We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcritical case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.
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Matthew D. Blair, Hart F. Smith, Christopher D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, pp. 1817–1829DOI 10.1016/J.ANIHPC.2008.12.004