Blow-up and global existence for semilinear parabolic equations on infinite graphs

We investigate existence of global in time solutions versus blow-up ones for the semilinear heat equation posed on infinite graphs. The source term is a general function f ( u ), and the different behaviour of solutions is characterized by the behaviour of f near the origin and by the first eigenvalue λ ₁(G) λ 1 ( G ) of the negative Laplacian on the graph, which is assumed to satisfy λ ₁(G)>0 λ 1 ( G ) > 0 . In particular, if f'(0)>λ ₁(G) f ′ ( 0 ) > λ 1 ( G ) than all positive nontrivial solution blows up in finite time, whereas if f'(0)<λ ₁(G) f ′ ( 0 ) < λ 1 ( G ) , or if a weaker condition involving the Lipschitz constant of f in a neighborhood of the origin holds, then there exist global in time, bounded solutions.