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

Abstract

We investigate the Cauchy problem for a heat equation driven by the mixed local-nonlocal operator $\mathcal{L}:=-\hat{I}”+(-\hat{I}”{)}^s$, $s\in (0,1)$, 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.