Heat equations driven by mixed local-nonlocal operators with exponential nonlinearity

TL;DR

This paper studies a heat equation with a mixed local-nonlocal operator and exponential nonlinearity, establishing local and global solutions, and analyzing their long-term decay behavior.

Contribution

It introduces a framework for well-posedness and decay estimates for heat equations with combined local and nonlocal operators and exponential growth nonlinearities.

Findings

Established local well-posedness in Orlicz spaces for exponential nonlinearities.

Proved global existence of solutions for small initial data under polynomial growth conditions.

Derived decay estimates showing the influence of nonlinearity near zero on long-term behavior.

Abstract

We investigate the Cauchy problem for a heat equation driven by the mixed local-nonlocal operator $\mathcal{L}:=-\Delta+(-\Delta)^s$, $s\in(0,1)$, with exponential nonlinearity \[ \partial_tu(x,t)+\mathcal{L}u(x,t)=f(u(x,t)), \qquad (x,t)\in \mathbb{R}^{d}\times(0,\infty), \] where $f:\mathbb{R}\to\mathbb{R}$ exhibits exponential growth at infinity and satisfies $f(0)=0$. We establish local well-posedness in a suitable Orlicz space in the case where $f(u)\sim e^{|u|^p}$ as $|u|\to\infty$, with $p>1$. We further prove the existence of global solutions for small initial data under the assumption that $f$ satisfies the growth condition $|f(u)|\sim |u|^m$ near the origin. Moreover, we derive large-time decay estimates in Lebesgue spaces, showing that the behavior of the nonlinearity near the origin determines the decay rate of solutions and highlights a unique asymptotic transition that bridges local and non-local diffusion theories.