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

Abstract 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 ₁ (G) λ 1 (G) of the negative Laplacian on the graph, which is assumed to satisfy ₁ (G) >0 λ 1 (G) > 0. In particular, if f' (0) > ₁ (G) f ′ (0) > λ 1 (G) than all positive nontrivial solution blows up in finite time, whereas if f' (0) 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.