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Strichartz estimates for the wave equation on manifolds with boundary. (English) Zbl 1198.58012
The authors consider the wave equation \(\partial^2_t u(t,x)- \Delta_g u(t,x)= 0\), \(u(0,x)= f(x)\), \(\partial_t u(0,x)= g(x)\) on a compact Riemannian manifold with boundary, where \(\Delta_g\) denotes the Laplace-Beltrami operator.
Assuming \(2\leq q\leq\infty\), \(2\leq q<\infty\), \({1\over p}+{n\over q}={n\over 2}-\gamma\), \({3\over p}+{n-1\over q}\leq {n-1\over 2}\) for \(2\leq n\leq 4\), and \({1\over p}+{1\over q}\leq {1\over 2}\) for \(n\geq 4\), they obtain the following Strichartz estimates on solutions \(u: (-T,T)\times M\to\mathbb{C}\) satisfying either Dirichlet or Neumann homogeneous boundary conditions:
\[ \| u\|_{L^p([-T,T];L^q(M))}\leq C(\| f\|_{H^\gamma(M)}+\| g\|_{H^{\gamma-1}(M)}). \] Here \(n\) is the dimension of \(M\), \(H^\gamma(M)\) denotes the \(L^2\) Sobolev space over \(M\) of order \(\gamma\) and \(C\) is some constant depending on \(M\) and \(T\).
Next, they investigate the semilinear wave equation \(\partial^2_t u-\Delta u+|u|^{r-1} u= 0\), \((u,\partial_t u)|_{t=0}= (f,g)\), \(u|_{\partial M}= 0\) (or \(\partial_\nu u|_{\partial M}= 0\), where \(\nu= \nu(x)\) denotes the outward pointing unit normal vector to the boundary \(\partial M\) at \(x\)). Here two cases are distinguished: \(r< 1+{4\over n-2}\) (the subcritical case) and \(r= 1+{4\over n-2}\) (the critical case).
For \(\Omega\subset\mathbb{R}^3\) with smooth compact boundary under appropriate assumptions on \(f\) and \(g\), the authors prove the existence of a global unique solution both for the subcritical case and for the critical case, the latter one for \(f\) and \(g\) sufficiently small with respect to norms in suitable spaces.
Finally, the equation \(\partial^2_t u-\Delta u+ u^5= 0\) on \(\Omega= \mathbb{R}^3\setminus K\) (\(n=3\), \(r= 5\)) is considered, when \(K\) is a compact star-shaped with respect to the origin (i.e., \(\nu(x)\cdot x\geq 0\) on \(\partial K\)) obstacle. Under Dirichlet boundary conditions, scattering results for a solution to the homogeneous equation \(\partial^2_t u-\Delta u= 0\) are proved.
Reviewer: S. Burys (Kraków)

58J45 Hyperbolic equations on manifolds
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