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On the critical semilinear wave equation outside convex obstacles. (English) Zbl 0860.35081
The main result of this paper is the extension to the case of the exterior domains of the theorem M. G. Grillakis [Ann. Math. II. Ser. 132, No. 3, 485-509 (1990; Zbl 0736.35067)] on the global solvability to the critical semilinear wave equation in three space dimensions. More precisely, the authors prove that the Cauchy problem \[ \square u= -u^5, \quad u(0,x) = f(x), \quad u_t(0,x) = g(x), \quad u=0 \text{ on } \partial \Omega, \] where \(\Omega= \mathbb{R}^3 \backslash K\) with \(K\) compact, strictly convex, smooth set, and \((f,g)\) are smooth functions satisfying infinitely many (necessary) compatibility conditions at \(\{0\} \times\Omega\), admits a unique, global, smooth solution on \(\mathbb{R}^+ \times \Omega\).
Another result is the estimate \(|f_j |_{L^q} \leq\lambda_j^{\sigma (q)} |f_j |_{L^2}\), where \(f_j\) is an eigenfunction, with eigenvalue \(\lambda^2_j\), of the Dirichlet Laplacian operator \(-\Delta_g\) on a \(n\)-dimensional compact manifold \((M_n,g)\) with strictly geodesically concave boundary. Here, \(\sigma (q)\) is equal to \(n(1/2- 1/q)- 1/2\) if \(2(n+1) (n-1)^{-1} \leq q\leq \infty\), and to \((n-1) (1/2-1/q)/2\) if \(2\leq q\leq 2 (n+1) (n-1)^{-1}\). In order to prove both of these results, the authors firstly prove that the Strichartz estimate \[ |v |_{L^{2(n+ 1)/(n-1)} ([0,T] \times \Omega)} \leq C_T \bigl(|f |_{\dot H^{1/2} (\Omega)} + |g|_{\dot H^{-1/2} (\Omega)} \bigr) \] holds on any Riemannian manifold \((\Omega,g)\) with strictly geodesically concave boundary. Here, \(v\) is the solution of \(v_{tt} - \Delta_g v=0\) on \(\Omega\) with initial-boundary conditions \((v,v_t)= (f,g)\) at \(t=0\), \(v=0\) on \(\Omega\). In the case when \(\Omega\) is a manifold without boundary, the Strichartz inequality was firstly proved by L. V. Kapitanski [Leningr. Math. J. 1, No. 3, 693-726 (1990); translation from Algebra Anal. 1, No. 3, 127-159 (1989; Zbl 0732.35118)].

MSC:
35L70 Second-order nonlinear hyperbolic equations
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