The existence and nonexistence of global solutions for a semilinear heat equation on graphs

2017/02/12 by Yong Lin, Lin, Yong, Yiting Wu +1 · 6 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1702.03531

openalex publication_date 2017/02/12 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

Let G=(V,E) be a finite or locally finite connected weighted graph, Δ be the usual graph Laplacian. Using heat kernel estimate, we prove the existence and nonexistence of global solutions for the following semilinear heat equation on G \ u_t=Δu + u^1+α · amp; in (0,+∞)× V,
u(0,x)=a(x) · amp; in V. . We conclude that, for a graph satisfying curvature dimension condition CDE'(n,0) and V(x,r)≃ r^m, if 02, then there is a non-negative global solution u provided that the initial value is small enough. In particular, these results are true on lattice ℤ^m.

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