Time regularity and long-time behavior of parabolic \(p\)-Laplace equations on infinite graphs. (English) Zbl 1323.39005
Summary: We consider the so-called discrete \(p\)-Laplacian, a nonlinear difference operator that acts on functions defined on the nodes of a possibly infinite graph. We study the corresponding Cauchy problem and identify the generator of the associated nonlinear semigroups. We prove higher order time regularity of the solutions. We investigate the long-time behavior of the solutions and discuss in particular finite extinction time and conservation of mass. Namely, on one hand, for small \(p\) if an infinite graph satisfies some isoperimetric inequality, then the solution to the parabolic \(p\)-Laplace equation vanishes in finite time; on the other hand, for large \(p\), these parabolic \(p\)-Laplace equations always enjoy conservation of mass.
MSC:
| 39A12 | Discrete version of topics in analysis |
| 35K92 | Quasilinear parabolic equations with \(p\)-Laplacian |
| 39A10 | Additive difference equations |
| 47H20 | Semigroups of nonlinear operators |
| 05C63 | Infinite graphs |
Keywords:
nonlinear semigroups generated by subdifferentials; operators on graphs; gradient systems; parabolic equations with finite extinction time; long-time behavior; inifinite graph; isoperimetric inequality; parabolic \(p\)-Laplace equationsReferences:
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