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Scattering and exponential decay of the local energy for the solutions of semilinear and subcritical wave equation outside convex obstacle. (English) Zbl 1063.35033
The authors establish existence and asymptotic completeness of wave operators for large data energy class solutions for the localized subcritical defocusing wave equation outside of a convex obstacle \(O\), or more precisely for the Cauchy problem \[ \square u + \chi(x) g(u) = 0 \text{ on } \mathbb R \times (\mathbb R^3 \backslash 0) \]
\[ u = 0 \text{ on } \mathbb R \times \partial \Omega \]
\[ (u(0),u_t(0)) \in (H^1 \times L^2)(\mathbb R^3 \backslash 0) \] where we of course require \(u(0)\) to vanish on the boundary. Here \(g\) is a nonlinearity such as \(| u| ^{p-1} u\) for some \(2 < p < 5\); more general nonlinearities of similar coercive subcritical type are treated in the paper. The key tools are a Morawetz-type exponential decay estimate near the convex obstacle, the Smith-Sogge Strichartz estimate outside of this obstacle, and some modified Lax-Phillips semigroup theory using defect measures.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35L05 Wave equation
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