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.

@article{JEDP_2001____A4_0,
     author = {Georgiev, Vladimir},
     title = {Resolvent estimates and the decay of the solution to the wave equation with potential},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     publisher = {Universit\'e de Nantes},
     year = {2001},
     doi = {10.5802/jedp.588},
     zbl = {1021.35071},
     mrnumber = {1843405},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_2001____A4_0}
}
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://www.numdam.org/item/JEDP_2001____A4_0/

[A75] Agmon, S. Spectral Properties of Schrödinger Operators and Scattering Theory. Ann. Scuola Norm. Sup. Pisa 1975, 4 (2), 151-218. | Numdam | MR 397194 | Zbl 0315.47007

[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 1233198 | Zbl 0795.35059

[GHK] Georgiev, V., Heiming, Ch., Kubo, H. Supercritical semilinear wave equation with non-negative potential, will appear in Comm. Part. Diff. Equations. | MR 1876418 | Zbl 0994.35091

[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 128355 | Zbl 0145.36902

[I85]Isozaki, H., Differentiability of Generalized Fourier Transforms associated with Schrödinger Operators. J. Math Kyoto Univ. 1985, 25 (4), 789-806. | MR 810981 | Zbl 0612.35004

[J79] John, F. Blow-up of Solutions of Nonlinear Wave Equations in Three Space Dimensions. Manuscripta Math.1979, 28, 235 - 268. | MR 535704 | Zbl 0406.35042

[J81] John, F. Blow-up of Quasi-linear Wave Equations in Three Space Dimensions. Comm. Pure Appl. Math. 1981, 34, 29-51. | Zbl 0453.35060

[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 705278 | Zbl 0521.35002

[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 1696215 | Zbl 0942.35052

[M75] Morawetz, C. Notes on Time Decay and Scattering for some Hyperbolic Problems. Society for Industrial and Applied Mathematics, Philadelphia, 1975. | MR 492919 | Zbl 0303.35002

[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 1432072 | Zbl 0948.35084