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Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential. (English) Zbl 1030.35024
Consider the equations \[ i\partial_{t}u + \Delta u - {a\over |x|^{2}}u = 0, \qquad u(0,x) = f(x), \tag{1} \]
\[ \partial_{t}^{2}u + \Delta u - {a\over |x|^{2}}u = 0, \qquad u(0,x) = f(x),\;\partial_{t}u(0,x) = g(x),\tag{2} \] where \(\Delta\) denotes the \(n\)-dimensional Laplacian and \(a\) is a real number. Let \(n\geq 2\), \(\lambda(n) = {n-2\over 2}\) and for \(a\geq -\lambda(n)^{2}\) set \(\nu_{d}= \sqrt{(\lambda(n) +d)^{2} + a}\).
In the case that \(u\) solves (1) the authors prove the following estimate for \(f\in L^{2}(\mathbb R^{n})\) \[ \|u\|_{L^{p}_{t}(L^{q}_{x})} \leq C\|f\|_{L^{2}} \] where \(p\geq 2\) and \(q\) such that \({2\over p} + {n\over q} = {n\over 2}\) with \((n,p) \neq (2,2)\). The proof is based on a weighted \(L^{p}\)-estimate for the operator \(P_{a} = -\Delta + {a\over |x|^{2}}\).
If \(u\) solves the wave equation (2) it is proven that for Cauchy data \((f,g) \in {\dot H}^{1/2} \times {\dot H}^{-1/2}\) \[ \|(-\Delta)^{\sigma/2}u\|_{L_{t}^{p}(L_{x}^{q})} \leq C(\|f\|_{{\dot H}^{1/2}} + \|g\|_{{\dot H}^{-1/2}}) \] holds, where \(p\geq 2\) and \(q\) such that \({2\over p} + {n-1\over q} = {n-1 \over 2}\) and \(a+\lambda^{2}>0\), \(\sigma = {1\over p} +{n\over q} - {n-1\over 2}\). This result is based on a generalized Morawetz estimate again for \(P_{a}\).
The last section contains estimates of \((-\Delta)^{s/2}u\) in suitable norms. This is applied to show that the nonlinear initial value problem for \(\partial_{t}^{2}u + P_{a}u = \pm |u|^{\kappa}\), for \(\kappa \geq {n+3 \over n-1}\) with appropriate Cauchy data has a unique global solution.

MSC:
35G10 Initial value problems for linear higher-order PDEs
35J10 Schrödinger operator, Schrödinger equation
35L05 Wave equation
35L15 Initial value problems for second-order hyperbolic equations
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References:
[1] Ben-Artzi, M.; Klainerman, S., Decay and regularity for the Schrödinger equation, J. anal. math., 58, 25-37, (1992), (Festschrift on the occasion of the 70th birthday of Shmuel Agmon) · Zbl 0802.35057
[2] Case, K.M., Singular potentials, Phys. rev., 80, 2, 797-806, (1950) · Zbl 0039.22403
[3] Cheeger, J.; Taylor, M., On the diffraction of waves by conical singularities—I, Comm. pure appl. math., 35, 3, 275-331, (1982) · Zbl 0526.58049
[4] Christ, M.; Kiselev, A., Maximal functions associated to filtrations, J. funct. anal., 179, 2, 409-425, (2001) · Zbl 0974.47025
[5] V. Georgiev, N. Visciglia, Decay estimates for the wave equation with potential, preprint, 2002. · Zbl 1035.35059
[6] J. Ginibre, personal communication.
[7] Jensen, A.; Kato, T., Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke math. J., 46, 3, 583-611, (1979) · Zbl 0448.35080
[8] John, F., Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta math., 28, 1-3, 235-268, (1979) · Zbl 0406.35042
[9] Journé, J.-L.; Soffer, A.; Sogge, C.D., Decay estimates for Schrödinger operators, Comm. pure appl. math., 44, 5, 573-604, (1991) · Zbl 0743.35008
[10] H. Kalf, U.-W. Schmincke, J. Walter, R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, in: Spectral Theory and Differential Equations (Proceedings Symposium Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Mathematics, Vol. 448, Springer, Berlin, 1975, pp. 182-226.
[11] Kapitanski, L., Weak and yet weaker solutions of semilinear wave equations, Comm. partial differential equations, 19, 9-10, 1629-1676, (1994) · Zbl 0831.35109
[12] Keel, M.; Tao, T., Endpoint Strichartz estimates, Amer. J. math., 120, 5, 955-980, (1998) · Zbl 0922.35028
[13] Lindblad, H.; Sogge, C.D., On existence and scattering with minimal regularity for semilinear wave equations, J. funct. anal., 130, 2, 357-426, (1995) · Zbl 0846.35085
[14] Moncrief, V., Odd-parity stability of a reissner – nordström black hole, Phys. rev. D, 9, 3, 2707, (1974)
[15] Morawetz, C.S., Time decay for the nonlinear klein – gordon equations, Proc. roy. soc. ser. A, 306, 291-296, (1968) · Zbl 0157.41502
[16] O’Neil, R., Convolution operators and \(L(p,q)\) spaces, Duke math. J., 30, 129-142, (1963) · Zbl 0178.47701
[17] F. Planchon, J. Stalker, A.S. Tahvildar-Zadeh, Dispersive estimate for the wave equation with the inverse-square potential, Discrete Continuous Dyn. Systems (2001), submitted. · Zbl 1047.35081
[18] Planchon, F.; Stalker, J.; Tahvildar-Zadeh, A.S., Lp estimates for the wave equation with the inverse-square potential, Discrete continuous dyn. systems, 9, 2, 427-442, (2003) · Zbl 1031.35092
[19] M. Reed, B. Simon, Scattering theory, Methods of Modern Mathematical Physics. III, Academic Press (Harcourt Brace Jovanovich Publishers), New York, 1979. · Zbl 0405.47007
[20] Regge, T.; Wheeler, J.A., Stability of a Schwarzschild singularity, Phys. rev., 108, 2, 1063-1069, (1957) · Zbl 0079.41902
[21] I. Rodnianski, W. Schlag, Time decay for solutions of Schrödinger equations with rough and time dependent potentials, preprint, 2001.
[22] Simon, B., Best constants in some operator smoothness estimates, J. funct. anal., 107, 1, 66-71, (1992) · Zbl 0815.47003
[23] Strauss, W.A., Nonlinear scattering theory at low energy, J. funct. anal., 41, 1, 110-133, (1981) · Zbl 0466.47006
[24] Strauss, W.A.; Tsutaya, K., Existence and blow up of small amplitude nonlinear waves with a negative potential, Discrete continuous dyn. systems, 3, 2, 175-188, (1997) · Zbl 0948.35084
[25] Luis Vazquez, J.; Zuazua, E., The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. funct. anal., 173, 1, 103-153, (2000) · Zbl 0953.35053
[26] Yajima, K., The Wk,p-continuity of wave operators for Schrödinger operators, J. math. soc. Japan, 47, 3, 551-581, (1995) · Zbl 0837.35039
[27] Zerilli, F.J., Gravitational field of a particle falling in a Schwarzschild geometry analyzed in tensor harmonics, Phys. rev. D, 2, 3, 2141-2160, (1970) · Zbl 1227.83025
[28] Zerilli, F.J., Perturbation analysis for gravitational and electromagnetic radiation in a reissner – nordström geometry, Phys. rev. D, 9, 3, 860, (1974)
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