Global solutions to semilinear parabolic equations driven by mixed local-nonlocal operators

TL;DR

This paper investigates the Cauchy problem for semilinear parabolic equations involving a combined local and nonlocal operator, demonstrating that the Fujita phenomenon and critical exponent are consistent with those for the fractional Laplacian.

Contribution

It establishes the Fujita phenomenon and determines the critical exponent for equations driven by mixed local-nonlocal operators, extending known results to this combined operator case.

Findings

Fujita phenomenon holds for the mixed local-nonlocal operator

Critical value matches that of the fractional Laplacian

Provides insight into blow-up behavior for such equations

Abstract

We are concerned with the Cauchy problem for the semilinear parabolic equation driven by the mixed local-nonlocal operator $\mathcal{L} = -\Delta+(-\Delta)^s$, with a power-like source term. We show that the so-called Fujita phenomenon holds, and the critical value is exactly the same as for the fractional Laplacian.