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On a semilinear heat equation on infinite graphs I: blow-up for large initial data

Fabio Punzo, Federico Zucchero

Abstract

We investigate finite-time blow-up of solutions to the Cauchy problem for a semilinear heat equation posed on infinite graphs. Assuming that the initial datum is sufficiently large, we establish a general blow-up criterion valid on arbitrary infinite graphs. We then apply this result to specific classes of graphs, including trees and the integer lattice. The approach developed in the paper can be regarded as a discrete counterpart of Kaplan's method, suitably adapted to the graph setting. In a companion paper, which is the second part of this work, we also complement the blow-up analysis by addressing arbitrary initial data and proving global existence for sufficiently small data.

On a semilinear heat equation on infinite graphs I: blow-up for large initial data

Abstract

We investigate finite-time blow-up of solutions to the Cauchy problem for a semilinear heat equation posed on infinite graphs. Assuming that the initial datum is sufficiently large, we establish a general blow-up criterion valid on arbitrary infinite graphs. We then apply this result to specific classes of graphs, including trees and the integer lattice. The approach developed in the paper can be regarded as a discrete counterpart of Kaplan's method, suitably adapted to the graph setting. In a companion paper, which is the second part of this work, we also complement the blow-up analysis by addressing arbitrary initial data and proving global existence for sufficiently small data.
Paper Structure (18 sections, 24 theorems, 252 equations)

This paper contains 18 sections, 24 theorems, 252 equations.

Table of Contents

  1. Introduction
  2. Mathematical framework
  3. The graph setting
  4. Difference and Laplace operators
  5. The combinatorial distance, inner and outer degrees
  6. Spherically symmetric functions
  7. Functional spaces
  8. Model trees
  9. The integer lattice
  10. Hypotheses and definition of solution
  11. Statement of the main results
  12. Blow-up on general graphs
  13. Blow-up on trees
  14. Blow-up on the integer lattice
  15. Proof of \ref{['theorem_blow_up_chapter_3']}
  16. ...and 3 more sections

Key Result

Proposition 2.9

Let $(X, \omega, \mu)$ be a locally finite weighted graph. Then

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 40 more