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

TL;DR

This work analyzes the large-time behavior of nonnegative solutions to the Cauchy problem $\partial_t u+t^{\beta}\mathcal{L}u=-h(t)u^p$ on $\mathbb{R}^N$ with $\mathcal{L}=-\Delta+(-\Delta)^{\alpha/2}$, $\alpha\in(0,2)$. Using the mixed diffusion framework and the heat kernel $E_\alpha$, the authors establish a sharp threshold $p_c=1+\frac{\alpha}{N(\beta+1)}$ that separates nonlinear-dominated dynamics from diffusion-dominated dynamics: for $p>p_c$ the mass tends to a positive limit $M_\infty$ and solutions converge to a self-similar profile $M_\infty E_\alpha(\cdot)$, while for $p\le p_c$ the mass decays to zero. The analysis combines $L^p-L^q$ estimates, mild-solution techniques, and a nonlinear capacity (rescaled test function) method to handle the critical regime. These results extend diffusion-dominated behavior from purely local or nonlocal settings to the mixed local-nonlocal operator, clarifying how local and anomalous diffusion interact with nonlinear damping. The findings have implications for understanding mass persistence and decay in systems with competing diffusion mechanisms.

Abstract

In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation $\partial_t u+t^β\mathcal{L} u= - h(t)u^p$ posed on $\mathbb{R}^N$, driven by the mixed local-nonlocal operator $\mathcal{L}=-Δ+(-Δ)^{α/2}$, $α\in(0,2)$, and supplemented with a nonnegative integrable initial data, where $p>1$, $β\geq 0$, 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 $p\leq 1+α/{N(β+1)},$ while the classical/anomalous diffusion effects win if $p>1+α/{N(β+1)}$.

Theorems & Definitions (22)

• Definition 1
• Proposition 1
• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• Lemma 5
• proof
• Lemma 6