Fujita exponent and blow-up rate for a mixed local and nonlocal heat equation

TL;DR

This paper investigates the blow-up behavior of a mixed local and nonlocal heat equation, establishing the Fujita exponent based on the nonlocal component and analyzing blow-up rates in certain cases.

Contribution

It determines the Fujita exponent for the mixed diffusion equation and analyzes blow-up rates, extending understanding of blow-up phenomena in nonlocal and local diffusion models.

Findings

Fujita exponent is given by 1+2s/N, determined by the nonlocal part.

Provides blow-up rate estimates in specific cases.

Highlights the influence of nonlocal diffusion on blow-up behavior.

Abstract

In this paper we consider the blow-up problem for a mixed local-nonlocal diffusion operator, \[ u_t=a\Delta u -b(-\Delta)^s u+u^p. \] We show that the Fujita exponent is given by the nonlocal part, $p_F=1+2s/N$. We also determinate, in some cases, the blow-up rate.