Global and Nonglobal Solutions for a Mixed Local-nonlocal Heat Equation

Abstract

In this work, we establish optimal criteria characterizing the global and nonglobal existence of solutions to a semilinear parabolic equation with a general source term f. The equation is governed by the mixed operator

$$ \mathcal {L} = -\Delta + (-\Delta )^s, \quad \text {with } 0< s < 1. $$

In our analysis, we study supersolutions and obtain estimates on the semigroup generated by \(\mathcal {L}\). In particular, we recover Fujita’s critical exponent \(1 + \tfrac{2s}{N}\), previously established by Biagi, Punzo, and Vecchi [2], and by Del Pezzo and Ferreira [18].