Exponential Decay Results for Semilinear Parabolic PDE with $C^0$ Potentials: A "Mean Value" Approach

TL;DR

This paper introduces a 'mean value' approach to analyze the asymptotic behavior of semilinear parabolic PDEs with $C^0$ potentials, establishing exponential decay and blow-up results through a priori estimates.

Contribution

It presents a novel 'mean value' method to derive exponential decay and blow-up criteria for semilinear parabolic PDEs with minimal regularity assumptions.

Findings

Established exponential decay for certain global solutions.

Analyzed finite time blow-up using the new method.

Applied results to PDEs with boundary degeneracy.

Abstract

The asymptotic behavior of some semilinear parabolic PDEs is analyzed by means of a "mean value" property. This property allows us to determine, by means of appropriate {\em{a priori}} estimates, some exponential decay results for suitable global solutions. We also apply the method to investigate a well-known finite time blow-up result. An application is given to a one-dimensional semilinear parabolic PDE with boundary degeneracy. Our results shed further light onto the problem of determining initial data for which the corresponding solution is guaranteed to exponentially decay to zero or blow-up in finite time.