Scattering theory for the wave equation of a Hartree type in three space dimensions

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May 2014, 34(5): 2261-2281. doi: 10.3934/dcds.2014.34.2261

Scattering theory for the wave equation of a Hartree type in three space dimensions

Kimitoshi Tsutaya ^1,

1.	Graduate School of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan

Received December 2012 Revised August 2013 Published October 2013

Abstract
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The paper concerns a scattering problem of the wave equation of a Hartree type with small initial data with fast decay. The equation is \[ \partial_t^2 u - \Delta u = V_1(x)u+ (V_2\ast |u|^{p-1})u , \qquad t\in {\bf R}, \; x \in {\bf R}^3, \] where $p\ge 3, \; V_1(x)=O(|x|^{-\gamma_1})$ with $\gamma_1>0$ as $|x|\to\infty, \; V_2(x) = \pm |x|^{-\gamma_2}$ with $\gamma_2>0$. We prove the existence of scattering operators under almost optimal conditions on the potentials and initial data in terms of decay, using pointwise estimates. Our result generalizes the one by [14, 15] for the case $p=3$.

Keywords: scattering problem., Hartree type, Wave equation.

Mathematics Subject Classification: Primary: 35L05, 35L70; Secondary: 35P2.

Citation: Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261

References:

[1]	F. Asakura, Existence of a global solution to a semi-linear wave equation with slowly decreasing initial data in three space dimensions, Comm. Partial Differential Equations, 11 (1986), 1459-1487. doi: 10.1080/03605308608820470.
[2]	K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcialaj Ekvacioj, 43 (2000), 559-588.
[3]	F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. doi: 10.1007/BF01647974.
[4]	P. Karageorgis and K. Tsutaya, On the asymptotic behavior of nonlinear waves in the presence of a short-range potential, Manuscripta Math., 119 (2006), 323-345. doi: 10.1007/s00229-005-0620-z.
[5]	P. Karageorgis and K. Tsutaya, On the Asymptotic Behavior of Solutions of the Wave Equation of Hartree Type, in preparation.
[6]	P. Karageorgis and K. Tsutaya, Existence and Blow Up for A Hartree-Type Wave Equation, in preparation.
[7]	H. Kubo, On Point-Wise Decay Estimates for the Wave Equation and Their Applications, Dispersive nonlinear problems in mathematical physics, 123-148, Quad. Mat., 15, Dept. Math., Seconda Univ. Napoli, Caserta, 2004.
[8]	G. Perla Menzala and W. A. Strauss, On a wave equation with a cubic convolution, J. Diff. Eq., 43 (1982), 93-105. doi: 10.1016/0022-0396(82)90076-6.
[9]	K. Mochizuki and T. Motai, On Small Data Scattering for Some Nonlinear Wave Equations, Patterns and waves, 543-560, Stud. Math. Appl., 18, North-Holland, Amsterdam, 1986. doi: 10.1016/S0168-2024(08)70145-0.
[10]	K. Mochizuki, On small data scattering with cubic convolution nonlinearity, J. Math. Soc. Japan, 41 (1989), 143-160. doi: 10.2969/jmsj/04110143.
[11]	H. Pecher, Scattering for semilinear wave equations with small data in three space dimensions, Math. Z., 198 (1988), 277-289. doi: 10.1007/BF01163296.
[12]	W. A. Strauss, Nonlinear invariant wave equations, Lecture Notes in Phys., 73, Springer, Berlin-New York, (1978), 197-249.
[13]	W. A. Strauss and K. Tsutaya, Existence and blow up of small amplitude nonlinear waves with a negative potential, Discrete Continuous Dynam. Systems, 3 (1997), 175-188. doi: 10.3934/dcds.1997.3.175.
[14]	K. Tsutaya, Global existence and blow up for a wave equation with a potential and a cubic convolution, Nonlinear Analysis and Applications: to V. Lakshmikantham on his 80th Birthday. Vol. 1, 2, 913-937, Kluwer Acad. Publ., Dordrecht, 2003.
[15]	K. Tsutaya, Scattering theory for a wave equation of Hartree type, Differential & Difference Equations And Applications, 1061-1065, Hindawi Publ. Corp., New York, 2006.
[16]	K. Tsutaya, Weighted estimates for a convolution appearing in the wave equation of Hartree type, to appear in J. Math. Anal. Appl..

show all references

References:

[1]	F. Asakura, Existence of a global solution to a semi-linear wave equation with slowly decreasing initial data in three space dimensions, Comm. Partial Differential Equations, 11 (1986), 1459-1487. doi: 10.1080/03605308608820470.
[2]	K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcialaj Ekvacioj, 43 (2000), 559-588.
[3]	F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. doi: 10.1007/BF01647974.
[4]	P. Karageorgis and K. Tsutaya, On the asymptotic behavior of nonlinear waves in the presence of a short-range potential, Manuscripta Math., 119 (2006), 323-345. doi: 10.1007/s00229-005-0620-z.
[5]	P. Karageorgis and K. Tsutaya, On the Asymptotic Behavior of Solutions of the Wave Equation of Hartree Type, in preparation.
[6]	P. Karageorgis and K. Tsutaya, Existence and Blow Up for A Hartree-Type Wave Equation, in preparation.
[7]	H. Kubo, On Point-Wise Decay Estimates for the Wave Equation and Their Applications, Dispersive nonlinear problems in mathematical physics, 123-148, Quad. Mat., 15, Dept. Math., Seconda Univ. Napoli, Caserta, 2004.
[8]	G. Perla Menzala and W. A. Strauss, On a wave equation with a cubic convolution, J. Diff. Eq., 43 (1982), 93-105. doi: 10.1016/0022-0396(82)90076-6.
[9]	K. Mochizuki and T. Motai, On Small Data Scattering for Some Nonlinear Wave Equations, Patterns and waves, 543-560, Stud. Math. Appl., 18, North-Holland, Amsterdam, 1986. doi: 10.1016/S0168-2024(08)70145-0.
[10]	K. Mochizuki, On small data scattering with cubic convolution nonlinearity, J. Math. Soc. Japan, 41 (1989), 143-160. doi: 10.2969/jmsj/04110143.
[11]	H. Pecher, Scattering for semilinear wave equations with small data in three space dimensions, Math. Z., 198 (1988), 277-289. doi: 10.1007/BF01163296.
[12]	W. A. Strauss, Nonlinear invariant wave equations, Lecture Notes in Phys., 73, Springer, Berlin-New York, (1978), 197-249.
[13]	W. A. Strauss and K. Tsutaya, Existence and blow up of small amplitude nonlinear waves with a negative potential, Discrete Continuous Dynam. Systems, 3 (1997), 175-188. doi: 10.3934/dcds.1997.3.175.
[14]	K. Tsutaya, Global existence and blow up for a wave equation with a potential and a cubic convolution, Nonlinear Analysis and Applications: to V. Lakshmikantham on his 80th Birthday. Vol. 1, 2, 913-937, Kluwer Acad. Publ., Dordrecht, 2003.
[15]	K. Tsutaya, Scattering theory for a wave equation of Hartree type, Differential & Difference Equations And Applications, 1061-1065, Hindawi Publ. Corp., New York, 2006.
[16]	K. Tsutaya, Weighted estimates for a convolution appearing in the wave equation of Hartree type, to appear in J. Math. Anal. Appl..

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