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 ${L}^{\infty }$ 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.

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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), article  no. 4, 7 p. doi : 10.5802/jedp.588. http://archive.numdam.org/articles/10.5802/jedp.588/

[A75] Agmon, S. Spectral Properties of Schrödinger Operators and Scattering Theory. Ann. Scuola Norm. Sup. Pisa 1975, 4 (2), 151-218. | Numdam | MR | Zbl

[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. | MR | Zbl

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

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

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

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

[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. | MR | Zbl

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

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

[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. | MR | Zbl

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