Decay estimates and blow up of solutions to a class of heat equations

TL;DR

This paper investigates a semi-linear heat equation with polynomial-logarithmic nonlinearity, establishing conditions for global existence, exponential decay, and finite-time blow-up across various energy levels.

Contribution

It introduces new decay estimates and blow-up criteria for solutions to a heat equation with complex nonlinearities, without restrictions on the exponent.

Findings

Proved global existence and exponential decay in $L^2$ norm.

Established finite-time blow-up conditions at different energy levels.

No restrictions on the exponent in the source term under certain initial data conditions.

Abstract

In this article, we study a semi-linear heat equation with the nonlinearity which is the product of polynomial and logarithmic functions. Using the invariance of the potential well(s), we have established the global existence and exponential decay estimates of solutions in $L^2$ - norm without having any restriction on the exponent in the source term under suitable conditions on the initial data. Moreover, finite time blow up of solutions at subcritical, critical and supercritical initial energy levels is also discussed.