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On a semilinear heat equation on infinite graphs II: blow-up for arbitrary initial data and global existence

Fabio Punzo, Federico Zucchero

Abstract

This paper is the second part of the study initiated in a companion work and is devoted to finite-time blow-up and global existence for a semilinear heat equation on infinite weighted graphs. We first establish basic results on mild and classical solutions (which, to the best of our knowledge, were not previously available in the setting of graphs) proving their equivalence under suitable assumptions and showing the existence of a solution between a given sub- and supersolution. We then analyze blow-up and global existence on $\mathbb Z^N$, providing proofs based on methods different from those used on $\mathbb Z^N$ in the existing literature. Moreover, for graphs with positive spectral gap, we prove global existence for small initial data. In contrast with previous functional analytic approaches yielding mild solutions, our method relies on the construction of global-in-time supersolutions and leads to the existence of classical solutions.

On a semilinear heat equation on infinite graphs II: blow-up for arbitrary initial data and global existence

Abstract

This paper is the second part of the study initiated in a companion work and is devoted to finite-time blow-up and global existence for a semilinear heat equation on infinite weighted graphs. We first establish basic results on mild and classical solutions (which, to the best of our knowledge, were not previously available in the setting of graphs) proving their equivalence under suitable assumptions and showing the existence of a solution between a given sub- and supersolution. We then analyze blow-up and global existence on , providing proofs based on methods different from those used on in the existing literature. Moreover, for graphs with positive spectral gap, we prove global existence for small initial data. In contrast with previous functional analytic approaches yielding mild solutions, our method relies on the construction of global-in-time supersolutions and leads to the existence of classical solutions.
Paper Structure (21 sections, 22 theorems, 358 equations)

This paper contains 21 sections, 22 theorems, 358 equations.

Table of Contents

  1. Introduction
  2. Mathematical framework
  3. The graph setting
  4. Difference and Laplace operators
  5. The combinatorial distance
  6. Functional spaces
  7. Homogeneous model trees
  8. The integer lattice
  9. Statement of the results
  10. Main results
  11. Auxiliary results
  12. Definition of classical and mild solutions
  13. Basic properties of classical and mild solutions
  14. The heat equation on infinite graphs
  15. The heat semigroup
  16. ...and 6 more sections

Key Result

Theorem 3.1

Let $1 < p \leq 1 + \frac{2}{N}$. Let $u$ be a solution of problem problem_HR_G with $X=\mathbb Z^N$ and $u_0\not\equiv 0$. Then $u$ blows up in finite time.

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 34 more