Fujita-type results for the semilinear heat equations driven by mixed local-nonlocal operators

Abstract

This paper explores the critical behavior of the semilinear heat equation $u_t+\mathcal{L}_{a, b}u=|u|^p+f(x)$, considering both the presence and absence of a forcing term $f(x).$ The mixed local-nonlocal operator $\mathcal{L}_{a, b}=-a\Delta+b(-\Delta)^s,\,a,\,b \in \mathbb{R}_+,$ incorporates both local and nonlocal Laplacians. We determine the Fujita-type critical exponents by considering the existence or nonexistence of global solutions. Interestingly, the critical exponent is determined by the nonlocal component of the operator and, as a result, coincides with that of the fractional Laplacian. In the case without a forcing term, our results improve upon recent findings by Biagi et al. [Bull. London Math. Soc. 57 (2025), 265-284] and Del Pezzo et al. [Nonlinear Analysis 255 (2025), 113761]. When a forcing term is included, our results refine those of Wang et al. [J. Math. Anal. Appl., 488 (1) (2020), 124067] and complement the work of Majdoub [La Matematica, 2 (2023), 340-361].