American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[2]

K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcialaj Ekvacioj, 43 (2000), 559-588.  Google Scholar

[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.  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), 323-345. 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, 123-148, Quad. Mat., 15, Dept. Math., Seconda Univ. Napoli, Caserta, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K. Mochizuki, On small data scattering with cubic convolution nonlinearity, J. Math. Soc. Japan, 41 (1989), 143-160. 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), 277-289. doi: 10.1007/BF01163296.  Google Scholar

[12]

W. A. Strauss, Nonlinear invariant wave equations, Lecture Notes in Phys., 73, Springer, Berlin-New York, (1978), 197-249.  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), 175-188. 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, 2, 913-937, Kluwer Acad. Publ., Dordrecht, 2003.  Google Scholar

[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.  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

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.  Google Scholar

[2]

K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcialaj Ekvacioj, 43 (2000), 559-588.  Google Scholar

[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.  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), 323-345. 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, 123-148, Quad. Mat., 15, Dept. Math., Seconda Univ. Napoli, Caserta, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K. Mochizuki, On small data scattering with cubic convolution nonlinearity, J. Math. Soc. Japan, 41 (1989), 143-160. 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), 277-289. doi: 10.1007/BF01163296.  Google Scholar

[12]

W. A. Strauss, Nonlinear invariant wave equations, Lecture Notes in Phys., 73, Springer, Berlin-New York, (1978), 197-249.  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), 175-188. 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, 2, 913-937, Kluwer Acad. Publ., Dordrecht, 2003.  Google Scholar

[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.  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

[1]

Tarek Saanouni. Energy scattering for the focusing fractional generalized Hartree equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021124
[2]

Changhun Yang. Scattering results for Dirac Hartree-type equations with small initial data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081
[3]

Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431
[4]

Kenji Nakanishi. Modified wave operators for the Hartree equation with data, image and convergence in the same space. Communications on Pure & Applied Analysis, 2002, 1 (2) : 237-252. doi: 10.3934/cpaa.2002.1.237
[5]

Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
[6]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557
[7]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004
[8]

Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048
[9]

Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134
[10]

Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731
[11]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. On small data scattering of Hartree equations with short-range interaction. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1809-1823. doi: 10.3934/cpaa.2016016
[12]

Anudeep Kumar Arora. Scattering of radial data in the focusing NLS and generalized Hartree equations. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6643-6668. doi: 10.3934/dcds.2019289
[13]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903
[14]

Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013
[15]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[16]

Gabriella Pinzari. Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result. Electronic Research Announcements, 2015, 22: 55-75. doi: 10.3934/era.2015.22.55
[17]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[18]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011
[19]

Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations & Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531
[20]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences