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

We investigate the Cauchy problem for a heat equation driven by the mixed local-nonlocal operator $\mathcal{L}:=-\mathrm{\Delta }+(-\mathrm{\Delta }{)}^s$, $s\in (0,1)$, with exponential nonlinearity $${\mathrm{\partial }}_{t}u(x,t)+\mathcal{L}u(x,t)=f(u(x,t)),(x,t)\in {\mathbb{R}}^d\times (0,\mathrm{\infty }),$$ 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 $f(u)\sim e^{\mathrm{\mid }u{\mathrm{\mid }}^p}$ as $\mathrm{\mid }u\mathrm{\mid }\to \mathrm{\infty }$, with $p>1$. We further prove the existence of global solutions for small initial data under the assumption that $f$ satisfies the growth condition $\mathrm{\mid }f(u)\mathrm{\mid }\sim \mathrm{\mid }u{\mathrm{\mid }}^m$ 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.