## 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
Full Text:

### References:

 [1] Bahouri, H.; Gérard, P., High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. math., 121, 131-175, (1999) · Zbl 0919.35089 [2] Bahouri, H.; Shatah, J., Decay estimates for the critical wave equation, Ann. inst. H. Poincaré anal. non lineáire, 15, 6, 783-789, (1998) · Zbl 0924.35084 [3] Bchatnia, A.; Daoulatli, M., Scattering and exponential decay of the local energy for the solutions of semilinear and subcritical wave equation outside convex obstacle, Math. Z., 247, 619-642, (2004) · Zbl 1063.35033 [4] Burq, N., Global Strichartz estimates for nontrapping geometries: about an article by H. Smith and C. sogge, Comm. partial differential equations, 28, 1675-1683, (2003) · Zbl 1026.35020 [5] Burq, N.; Lebeau, G.; Planchon, F., Global existence for energy critical waves in 3-D domains, J. amer. math. soc., 21, 831-845, (2008) · Zbl 1204.35119 [6] N. Burq, F. Planchon, Global existence for energy critical waves in 3-d domains: Neumann boundary conditions, Amer. J. Math., in press · Zbl 1184.35210 [7] Christ, M.; Kiselev, A., Maximal functions asociated to filtrations, J. funct. anal., 179, 409-425, (2001) · Zbl 0974.47025 [8] Ginibre, J.; Velo, G., Generalized Strichartz inequalities for the wave equation, J. funct. anal., 133, 50-68, (1995) · Zbl 0849.35064 [9] Grillakis, M.G., Regularity for the wave equation with a critical nonlinearity, Comm. pure appl. math., 45, 749-774, (1992) · Zbl 0785.35065 [10] Hidano, K.; Metcalfe, J.; Smith, H.F.; Sogge, C.D.; Zhou, Y., On abstract Strichartz estimates and the strauss conjecture for nontrapping obstacles · Zbl 1193.35100 [11] Ivanovici, O., Counter examples to Strichartz estimates for the wave equation in domains · Zbl 1201.35060 [12] Kapitanski, L.V., Norm estimates in Besov and lizorkin – treibel spaces for the solutions of second order linear hyperbolic equations, J. soviet math., 56, 2348-2389, (1991) · Zbl 0759.35014 [13] Keel, M.; Tao, T., Endpoint Strichartz estimates, Amer. J. math., 120, 955-980, (1998) · Zbl 0922.35028 [14] Koch, H.; Tataru, D., Dispersive estimates for principally normal operators, Comm. pure appl. math., 58, 217-284, (2005) · Zbl 1078.35143 [15] Lindblad, H.; Sogge, C.D., On existence and scattering with minimal regularity for semilinear wave equations, J. funct. anal., 130, 357-426, (1995) · Zbl 0846.35085 [16] Metcalfe, J., Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle, Trans. amer. math. soc., 356, 4839-4855, (2004) · Zbl 1060.35026 [17] Mockenhaupt, G.; Seeger, A.; Sogge, C.D., Local smoothing of Fourier integral operators and carleson – sjölin estimates, J. amer. math. soc., 6, 65-130, (1993) · Zbl 0776.58037 [18] Morawetz, C., Time decay for the nonlinear klein – gordon equation, Proc. roy. soc. A., 306, 291-296, (1968) · Zbl 0157.41502 [19] Shatah, J.; Struwe, M., Regularity for the wave equation with a critical nonlinearity, Internat. math. res. notices, 7, 303-310, (1994) [20] Smith, H.F., A parametrix construction for wave equations with $$C^{1, 1}$$ coefficients, Ann. inst. Fourier (Grenoble), 48, 797-835, (1998) · Zbl 0974.35068 [21] Smith, H.F., Spectral cluster estimates for $$C^{1, 1}$$ estimates, Amer. J. math., 128, 1069-1103, (2006) · Zbl 1284.35149 [22] Smith, H.F.; Sogge, C.D., On the critical semilinear wave equation outside convex obstacles, J. amer. math. soc., 8, 879-916, (1995) · Zbl 0860.35081 [23] Smith, H.F.; Sogge, C.D., Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. partial differential equations, 25, 2171-2183, (2000) · Zbl 0972.35014 [24] Smith, H.F.; Sogge, C.D., On the $$L^p$$ norm of spectral clusters for compact manifolds with boundary, Acta math., 198, 107-153, (2007) · Zbl 1189.58017 [25] Sogge, C., Lectures on nonlinear wave equations, (1995), International Press Boston, MA · Zbl 1089.35500 [26] Strichartz, R., Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation, Duke math. J., 44, 705-714, (1977) · Zbl 0372.35001 [27] Tao, T., Nonlinear dispersive equations: local and global analysis, (2006), American Mathematical Society Providence, RI · Zbl 1106.35001 [28] Tataru, D., Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients III, J. amer. math. soc., 15, 419-442, (2002) · Zbl 0990.35027 [29] Tataru, D., Phase space transforms and microlocal analysis, (), 505-524 · Zbl 1111.35143
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.