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This is a Quantitative Study study.

September 30, 2025Open Access

Blow-up criteria for the semilinear parabolic equations driven by mixed local-nonlocal operators

Key Points

The study identifies necessary and sufficient conditions for the nonexistence of a global solution.
It focuses on semilinear heat equations driven by mixed local and nonlocal operators, including a critical Fujita exponent.
This research builds on existing work, refining the understanding of global solutions with time-dependent nonlinearities.
The results enhance the framework for understanding solutions to equations with complex nonlocal aspects.

Abstract

The main goal of this paper is to establish necessary and sufficient conditions for the nonexistence of a global solution to the semilinear heat equation with a mixed local--nonlocal operator -Δ+ (-Δ) ^σ, under a general time-dependent nonlinearity. Our results complement the recent work of Carhuas-Torre et al. ArXiv, (2025), arXiv: 2505. 20401, in which the authors provide sufficient conditions for the existence and nonexistence of global solutions. In particular, our results recover the critical Fujita exponent for time-independent power-type nonlinearities, as obtained by Biagi et al. Bull. London Math. Soc. (2024), 1--20 and Del Pezzo et al. Nonlinear Anal. 255 (2025), 113761.

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