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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)

##### MSC:
 58J45 Hyperbolic equations on manifolds
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