Global existence for coupled Klein-Gordon equations with different speeds. (English. French summary) Zbl 1255.35162
This paper deals with the global existence of solutions of the Cauchy problem for a system of semilinear wave equations with two speeds, \(1\) and \(c\); the unknowns \(u^1\) and \(u^c\) solve
\[
\begin{align*}{ (\partial_{tt}-\Delta)u^1+u^1 &= Q^1(u_1,u^c) \cr (\partial_{tt}-c^2\Delta)u^c+u^c &= Q^c(u_1,u^c), }\end{align*}
\]
in three space dimensions, where the nonlinearities \(Q^1\) and \(Q^c\) are \({\mathcal O}(|u^1|^2+|u^c|^2)\) as \(u^1\) and \(u^c\to 0\). A condition of “separation of resonances” is assumed. It is checked numerically that it is satisfied for \(c=5\), and the author “believe[s] that it is always [satisfied] except maybe for exceptional values of \(c\).” Assuming that the data are small in \(H^{N+1}\times H^N\) (for \(N\) sufficiently large), and also small in a weighted \(H^3\times H^2\) space, the main result expresses that the solutions are global; in addition, as \(t\to\infty\), they converge to \((U^1,U^c)\) such that \(-\Delta U^1+U^1=-\Delta U^c+U^c=0\) in \(H^{N+1}\), and their \(H^3\) norms are \({\mathcal O}(1/\sqrt t)\).
Reviewer: Satyanad Kichenassamy (Reims)
MSC:
| 35L52 | Initial value problems for second-order hyperbolic systems |
| 35L71 | Second-order semilinear hyperbolic equations |
| 35B34 | Resonance in context of PDEs |
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