Global existence and decay estimates for the heat equation with exponential nonlinearity

TL;DR

This paper establishes global existence and decay estimates for solutions to a heat equation with exponential nonlinearity in Lebesgue spaces, under small initial data and specific growth conditions on the nonlinearity.

Contribution

It provides new results on global solutions and decay rates for the heat equation with exponential growth nonlinearities in the setting of exp L^p spaces.

Findings

Global existence of solutions under small initial data.

Decay estimates in Lebesgue spaces depending on the nonlinearity growth.

Conditions relating parameters for solution existence and decay.

Abstract

In this paper we consider the initial value {problem $\partial_{t} u- \Delta u=f(u),$ $u(0)=u_0\in exp\,L^p(\mathbb{R}^N),$} where $p>1$ and $f : \mathbb{R}\to\mathbb{R}$ having an exponential growth at infinity with $f(0)=0.$ Under smallness condition on the initial data and for nonlinearity $f$ {such that $|f(u)|\sim \mbox{e}^{|u|^q}$ as $|u|\to \infty$,} $|f(u)|\sim |u|^{m}$ as $u\to 0,$ $0<q\leq p\leq\,m,\;{N(m-1)\over 2}\geq p>1$, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m.$