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

Vishvesh Kumar, Berikbol T. Torebek

TL;DR

The paper analyzes Fujita-type threshold phenomena for the semilinear heat equation with a mixed local-nonlocal diffusion operator $\mathcal{L}_{a,b}=-a\Delta+b(-\Delta)^s$. It shows the critical exponent is dictated by the nonlocal component, yielding $p_F=1+\tfrac{2s}{d}$ in the forcing-free case, with a separate forcing-driven threshold $p_{\text{crit}}=\tfrac{d}{d-2s}$, and establishes sharp nonexistence and global existence results across subcritical, critical, and supercritical regimes under various data assumptions. The work develops a robust local well-posedness framework via heat-kernel estimates and fixed-point arguments, and employs test-function methods and weighted-function spaces to prove blow-up or global solvability, including instantaneous blow-up criteria under positive forcing. It extends and refines existing results for pure local or pure nonlocal models, clarifying how the mixed operator governs threshold behavior and providing a comprehensive map of when small data yield global solutions. The results have implications for understanding diffusion with mixed mechanisms in parabolic problems and inform the construction of global solutions under small data in complex media.

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 $\mathcal{L}_{a, b}=-aΔ+b(-Δ)^s,\,a,\,b \in \mathbb{R}_+,$ 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].

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

TL;DR

The paper analyzes Fujita-type threshold phenomena for the semilinear heat equation with a mixed local-nonlocal diffusion operator . It shows the critical exponent is dictated by the nonlocal component, yielding in the forcing-free case, with a separate forcing-driven threshold , and establishes sharp nonexistence and global existence results across subcritical, critical, and supercritical regimes under various data assumptions. The work develops a robust local well-posedness framework via heat-kernel estimates and fixed-point arguments, and employs test-function methods and weighted-function spaces to prove blow-up or global solvability, including instantaneous blow-up criteria under positive forcing. It extends and refines existing results for pure local or pure nonlocal models, clarifying how the mixed operator governs threshold behavior and providing a comprehensive map of when small data yield global solutions. The results have implications for understanding diffusion with mixed mechanisms in parabolic problems and inform the construction of global solutions under small data in complex media.

Abstract

This paper explores the critical behavior of the semilinear heat equation , considering both the presence and absence of a forcing term The mixed local-nonlocal operator 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].

Paper Structure

This paper contains 15 sections, 5 theorems, 106 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Existence, uniqueness, extension and continuity of local solutions
  5. Proof of Main results
  6. Proof of Theorem \ref{['thm:Main1']}
  7. The case $p<p_{Crit}$
  8. The case $p=p_{Crit}$
  9. The case $p>p_{Crit}$
  10. Proof of Theorem \ref{['thm:Main']}
  11. The case $p<p_{F}$ and $\int_{\mathbb{R}^d}u_0(x)\,dx>0$
  12. The case $p<p_{F}$ and $\int_{\mathbb{R}^d}u_0(x)\,dx=0,\,u_0\not\equiv 0$
  13. The case $p=p_{F}$
  14. The case $p>p_{F}$
  15. Proof of Theorem \ref{['thm:Main3']}

Key Result

Theorem 1.2

Let $a,b>0,$$s \in (0, 1),$$f(x)\equiv 0,$$u_0\in L^1_{loc}(\mathbb R^d),$ and let $1< p<\infty$. Then, the following facts hold.

Theorems & Definitions (12)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Definition 2.1: Local mild solution
  • Theorem 2.2
  • proof
  • Definition 2.3: Local weak solution
  • ...and 2 more