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

Vishvesh Kumar, Berikbol T. Torebek

TL;DR

The paper derives a complete blow-up criterion for the semilinear heat equation with a mixed local–nonlocal operator $\mathcal{L}=-\Delta+(-\Delta)^{\sigma}$ and general time-dependent nonlinearity $h(t)f(u)$. The authors introduce minorant and majorant tools $f_m$ and $f_M$ to capture growth and prove a necessary-and-sufficient condition (via Theorem $\th1$) that connects a diverging integral involving the heat semigroup to finite-time blow-up for all nontrivial nonnegative data. They recover the Fujita-type threshold $p_F=1+\frac{2\sigma}{d}$ for time-independent power nonlinearities and extend the framework to various $h(t)$ and nonlinearities, including logarithmic and exponential forcing terms. The results unify and extend prior works, offering a practical integral-test criterion for global existence versus blow-up and highlighting how the interplay between local and nonlocal diffusion shapes critical behavior. These findings have implications for understanding blow-up dynamics in models that couple Brownian and Lévy processes through the mixed operator.

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 $ -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].

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

TL;DR

The paper derives a complete blow-up criterion for the semilinear heat equation with a mixed local–nonlocal operator and general time-dependent nonlinearity . The authors introduce minorant and majorant tools and to capture growth and prove a necessary-and-sufficient condition (via Theorem ) that connects a diverging integral involving the heat semigroup to finite-time blow-up for all nontrivial nonnegative data. They recover the Fujita-type threshold for time-independent power nonlinearities and extend the framework to various and nonlinearities, including logarithmic and exponential forcing terms. The results unify and extend prior works, offering a practical integral-test criterion for global existence versus blow-up and highlighting how the interplay between local and nonlocal diffusion shapes critical behavior. These findings have implications for understanding blow-up dynamics in models that couple Brownian and Lévy processes through the mixed operator.

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 , 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].

Paper Structure

This paper contains 3 sections, 5 theorems, 68 equations.

Table of Contents

  1. Introduction and main result
  2. Proof of Main results
  3. Proofs of Corollaries \ref{['cor1']}-\ref{['cor4']}

Key Result

Theorem 1.2

Let $h \in C([0,\infty))$ with $h \geq 0$. Assume that $f \in C([0,\infty))$ is a nonnegative convex function with $f(0)=0$ and $f(u)>0$ for all $u>0$ such that (1.5) holds. Then the following statements are equivalent: Here $e^{-t\mathcal{L}}$ is the heat semigroup on $L^{q}\left(\mathbb{R}^{d}\right)(q \geq 1)$ generated by $\mathcal{L}$.

Theorems & Definitions (13)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Remark 1.8
  • proof : Proof of Theorem \ref{['th1']}
  • proof : Proof of Corollary \ref{['cor1']}
  • ...and 3 more