Asymptotic Behavior and Decay Estimates of the Solutions for a Nonlinear Parabolic Equation with Exponential Nonlinearity

TL;DR

This paper investigates the long-term behavior and decay rates of solutions to a nonlinear parabolic equation featuring exponential growth, establishing key estimates and solution properties for small initial data.

Contribution

It proves the equivalence of weak and mild solutions and derives decay estimates and asymptotic behavior for small global solutions.

Findings

Decay estimates for solutions

Equivalence of weak and mild solutions

Asymptotic behavior characterized

Abstract

We consider a nonlinear parabolic equation with an exponential nonlinearity which is critical with respect to the growth of the nonlinearity and the regularity of the initial data. After showing the equivalence of the notions of weak and mild solutions, we derive decay estimates and the asymptotic behavior for small global-in-time solutions.