Resolvent estimates and the decay of the solution to the wave equation with potential
Journées équations aux dérivées partielles (2001), article no. 4, 7 p.

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.

Vladimir Georgiev. Resolvent estimates and the decay of the solution to the wave equation with potential. Journées équations aux dérivées partielles (2001), article  no. 4, 7 p.. doi: 10.5802/jedp.588
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     doi = {10.5802/jedp.588},
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[A75] Agmon, S. Spectral Properties of Schrödinger Operators and Scattering Theory. Ann. Scuola Norm. Sup. Pisa 1975, 4 (2), 151-218. | Zbl | MR | Numdam

[BS93] Beals, M.; Strauss, W. L p Estimates for the Wave Equation with a Potential. Comm. Part. Diff. Eq. 1993, 18( 7,8), 1365-1397. | Zbl | MR

[GHK] Georgiev, V., Heiming, Ch., Kubo, H. Supercritical semilinear wave equation with non-negative potential, will appear in Comm. Part. Diff. Equations. | Zbl | MR

[I60] Ikebe, T. Eigenfunction Expansions associated with the Scrödinger Operator and their Applications to Scattering Theory. Arch. Rational Mech. Anal. 1960, 5, 1-34. | Zbl | MR

[I85]Isozaki, H., Differentiability of Generalized Fourier Transforms associated with Schrödinger Operators. J. Math Kyoto Univ. 1985, 25 (4), 789-806. | Zbl | MR

[J79] John, F. Blow-up of Solutions of Nonlinear Wave Equations in Three Space Dimensions. Manuscripta Math.1979, 28, 235 - 268. | Zbl | MR

[J81] John, F. Blow-up of Quasi-linear Wave Equations in Three Space Dimensions. Comm. Pure Appl. Math. 1981, 34, 29-51. | Zbl

[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. | Zbl | MR

[H99] Kerler, C. Perturbations of the Laplacian with Variable Coefficients in Exterior Domains and Differentiability of the Resolvent. Asymptotic Analysis 1999, 19, 209-232. | Zbl | MR

[M75] Morawetz, C. Notes on Time Decay and Scattering for some Hyperbolic Problems. Society for Industrial and Applied Mathematics, Philadelphia, 1975. | Zbl | MR

[Sch83] Schaeffer, J. Wave Wquation with Positive Nonlinearities. Ph. D. thesis, Indiana Univ. 1983.

[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. | Zbl | MR

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