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

2017/04/19 by Yong Lin, Lin, Yong, Yiting Wu +1 · 4 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1704.05702

Abstract

Let G=(V,E) be a locally finite connected weighted graph, Δ be the usual graph Laplacian. In this paper, we study the blow-up problems for the nonlinear parabolic equation u_t=Δu + 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.

Cited by

• The lifespan of positive solutions of heat equation with power-logarithmic nonlinearity on locally finite graph
• Blow up results and lifespan estimates for nonlinear damped wave equations on weighted graphs
• Blow-up solutions of parabolic p-Laplacian inequalities on locally finite graphs
• A note on Liouville type equations on graphs

Related

• Blow-up solutions of parabolic p-Laplacian inequalities on locally finite graphs
• Blow-up for a semilinear heat equation with Fujita's critical exponent on locally finite graphs
• Blow-up and global existence for semilinear parabolic equations on infinite graphs
• Existence of Nontrivial Solutions for the Nonlinear Equation on Locally Finite Graphs
• Existence and convergence of solutions to p-Laplace equations on locally finite graphs