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 λ1(G) of the negative Laplacian on the graph, which is assumed to satisfy λ1(G)>0. In particular, if f′(0)>λ1(G) than all positive nontrivial solution blows up in finite time, whereas if f′(0)<λ1(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.
Blow-up and global existence for semilinear parabolic equations on infinite graphs
Grillo G.;Meglioli G.;Punzo F.
2026-01-01
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 λ1(G) of the negative Laplacian on the graph, which is assumed to satisfy λ1(G)>0. In particular, if f′(0)>λ1(G) than all positive nontrivial solution blows up in finite time, whereas if f′(0)<λ1(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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
