American Institute of Mathematical Sciences

Advanced
March  2003, 9(2): 471-482. doi: 10.3934/dcds.2003.9.471

Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions

Hideo Kubo 1, and  Kotaro Tsugawa 2,

1. 

Department of Applied Mathematics, Faculty of Engineering, Shizuoka University, Hamamatsu 432-8561, Japan
2. 

Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Received  September 2001 Revised  April 2002 Published  December 2002

In this paper, we treat the coupled system of wave equations whose nonlinearities are $|u|^{p_j}|v|^{q_j}$ and propagation speeds may be different from each other. We study the lower bounds of $p_j$ and $q_j$ to assure the global existence of a class of small amplitude solutions which includes self-similar solutions. The exponent of self-similar solutions plays crucial role to find the lower bounds. Moreover, we prove that the discrepancy of propagation speeds allow us to bring them down. Conversely, if such conditions for the global existence do not hold, then no self-similar solution exists even for small initial data.
Keywords: Semilinear wave equations, self-similar solutions, global solutions, different speeds of propagation..
Mathematics Subject Classification: 35L7.
Citation: Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471
[1]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002
[2]

Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897
[3]

Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703
[4]

Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101
[5]

F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91
[6]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[7]

Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003
[8]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801
[9]

Jochen Merker, Aleš Matas. Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations. Conference Publications, 2015, 2015 (special) : 817-825. doi: 10.3934/proc.2015.0817
[10]

Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837
[11]

Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036
[12]

Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks & Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857
[13]

Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313
[14]

K. T. Joseph, Philippe G. LeFloch. Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits. Communications on Pure & Applied Analysis, 2002, 1 (1) : 51-76. doi: 10.3934/cpaa.2002.1.51
[15]

Meiyue Jiang, Juncheng Wei. $2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 785-803. doi: 10.3934/dcds.2016.36.785
[16]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
[17]

Francis Hounkpe, Gregory Seregin. An approximation of forward self-similar solutions to the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4823-4846. doi: 10.3934/dcds.2021059
[18]

Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651
[19]

Dongbing Zha, Yi Zhou. The lifespan for quasilinear wave equations with multiple propagation speeds in four space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1167-1186. doi: 10.3934/cpaa.2014.13.1167
[20]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences