Blow-up for a semilinear heat equation with Fujita's critical exponent on locally finite graphs

TL;DR

This paper investigates finite-time blow-up of solutions to a semilinear heat equation with critical exponent on locally finite graphs satisfying specific curvature and volume growth conditions.

Contribution

It establishes blow-up results for solutions with critical exponent on graphs under curvature and volume growth assumptions, extending previous work by Lin and Wu.

Findings

Solutions blow up in finite time when alpha = 2/m.

Results apply to graphs with curvature dimension condition CDE'(n,0).

Provides conditions under which initial data lead to blow-up.

Abstract

Let $G=(V,E)$ be a locally finite, connected and weighted graph. We prove that, for a graph satisfying curvature dimension condition $CDE'(n,0)$ and uniform polynomial volume growth of degree $m$, all non-negative solutions of the equation $\partial_tu=\Delta u+u^{1+\alpha}$ blow up in a finite time provided that $\alpha=\frac{2}{m}$. We also consider the blow-up problem under certain conditions for volume growth and initial value. The obtained results provide a significant complement to the work by Lin and Wu in earlier paper.