Decay of mass for a semilinear heat equation with mixed local-nonlocal operators

TL;DR

This paper investigates the long-term behavior of solutions to a reaction-diffusion equation involving mixed local and nonlocal operators, revealing how the nonlinear term or diffusion effects dominate depending on parameter regimes.

Contribution

It provides a detailed analysis of the asymptotic behavior of solutions to a semilinear heat equation with mixed operators, highlighting the influence of parameters on dominant effects.

Findings

Nonlinear term dominates for certain parameter ranges.

Diffusion effects prevail when parameters exceed critical thresholds.

Large time asymptotics depend on the relation between p and the critical exponent.

Abstract

In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation $\partial_t u+t^\beta\mathcal{L} u= - h(t)u^p$ posed on $\mathbb{R}^N$, driven by the mixed local-nonlocal operator $\mathcal{L}=-\Delta+(-\Delta)^{\alpha/2}$, $\alpha\in(0,2)$, and supplemented with a nonnegative integrable initial data, where $p>1$, $\beta\geq 0$, and $h:(0,\infty)\to(0,\infty)$ is a locally integrable function. We study the large time behavior of non-negative solutions and show that the nonlinear term determines the large time asymptotic for $p\leq 1+{\alpha}/{N(\beta+1)},$ while the classical/anomalous diffusion effects win if $p>1+{\alpha}/{N(\beta+1)}$.