uniqueness of limit flow for a class of quasi-linear parabolic equations

TL;DR

This paper studies the uniqueness of long-term behavior for a class of quasi-linear parabolic equations, focusing on conditions for solutions to decay, converge, or blow-up, and characterizing bifurcation phenomena.

Contribution

It provides new conditions ensuring the uniqueness of the limit flow and characterizes solution behaviors near bifurcation points for these equations.

Findings

Global solutions decay uniformly at infinity under certain conditions.

Solutions approach a single steady state as time tends to infinity.

Characterization of blow-up, vanishing, or convergence based on initial data and bifurcation analysis.

Abstract

We investigate the issue of uniqueness of the limit flow for a relevant class of quasi-linear parabolic equations defined on the whole space. More precisely, we shall investigate conditions which guarantee that the global solutions decay at infinity uniformly in time and their entire trajectory approaches a single steady state as time goes to infinity. Finally, we obtain a characterization of solutions which blow-up, vanish or converge to a stationary state for initial data of the form $\lambda \varphi_0$ while $\lambda>0$ crosses a bifurcation value $\lambda_0$.