# American Institute of Mathematical Sciences

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

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

Received  December 2012 Revised  August 2013 Published  October 2013

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 $p=3$.
Citation: Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1080/03605308608820470.  Google Scholar [2] K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution,, Funkcialaj Ekvacioj, 43 (2000), 559.   Google Scholar [3] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions,, Manuscripta Math., 28 (1979), 235.  doi: 10.1007/BF01647974.  Google Scholar [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.  doi: 10.1007/s00229-005-0620-z.  Google Scholar [5] P. Karageorgis and K. Tsutaya, On the Asymptotic Behavior of Solutions of the Wave Equation of Hartree Type,, in preparation., ().   Google Scholar [6] P. Karageorgis and K. Tsutaya, Existence and Blow Up for A Hartree-Type Wave Equation,, in preparation., ().   Google Scholar [7] H. Kubo, On Point-Wise Decay Estimates for the Wave Equation and Their Applications,, Dispersive nonlinear problems in mathematical physics, (2004), 123.   Google Scholar [8] G. Perla Menzala and W. A. Strauss, On a wave equation with a cubic convolution,, J. Diff. Eq., 43 (1982), 93.  doi: 10.1016/0022-0396(82)90076-6.  Google Scholar [9] K. Mochizuki and T. Motai, On Small Data Scattering for Some Nonlinear Wave Equations,, Patterns and waves, (1986), 543.  doi: 10.1016/S0168-2024(08)70145-0.  Google Scholar [10] K. Mochizuki, On small data scattering with cubic convolution nonlinearity,, J. Math. Soc. Japan, 41 (1989), 143.  doi: 10.2969/jmsj/04110143.  Google Scholar [11] H. Pecher, Scattering for semilinear wave equations with small data in three space dimensions,, Math. Z., 198 (1988), 277.  doi: 10.1007/BF01163296.  Google Scholar [12] W. A. Strauss, Nonlinear invariant wave equations,, Lecture Notes in Phys., 73 (1978), 197.   Google Scholar [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.  doi: 10.3934/dcds.1997.3.175.  Google Scholar [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, (2003), 913.   Google Scholar [15] K. Tsutaya, Scattering theory for a wave equation of Hartree type,, Differential & Difference Equations And Applications, (2006), 1061.   Google Scholar [16] K. Tsutaya, Weighted estimates for a convolution appearing in the wave equation of Hartree type,, to appear in J. Math. Anal. Appl.., ().   Google Scholar

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##### 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.  doi: 10.1080/03605308608820470.  Google Scholar [2] K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution,, Funkcialaj Ekvacioj, 43 (2000), 559.   Google Scholar [3] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions,, Manuscripta Math., 28 (1979), 235.  doi: 10.1007/BF01647974.  Google Scholar [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.  doi: 10.1007/s00229-005-0620-z.  Google Scholar [5] P. Karageorgis and K. Tsutaya, On the Asymptotic Behavior of Solutions of the Wave Equation of Hartree Type,, in preparation., ().   Google Scholar [6] P. Karageorgis and K. Tsutaya, Existence and Blow Up for A Hartree-Type Wave Equation,, in preparation., ().   Google Scholar [7] H. Kubo, On Point-Wise Decay Estimates for the Wave Equation and Their Applications,, Dispersive nonlinear problems in mathematical physics, (2004), 123.   Google Scholar [8] G. Perla Menzala and W. A. Strauss, On a wave equation with a cubic convolution,, J. Diff. Eq., 43 (1982), 93.  doi: 10.1016/0022-0396(82)90076-6.  Google Scholar [9] K. Mochizuki and T. Motai, On Small Data Scattering for Some Nonlinear Wave Equations,, Patterns and waves, (1986), 543.  doi: 10.1016/S0168-2024(08)70145-0.  Google Scholar [10] K. Mochizuki, On small data scattering with cubic convolution nonlinearity,, J. Math. Soc. Japan, 41 (1989), 143.  doi: 10.2969/jmsj/04110143.  Google Scholar [11] H. Pecher, Scattering for semilinear wave equations with small data in three space dimensions,, Math. Z., 198 (1988), 277.  doi: 10.1007/BF01163296.  Google Scholar [12] W. A. Strauss, Nonlinear invariant wave equations,, Lecture Notes in Phys., 73 (1978), 197.   Google Scholar [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.  doi: 10.3934/dcds.1997.3.175.  Google Scholar [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, (2003), 913.   Google Scholar [15] K. Tsutaya, Scattering theory for a wave equation of Hartree type,, Differential & Difference Equations And Applications, (2006), 1061.   Google Scholar [16] K. Tsutaya, Weighted estimates for a convolution appearing in the wave equation of Hartree type,, to appear in J. Math. Anal. Appl.., ().   Google Scholar
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