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Resolvent estimates and the decay of the solution to the wave equation with potential

Vladimir Georgiev

Journées équations aux dérivées partielles (2001)

  • page 1-7
  • ISSN: 0752-0360

Abstract

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We prove a weighted L ∞ estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential.

How to cite

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Georgiev, Vladimir. "Resolvent estimates and the decay of the solution to the wave equation with potential." Journées équations aux dérivées partielles (2001): 1-7. <http://eudml.org/doc/93415>.

@article{Georgiev2001,
abstract = {We prove a weighted $L^\infty $ estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential.},
author = {Georgiev, Vladimir},
journal = {Journées équations aux dérivées partielles},
keywords = {generalized Fourier transform; finite dependence domain argument; global small data solution; supercritical semilinear wave equations},
language = {eng},
pages = {1-7},
publisher = {Université de Nantes},
title = {Resolvent estimates and the decay of the solution to the wave equation with potential},
url = {http://eudml.org/doc/93415},
year = {2001},
}

TY - JOUR
AU - Georgiev, Vladimir
TI - Resolvent estimates and the decay of the solution to the wave equation with potential
JO - Journées équations aux dérivées partielles
PY - 2001
PB - Université de Nantes
SP - 1
EP - 7
AB - We prove a weighted $L^\infty $ estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential.
LA - eng
KW - generalized Fourier transform; finite dependence domain argument; global small data solution; supercritical semilinear wave equations
UR - http://eudml.org/doc/93415
ER -

References

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  3. [GHK] Georgiev, V., Heiming, Ch., Kubo, H.Supercritical semilinear wave equation with non-negative potential, will appear in Comm. Part. Diff. Equations. Zbl0994.35091MR1876418
  4. [I60] Ikebe, T.Eigenfunction Expansions associated with the Scrödinger Operator and their Applications to Scattering Theory. Arch. Rational Mech. Anal. 1960, 5, 1-34. Zbl0145.36902MR128355
  5. [I85]Isozaki, H., Differentiability of Generalized Fourier Transforms associated with Schrödinger Operators. J. Math Kyoto Univ. 1985, 25 (4), 789-806. Zbl0612.35004MR810981
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  7. [J81] John, F.Blow-up of Quasi-linear Wave Equations in Three Space Dimensions. Comm. Pure Appl. Math. 1981, 34, 29-51. Zbl0453.35060
  8. [Ho83] Hörmander, L. The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients; Eds.; Springer-Verlag: Berlin, Heidelberg, New York, Tokyo, 1983. Zbl0521.35002MR705278
  9. [H99] Kerler, C.Perturbations of the Laplacian with Variable Coefficients in Exterior Domains and Differentiability of the Resolvent. Asymptotic Analysis 1999, 19, 209-232. Zbl0942.35052MR1696215
  10. [M75] Morawetz, C.Notes on Time Decay and Scattering for some Hyperbolic Problems. Society for Industrial and Applied Mathematics, Philadelphia, 1975. Zbl0303.35002MR492919
  11. [Sch83] Schaeffer, J.Wave Wquation with Positive Nonlinearities. Ph. D. thesis, Indiana Univ. 1983. 
  12. [ST97] Strauss, W.; Tsutaya, K.Existence and blow up of small amplitude nonlinear waves with a negative potential. Discrete and Cont. Dynam. Systems 1997, 3 (2), 175-188. Zbl0948.35084MR1432072

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