Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Weighted decay estimate for the wave equation


Authors: Valéry Covachev and Vladimir Georgiev
Journal: Proc. Amer. Math. Soc. 112 (1991), 393-402
MSC: Primary 35L05; Secondary 35B45, 35Q40
MathSciNet review: 1055769
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Abstract: The work is devoted to the proof of a new $ {L^\infty } - {L^2}$ weighted estimate for the solution to the nonhomogeneous wave equation in $ \left( {3 + 1} \right)$-dimensional space-time. The weighted Sobolev spaces are associated with the generators of the Poincaré group. The estimate obtained is applied to prove the global existence of a solution to a nonlinear system of wave and Klein-Gordon equations with small initial data.


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  • [1] Alain Bachelot, Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon, Ann. Inst. H. Poincaré Phys. Théor. 48 (1988), no. 4, 387–422 (French, with English summary). MR 969173
  • [2] Vladimir Georgiev, Global solution of the system of wave and Klein-Gordon equations, Math. Z. 203 (1990), no. 4, 683–698. MR 1044072, 10.1007/BF02570764
  • [3] Vladimir Georgiev, L’existence des solutions globales pour des systèmes non linéaires avec champs massifs et sans masse, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 18, 529–532 (French, with English summary). MR 1001046
  • [4] Vladimir Georgiev, Existence des solutions globales pour le système de Maxwell-Dirac, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 7, 569–572 (French, with English summary). MR 1050133
  • [5] Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
  • [6] L. Hörmander, On Sobolev spaces associated with some Lie algebras, preprint.
  • [7] -, On global existence of solutions of non-linear hyperbolic equations in $ {{\mathbf{R}}^{1 + 3}}$, Inst. Mittag-Leffler report no. 9, 1985.
  • [8] -, Non-linear hyperbolic differential equations, Lectures 1986-1987, vol. 2, Lund, 1988.
  • [9] Sergiu Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), no. 3, 321–332. MR 784477, 10.1002/cpa.3160380305
  • [10] Sergiu Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math. 38 (1985), no. 5, 631–641. MR 803252, 10.1002/cpa.3160380512
  • [11] S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984) Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293–326. MR 837683
  • [12] Thomas C. Sideris, Decay estimates for the three-dimensional inhomogeneous Klein-Gordon equation and applications, Comm. Partial Differential Equations 14 (1989), no. 10, 1421–1455. MR 1022992, 10.1080/03605308908820660
  • [13] Wolf von Wahl, 𝐿^{𝑝}-decay rates for homogeneous wave-equations, Math. Z. 120 (1971), 93–106. MR 0280885

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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1055769-7
Keywords: Decay estimate, wave equation
Article copyright: © Copyright 1991 American Mathematical Society