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Dispersion for the wave equation inside strictly convex domains. I: The Friedlander model case. (English) Zbl 1310.35151
The IBV problem with Dirichlet boundary conditions for the linear homogeneous wave equation, on a manifold, is investigated. The considered domain is strictly convex and has a smooth boundary. The Friedlander’s model and Friedlander operator are introduced. As main result one presents a dispersion estimate for the solution to the stated problem, where for the initial value \(u(0,x)\) one takes the Dirac mass at point \((a,0,\dots,0)\), \(a\in{]0,1]}\), and for the initial value of the time derivative of solution the value zero. The estimate depends on a parameter \(h \in {(0,1]}\) and is proved for \(t \in { (0,T_0]}\). Compared to an earlier estimate, there is a \(1/4\) loss in the \(h/t\) exponent which is related to the presence of caustics for small values of parameter \(a\). One indicates values of parameter \(a\) when estimate is optimal and one shows that the optimal loss is due to swallowtail type singularities in the wave front set. Combined with some classical arguments, the dispersion estimate leads to a set of Strichartz estimates which further are used in the theory of nonlinear wave equations, for proving existence and uniqueness (local, global) of Cauchy problem solutions. The main steps for proving the dispersion estimate have been described in the paper of the second author [“Estimation de dispersion pour les ondes dans un convexe”, J. Équ. aux Dériv. Part., Exp. No. 6, 1–18 (2006)]. In the present paper one shows a complete construction of the parametrix for the wave equation which is then used to obtain decay estimates. Parametrix is constructed for different values of parameter \(a\). At the end there is a discussion on the extension of the results to dimensions higher or equal than three. Appendix A is dedicated to the energy critical nonlinear wave equation.

MSC:
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L05 Wave equation
35B45 A priori estimates in context of PDEs
58J45 Hyperbolic equations on manifolds
35A18 Wave front sets in context of PDEs
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