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On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles. (English) Zbl 1193.35100

This paper deals with the Strauss conjecture concerning the small data global existence in time for semilinear wave equations studied in the exterior domain \(\mathbb{R}_+\times\Omega\) of a compact obstacle \(\Omega\subset\mathbb{R}^n\setminus K\), \(K\)-compact, for space dimensions \(n= 3\) and \(n= 4\). The obstacle is nontrapping, the boundary condition on \(\mathbb{R}_+\times\partial\Omega\) is either Dirichlet or Neumann type and the nonlinear term \(F_p(u)\) behaves like \(|u|^p\) for \(|u|\) small. The authors formulate local energy decay assumptions Hyp. 1.1 for linear wave equations in the above-mentioned exterior domain and review some important cases when Hyp. 1.1 holds. The main existence result is given by Theorem 1.2. The key point in the proof of Theorem 1.2. is establishing an appropriate “abstract Strichartz estimates” for linear wave equations (see Theorem 1.4.). To do this the authors apply Hyp. 1.1, the global Minkowski abstract Strichartz estimates and the local abstract Strichartz estimates for \(\Omega\).

MSC:

35L05 Wave equation
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
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