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

2025/09/12 by Vishvesh Kumar, Kumar, Vishvesh, Berikbol T. Torebek +1
Computer Science · Mathematics · #35A01 #35B33 #35B44 #35K58 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2509.10014

openalex publication_date 2025/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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].

Citations

• Blow-up for a Nonlocal Diffusion Equation with Time Regularly Varying Nonlinearity and Forcing
• Global and nonglobal solutions for a mixed local-nonlocal heat equation
• Fujita-type results for the semilinear heat equations driven by mixed local-nonlocal operators
• Decay of mass for a semilinear heat equation with mixed local-nonlocal operators
• Fujita exponent and blow-up rate for a mixed local and nonlocal heat equation
• Global solutions to semilinear parabolic equations driven by mixed local-nonlocal operators
• On global solutions of heat equations with time-dependent nonlinearities on unimodular Lie groups
• Description of an ecological niche for a mixed local/nonlocal dispersal:\n an evolution equation and a new Neumann condition arising from the\n superposition of Brownian and L 'evy processes
• (Non)local logistic equations with Neumann conditions
• Heat kernel estimates for Δ+Δ^α/2 under gradient perturbation

Related

• Global and nonglobal solutions for a mixed local-nonlocal heat equation
• Fujita-type results for the semilinear heat equations driven by mixed local-nonlocal operators
• Global solutions to semilinear parabolic equations driven by mixed local-nonlocal operators
• Blow-up for a Nonlocal Diffusion Equation with Time Regularly Varying Nonlinearity and Forcing
• Global existence and blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition