Blow-up problems for nonlinear parabolic equations on locally finite graphs

TL;DR

This paper investigates conditions under which solutions to nonlinear parabolic equations on locally finite graphs blow up in finite time, considering different initial and boundary conditions.

Contribution

It establishes blow-up criteria for nonlinear parabolic equations on graphs, extending classical PDE results to discrete graph structures.

Findings

Solutions blow up in finite time under certain conditions on f

Blow-up behavior depends on initial and boundary conditions

Provides theoretical criteria for blow-up in graph-based PDEs

Abstract

Let $G=(V,E)$ be a locally finite connected weighted graph, $\Delta$ be the usual graph Laplacian. In this paper, we study the blow-up problems for the nonlinear parabolic equation $u_t=\Delta u + f(u)$ on $G$. The blow-up phenomenons of the equation are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if $f$ satisfies appropriate conditions, then the solution of the equation blows up in a finite time.