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Blow-up and global existence for semilinear parabolic equations on infinite graphs

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arxiv 2406.15069 v2 pith:NOMPFB36 submitted 2024-06-21 math.AP

Gabriele Grillo , Giulia Meglioli , Fabio Punzo This is my paper · ORCID
classification math.AP
keywords blow-upexistenceglobalgraphsinfinitelambdasemilinearsolution
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abstract

We investigate existence of global in time solutions and blow-up of solutions to the semilinear heat equation posed on infinite graphs. The source term is a general function $f(u)$. We always assume that the infimum of the spectrum of the Laplace operator $\lambda_1(G)$ on the graph is positive. According to an interaction between the behavior of $f$ close to $0$ and the value $\lambda_1(G)$, we get the existence of a global in time solution or blow-up of any nonnegative solution, provided that the initial datum is nontrivial.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Blow-up solutions of parabolic $p$-Laplacian inequalities on locally finite graphs

    math.AP 2025-07 conditional novelty 6.0 of 10

    For parabolic p-Laplacian inequalities on locally finite graphs, sufficiently fast growing nonlinear sources force finite-time blow-up of solutions.

  2. Nonexistence results for the semilinear wave equation on graphs

    math.AP 2025-06 reject novelty 6.0 of 10

    On infinite weighted graphs, if the graph Laplacian of the distance grows slowly and suitable weighted volume growth holds, every very weak solution of the semilinear wave inequality is identically zero.

  3. On a semilinear parabolic equation with time-dependent source term on infinite graphs

    math.AP 2025-02 reject novelty 4.0 of 10

    For u_t = Δu + h(t)u^q on infinite graphs with λ1(G)>0, the paper claims blow-up for fast-growing h and global small data when ∫ h(t)e^{-λ1(q-1)t}dt is finite, but the general blow-up theorem is vitiated by a sign error.

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