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arXiv · 2509.10014

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

Vishvesh Kumar·Berikbol T. Torebek
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Abstract

The main goal of this paper is to establish \emph{necessary and sufficient conditions} for the nonexistence of a global solution to the semilinear heat equation with a mixed local--nonlocal operator $ -\Delta + (-\Delta)^\sigma$, 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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Vishvesh Kumar, Berikbol T. Torebek. 2025-09-12. Blow-up criteria for the semilinear parabolic equations driven by mixed local-nonlocal operators. https://arxiv.org/abs/2509.10014

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