American Institute of Mathematical Sciences

Advanced
April  1997, 3(2): 175-188. doi: 10.3934/dcds.1997.3.175

Existence and blow up of small amplitude nonlinear waves with a negative potential

Walter A. Strauss 1, and  Kimitoshi Tsutaya 2,

1. 

Department of Mathematics and LCDS, Brown University, Providence, RI 02912, United States
2. 

Department of Mathematics, Brown University, Providence, R.I. 02912, United States

Received  November 1996 Published  January 1997

Consider a nonlinear wave equation in three space dimensions with zero mass together with a negative potential. If the potential is sufficiently short-range, then it does not alter the global existence of small-amplitude solutions. On the other hand, if the potential is sufficiently large, it will force some solutions to blow up in a finite time.
Keywords: Nonlinear wave equation, negative potential..
Mathematics Subject Classification: 35L70, 35B3.
Citation: Walter A. Strauss, Kimitoshi Tsutaya. Existence and blow up of small amplitude nonlinear waves with a negative potential. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 175-188. doi: 10.3934/dcds.1997.3.175
[1]

Yutian Lei. Wolff type potential estimates and application to nonlinear equations with negative exponents. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2067-2078. doi: 10.3934/dcds.2015.35.2067
[2]

Kazuhiro Ishige, Y. Kabeya. Hot spots for the two dimensional heat equation with a rapidly decaying negative potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 833-849. doi: 10.3934/dcdss.2011.4.833
[3]

Genni Fragnelli, Dimitri Mugnai. Stability of solutions for nonlinear wave equations with a positive--negative damping. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 615-622. doi: 10.3934/dcdss.2011.4.615
[4]

Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391
[5]

Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201
[6]

Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51
[7]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[8]

Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615
[9]

Q-Heung Choi, Tacksun Jung. A nonlinear wave equation with jumping nonlinearity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 797-802. doi: 10.3934/dcds.2000.6.797
[10]

Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099
[11]

Jorge A. Esquivel-Avila. Qualitative analysis of a nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 787-804. doi: 10.3934/dcds.2004.10.787
[12]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427
[13]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387
[14]

Weiguo Zhang, Yujiao Sun, Zhengming Li, Shengbing Pei, Xiang Li. Bounded traveling wave solutions for MKdV-Burgers equation with the negative dispersive coefficient. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2883-2903. doi: 10.3934/dcdsb.2016078
[15]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
[16]

Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525
[17]

Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic & Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831
[18]

Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215
[19]

Liu Rui. The explicit nonlinear wave solutions of the generalized $b$-equation. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1029-1047. doi: 10.3934/cpaa.2013.12.1029
[20]

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

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences