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

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 $p>1$, $\beta \ge 0$, 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 $p\le 1+\alpha /N(\beta +1),$ while the classical/anomalous diffusion effects win if $p>1+\alpha /N(\beta +1)$.