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December  2021, 41(12): 5707-5742. doi: 10.3934/dcds.2021093

Dispersive estimates for the wave and the Klein-Gordon equations in large time inside the Friedlander domain

Oana Ivanovici

Sorbonne Université, CNRS, LJLL, F-75005 Paris, France

Received  January 2021 Revised  April 2021 Published  December 2021 Early access  June 2021

Fund Project: The author is supported by ERC grant ANADEL 757 996

We prove global in time dispersion for the wave and the Klein-Gordon equation inside the Friedlander domain by taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at a given, large time. Moreover, we uncover a significant difference between Klein-Gordon and the wave equation in the low frequency, large time regime, where Klein-Gordon exhibits a worse decay than the wave, unlike in the flat space.

Keywords: Dispersive estimates, wave and Klein Gordon equations, Dirichlet boundary condition.
Mathematics Subject Classification: Primary: 35Rxx; Secondary: 58Jxx.
Citation: Oana Ivanovici. Dispersive estimates for the wave and the Klein-Gordon equations in large time inside the Friedlander domain. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5707-5742. doi: 10.3934/dcds.2021093
References:
[1]

M. D. Blair, H. F. Smith and Ch. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1817-1829.  doi: 10.1016/j.anihpc.2008.12.004.  Google Scholar

[2]

G. Eskin, Parametrix and propagation of singularities for the interior mixed hyperbolic problem, J. Analyse Math., 32 (1977), 17-62.  doi: 10.1007/BF02803574.  Google Scholar

[3]

O. Ivanovici, R. Lascar, G. Lebeau and F. Planchon., Dispersion for the wave equation inside strictly convex domains II: The general case, preprint, https://arXiv.org/abs/1605.08800. Google Scholar

[4]

O. Ivanovici, G. Lebeau and F. Planchon, Strichartz estimates for the wave equation inside strictly convex 2d model domain, preprint, https://arXiv.org/abs/2008.03598. Google Scholar

[5]

O. Ivanovici, G. Lebeau and F. Planchon, Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case, Ann. of Math. (2), 180 (2014), 323-380.  doi: 10.4007/annals.2014.180.1.7.  Google Scholar

[6]

O. Ivanovici, G. Lebeau and F. Planchon, New counterexamples to Strichartz estimates for the wave equation on a 2D model convex domain, to appear in J. Ec. polytech. Math., https://arXiv.org/abs/2008.02716. Google Scholar

[7]

J. Kato and T. Ozawa, Endpoint strichartz estimates for the Klein-Gordon equation in two space dimensions and some applications, J. Math. Pures Appl., 95 (2011), 48-71.  doi: 10.1016/j.matpur.2010.10.001.  Google Scholar

[8]

S. Machihara, K. Nakanishi and T. Ozawa, Small global solutions and the nonrelativistic limit for the nonlinear dirac equation, Rev. Mat. Iberoamericana, 19 (2003), 179-194.  doi: 10.4171/RMI/342.  Google Scholar

[9]

R. B. Melrose, Equivalence of glancing hypersurfaces, Invent. Math., 37 (1976), 165-191.  doi: 10.1007/BF01390317.  Google Scholar

[10]

R. B. Melrose and J. Sjöstrand, Singularities of boundary value problems. I, Comm. Pure Appl. Math., 31 (1978), 593-617.  doi: 10.1002/cpa.3160310504.  Google Scholar

[11]

R. B. Melrose and J. Sjöstrand, Singularities of boundary value problems. II, Comm. Pure Appl. Math., 35 (1982), 129-168.  doi: 10.1002/cpa.3160350202.  Google Scholar

[12]

R. B. Melrose and M. E. Taylor, Boundary Problems for Wave Equations With Grazing and Gliding Rays, Available at https://www.unc.edu/math/Faculty/met/wavep.html. Google Scholar

[13]

R. B. Melrose and M. E. Taylor, The radiation pattern of a diffracted wave near the shadow boundary, Comm. Partial Differential Equations, 11 (1986), 599-672.  doi: 10.1080/03605308608820439.  Google Scholar

[14]

O. Vallée and M. Soares, Airy Functions and Applications to Physics, Imperial College Press, London; Distributed by World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004. doi: 10.1142/p345.  Google Scholar

show all references

References:
[1]

M. D. Blair, H. F. Smith and Ch. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1817-1829.  doi: 10.1016/j.anihpc.2008.12.004.  Google Scholar

[2]

G. Eskin, Parametrix and propagation of singularities for the interior mixed hyperbolic problem, J. Analyse Math., 32 (1977), 17-62.  doi: 10.1007/BF02803574.  Google Scholar

[3]

O. Ivanovici, R. Lascar, G. Lebeau and F. Planchon., Dispersion for the wave equation inside strictly convex domains II: The general case, preprint, https://arXiv.org/abs/1605.08800. Google Scholar

[4]

O. Ivanovici, G. Lebeau and F. Planchon, Strichartz estimates for the wave equation inside strictly convex 2d model domain, preprint, https://arXiv.org/abs/2008.03598. Google Scholar

[5]

O. Ivanovici, G. Lebeau and F. Planchon, Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case, Ann. of Math. (2), 180 (2014), 323-380.  doi: 10.4007/annals.2014.180.1.7.  Google Scholar

[6]

O. Ivanovici, G. Lebeau and F. Planchon, New counterexamples to Strichartz estimates for the wave equation on a 2D model convex domain, to appear in J. Ec. polytech. Math., https://arXiv.org/abs/2008.02716. Google Scholar

[7]

J. Kato and T. Ozawa, Endpoint strichartz estimates for the Klein-Gordon equation in two space dimensions and some applications, J. Math. Pures Appl., 95 (2011), 48-71.  doi: 10.1016/j.matpur.2010.10.001.  Google Scholar

[8]

S. Machihara, K. Nakanishi and T. Ozawa, Small global solutions and the nonrelativistic limit for the nonlinear dirac equation, Rev. Mat. Iberoamericana, 19 (2003), 179-194.  doi: 10.4171/RMI/342.  Google Scholar

[9]

R. B. Melrose, Equivalence of glancing hypersurfaces, Invent. Math., 37 (1976), 165-191.  doi: 10.1007/BF01390317.  Google Scholar

[10]

R. B. Melrose and J. Sjöstrand, Singularities of boundary value problems. I, Comm. Pure Appl. Math., 31 (1978), 593-617.  doi: 10.1002/cpa.3160310504.  Google Scholar

[11]

R. B. Melrose and J. Sjöstrand, Singularities of boundary value problems. II, Comm. Pure Appl. Math., 35 (1982), 129-168.  doi: 10.1002/cpa.3160350202.  Google Scholar

[12]

R. B. Melrose and M. E. Taylor, Boundary Problems for Wave Equations With Grazing and Gliding Rays, Available at https://www.unc.edu/math/Faculty/met/wavep.html. Google Scholar

[13]

R. B. Melrose and M. E. Taylor, The radiation pattern of a diffracted wave near the shadow boundary, Comm. Partial Differential Equations, 11 (1986), 599-672.  doi: 10.1080/03605308608820439.  Google Scholar

[14]

O. Vallée and M. Soares, Airy Functions and Applications to Physics, Imperial College Press, London; Distributed by World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004. doi: 10.1142/p345.  Google Scholar

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