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Nonexistence of solutions to parabolic problems with a potential on weighted graphs

Dario D. Monticelli, Fabio Punzo, Jacopo Somaglia

TL;DR

This work extends Fujita-type nonexistence results for semilinear parabolic equations to the setting of weighted graphs. Using a test-function method with space-time cutoffs, the authors derive bounds tied to a distance-Laplacian condition $\Delta d(x,x_0)\le C/d^{\alpha}(x,x_0)$ and a space-time weighted volume-growth hypothesis, demonstrating that no nontrivial nonnegative global solutions exist for $\sigma>1$ under these conditions. They prove a sharp threshold on infinite graphs, with sharpness illustrated on $\mathbb{Z}^N$ for $v\equiv1$, recovering the classical Fujita exponent $1+2/N$ in this discrete setting. The finite-graph case is also treated, providing a parallel nonexistence result under a compact-graph volume condition. Overall, the paper bridges discrete graph settings with continuous Fujita-type phenomena and clarifies how distance-Laplacian bounds and volume-growth govern global blow-up behavior on graphs.

Abstract

We investigate nonexistence of nontrivial nonnegative solutions to a class of semilinear parabolic equations with a positive potential, posed on weighted graphs. Assuming an upper bound on the Laplacian of the distance and a suitable weighted space-time volume growth condition, we show that no global solutions exists. We also discuss the optimality of the hypotheses, thus recovering a critical exponent phenomenon of Fujita type.

Nonexistence of solutions to parabolic problems with a potential on weighted graphs

TL;DR

This work extends Fujita-type nonexistence results for semilinear parabolic equations to the setting of weighted graphs. Using a test-function method with space-time cutoffs, the authors derive bounds tied to a distance-Laplacian condition and a space-time weighted volume-growth hypothesis, demonstrating that no nontrivial nonnegative global solutions exist for under these conditions. They prove a sharp threshold on infinite graphs, with sharpness illustrated on for , recovering the classical Fujita exponent in this discrete setting. The finite-graph case is also treated, providing a parallel nonexistence result under a compact-graph volume condition. Overall, the paper bridges discrete graph settings with continuous Fujita-type phenomena and clarifies how distance-Laplacian bounds and volume-growth govern global blow-up behavior on graphs.

Abstract

We investigate nonexistence of nontrivial nonnegative solutions to a class of semilinear parabolic equations with a positive potential, posed on weighted graphs. Assuming an upper bound on the Laplacian of the distance and a suitable weighted space-time volume growth condition, we show that no global solutions exists. We also discuss the optimality of the hypotheses, thus recovering a critical exponent phenomenon of Fujita type.
Paper Structure (8 sections, 6 theorems, 101 equations)

This paper contains 8 sections, 6 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. The graph setting
  4. The weighted Laplacian
  5. Main results
  6. Proofs of the main results
  7. Examples
  8. Finite graphs

Key Result

Theorem 2.6

Let assumption e7f be satisfied. Assume the graph $(V,\omega,\mu)$ is infinite. Let $v\colon V\times [0,\infty)\to \mathbb{R}$ be a positive function, $\sigma>1$ and $\theta_1\geqslant 2$, $\theta_2\geqslant 1$ such that $\frac{\theta_1}{\theta_2}\geqslant 1 + \alpha$, with $\alpha\in[0,1]$ as in e7 where and $x_0\in V$ as in e7f. Let $u\colon V\times [0,\infty)\to \mathbb{R}$ be a non-negative v

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Corollary 2.8
  • Corollary 2.9
  • proof : Proof of Theorem \ref{['teo1']}
  • ...and 11 more