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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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References:
 [1] M. V. Berry, ”Waves and Thom’s theorem,” Adv. in Phys., vol. 25, pp. 1-26, 1976. [2] M. D. Blair, H. F. Smith, and C. D. Sogge, ”On Strichartz estimates for Schrödinger operators in compact manifolds with boundary,” Proc. Amer. Math. Soc., vol. 136, iss. 1, pp. 247-256, 2008. · Zbl 1169.35012 [3] M. D. Blair, H. F. Smith, and C. D. Sogge, ”Strichartz estimates for the wave equation on manifolds with boundary,” Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 26, iss. 5, pp. 1817-1829, 2009. · Zbl 1198.58012 [4] N. Burq, G. Lebeau, and F. Planchon, ”Global existence for energy critical waves in 3-D domains,” J. Amer. Math. Soc., vol. 21, iss. 3, pp. 831-845, 2008. · Zbl 1204.35119 [5] G. Eskin, ”Parametrix and propagation of singularities for the interior mixed hyperbolic problem,” J. Analyse Math., vol. 32, pp. 17-62, 1977. · Zbl 0375.35037 [6] L. Hörmander, ”The analysis of linear partial differential operators. I,” in Distribution Theory and Fourier Analysis, New York: Springer-Verlag, 2003, p. x. · Zbl 1028.35001 [7] O. Ivanovici, ”Counterexamples to Strichartz estimates for the wave equation in domains,” Math. Ann., vol. 347, iss. 3, pp. 627-673, 2010. · Zbl 1201.35060 [8] O. Ivanovici, ”Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary,” J. Eur. Math. Soc. $$($$JEMS$$)$$, vol. 14, iss. 5, pp. 1357-1388, 2012. · Zbl 1254.35035 [9] O. Ivanovici and F. Planchon, Square function and heat flow estimates on domains, 2008. · Zbl 1200.35066 [10] O. Ivanovici and F. Planchon, ”On the energy critical Schrödinger equation in $$3D$$ non-trapping domains,” Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 27, iss. 5, pp. 1153-1177, 2010. · Zbl 1200.35066 [11] G. Lebeau, Estimation de dispersion pour les ondes dans un convexe, 2006. [12] R. B. Melrose and J. Sjöstrand, ”Singularities of boundary value problems. I,” Comm. Pure Appl. Math., vol. 31, iss. 5, pp. 593-617, 1978. · Zbl 0368.35020 [13] R. B. Melrose and M. E. Taylor, ”The radiation pattern of a diffracted wave near the shadow boundary,” Comm. Partial Differential Equations, vol. 11, iss. 6, pp. 599-672, 1986. · Zbl 0632.35056 [14] F. W. J. Olver, Asymptotics and Special Functions, New York: Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], 1974. · Zbl 0303.41035 [15] J. Shatah and M. Struwe, ”Well-posedness in the energy space for semilinear wave equations with critical growth,” Internat. Math. Res. Notices, vol. 1994, p. 7. · Zbl 0830.35086 [16] H. F. Smith, ”A parametrix construction for wave equations with $$C^{1,1}$$ coefficients,” Ann. Inst. Fourier $$($$Grenoble$$)$$, vol. 48, iss. 3, pp. 797-835, 1998. · Zbl 0974.35068 [17] H. F. Smith and C. D. Sogge, ”On the critical semilinear wave equation outside convex obstacles,” J. Amer. Math. Soc., vol. 8, iss. 4, pp. 879-916, 1995. · Zbl 0860.35081 [18] H. F. Smith and C. D. Sogge, ”On the $$L^p$$ norm of spectral clusters for compact manifolds with boundary,” Acta Math., vol. 198, iss. 1, pp. 107-153, 2007. · Zbl 1189.58017 [19] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton, NJ: Princeton Univ. Press, 1993, vol. 43. · Zbl 0821.42001 [20] D. Tataru, ”Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III,” J. Amer. Math. Soc., vol. 15, iss. 2, pp. 419-442, 2002. · Zbl 0990.35027
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