Long time dynamics for damped Klein-Gordon equations
| Autor: | Geneviève Raugel, Wilhelm Schlag, Nicolas Burq |
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| Přispěvatelé: | Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics, University of Chicago, ANR: ANR-13-BS01-0010-03 (ANA\'E).NSF: DMS-1160817, ANR-13-BS01-0010,ANAÉ,Analyse asymptotique des Equations aux dérivées partielles d'évolution(2013) |
| Rok vydání: | 2017 |
| Předmět: |
Dynamical systems theory
General Mathematics [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] Invariant manifold Mathematics::Analysis of PDEs Banach space Dynamical Systems (math.DS) Type (model theory) 01 natural sciences symbols.namesake Mathematics - Analysis of PDEs 0103 physical sciences FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] Mathematics - Dynamical Systems 0101 mathematics Klein–Gordon equation Mathematical physics Physics 010102 general mathematics Nonlinear system Bounded function symbols MSC: 35L05 74J20 35H15 37L30 37L50 010307 mathematical physics Analysis of PDEs (math.AP) Sign (mathematics) |
| Zdroj: | Annales scientifiques de l'École normale supérieure. 50:1447-1498 |
| ISSN: | 1873-2151 0012-9593 |
| Popis: | For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global solution is bounded. The result applies to standard energy subcritical focusing nonlinearities $|u|^{p-1} u$, $1\textless{}p\textless{}(d+2)/(d-2)$ as well as any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems). |
| Databáze: | OpenAIRE |
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