On the blow-up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups. (English) Zbl 1459.35050
Summary: In this note, we prove a blow-up result for the semilinear damped wave equation in a compact Lie group with power nonlinearity \(|u|^p\) for any \(p > 1\), under suitable integral sign assumptions for the initial data, by using an iteration argument. A byproduct of this method is the upper bound estimate for the lifespan of a local in time solution. As a preliminary result, a local (in time) existence result is proved in the energy space via Fourier analysis on compact Lie groups.
MSC:
| 35B44 | Blow-up in context of PDEs |
| 35L71 | Second-order semilinear hyperbolic equations |
| 43A30 | Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. |
| 43A77 | Harmonic analysis on general compact groups |
| 58J45 | Hyperbolic equations on manifolds |
| 35L52 | Initial value problems for second-order hyperbolic systems |
| 35R09 | Integro-partial differential equations |
| 35B33 | Critical exponents in context of PDEs |
Keywords:
Fujita exponent; upper bound estimates for the lifespan; compact Lie group; local existenceReferences:
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