Blow-up and global existence for semilinear parabolic equations on infinite graphs
Abstract
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 $$\lambda _1(G)$$ λ 1 ( G ) of the negative Laplacian on the graph, which is assumed to satisfy $$\lambda _1(G)>0$$ λ 1 ( G ) > 0 . In particular, if $$f'(0)>\lambda _1(G)$$ f ′ ( 0 ) > λ 1 ( G ) than all positive nontrivial solution blows up in finite time, whereas if $$f'(0)
Fields of Study & Subjects
Nonlinear Partial Differential Equations
Stability and Controllability of Differential Equations
Geometric Analysis and Curvature Flows
Publication details
- Journal / source
- Calculus of Variations and Partial Differential Equations
- Publisher
- Springer Science and Business Media LLC
- Volume
- 65
- Issue
- 4
- Publication date
- 28 February 2026
- Language
- EN