American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stability of a Vlasov-Boltzmann binary mixture at the phase transition on an interval
  • KRM Home
  • This Issue
  • Next Article
    A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation
December  2013, 6(4): 729-760. doi: 10.3934/krm.2013.6.729

Global stability of stationary waves for damped wave equations

Lili Fan 1, , Hongxia Liu 2, , Huijiang Zhao 3, and  Qingyang Zou 4,

1. 

Department of Mathematics and Physics, Wuhan Polytechnic University, Wuhan 430023, China
2. 

Department of Mathematics, Jinan University, Guangzhou 510632, China
3. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072
4. 

College of Science, Wuhan University of Science and Technology, Wuhan 430081, China

Received  March 2013 Revised  June 2013 Published  November 2013

Nonlinear stability of stationary waves to damped wave equations has been studied by many authors in recent years, the main difficulty lies in how to control the possible growth of its solutions caused by the nonlinearity of the equation under consideration. An effective way to overcome such a difficulty is to use the smallness of the initial perturbation and/or the smallness of the strength of the stationary waves and based on this argument, local stability of strong increasing stationary waves for convex flux functions and weak decreasing stationary waves for general flux functions are well-established, cf. [11,13,15]. As to the nonlinear stability result with large initial perturbation, the only results available now are [3,4] which use the smallness of the stationary waves to overcome the above mentioned difficulty and consequently, although the initial perturbation can be large, it does require that it satisfies certain growth condition as the strength of the stationary waves tends to zero. Thus a natural question of interest is, for any initial perturbation lying in certain Sobolev space $H^s\left({\bf R}_+\times{\bf R}^{n-1}\right)$, how to obtain the global stability of stationary waves to the damped wave equations? The main purpose of this manuscript is devoted to this problem.
Keywords: damped wave equations, large initial perturbation., Nonlinear stability of stationary waves.
Mathematics Subject Classification: Primary: 35L70; Secondary: 35B35, 35B40, 35L0.
Citation: Lili Fan, Hongxia Liu, Huijiang Zhao, Qingyang Zou. Global stability of stationary waves for damped wave equations. Kinetic & Related Models, 2013, 6 (4) : 729-760. doi: 10.3934/krm.2013.6.729
References:
[1]

R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (1975).   Google Scholar

[2]

L.-L. Fan, H.-X. Liu and H. Yin, Decay estimates of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space,[Decay estimates of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space], Acta Mathematica Scientia Ser. B Engl. Ed., 31 (2011), 1389.  doi: 10.1016/S0252-9602(11)60326-3.  Google Scholar

[3]

L.-L. Fan, H.-X. Liu and H.-J. Zhao, One-dimensional damped wave equation with large initial perturbation,, Analysis and Applications, 11 (2013).  doi: 10.1142/S0219530513500139.  Google Scholar

[4]

L.-L. Fan, H.-X. Liu and H.-J. Zhao, Nonlinear stability of planar boundary layer solutions for damped wave equation,, J. Hyperbolic Differ. Equ., 8 (2011), 545.  doi: 10.1142/S0219891611002494.  Google Scholar

[5]

L.-L. Fan, H. Yin and H.-J. Zhao, Decay rates toward stationary waves of solutions for damped wave equations,, J. Partial Differential Equations, 21 (2008), 141.   Google Scholar

[6]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion,, Comm. Math. Phys., 101 (1985), 97.  doi: 10.1007/BF01212358.  Google Scholar

[7]

S. Kawashima, S. Nishibata and M. Nishikawa, $L^p$ energy method for multi-dimensional viscous coonservation laws and applications to the stability of planar waves,, J. Hyperbolic Differential Equations, 1 (2004), 581.  doi: 10.1142/S0219891604000196.  Google Scholar

[8]

T.-P. Liu, A. Matsumura and K. Nishihara, Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves,, SIAM J. Math. Anal., 29 (1998), 293.  doi: 10.1137/S0036141096306005.  Google Scholar

[9]

T.-P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect,, J. Differential Equations, 133 (1997), 296.  doi: 10.1006/jdeq.1996.3217.  Google Scholar

[10]

R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws,, Comm. Pure Appl. Math., 49 (1996), 795.  doi: 10.1002/(SICI)1097-0312(199608)49:8<795::AID-CPA2>3.0.CO;2-3.  Google Scholar

[11]

Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term,, Adv. Math. Sci. Appl., 18 (2008), 329.   Google Scholar

[12]

Y. Ueda and S. Kawashima, Large time behavior of solutions to a semilinear hyperbolic system with relaxation,, J. Hyperbolic Differ. Equ., 4 (2007), 147.   Google Scholar

[13]

Y. Ueda, T. Nakamura and S. Kawashima, Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space,, Kinetic and Related Models, 1 (2008), 49.  doi: 10.3934/krm.2008.1.49.  Google Scholar

[14]

Y. Ueda, T. Nakamura and S. Kawashima, Stability of degenerate stationary waves for viscous gases,, Arch. Ration. Mech. Anal., 198 (2010), 735.  doi: 10.1007/s00205-010-0369-8.  Google Scholar

[15]

Y. Ueda, T. Nakamura and S. Kawashima, Energy method in the partial Fourier space and application to stability problems in the half space,, J. Differential Equations, 250 (2011), 1169.  doi: 10.1016/j.jde.2010.10.003.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (1975).   Google Scholar

[2]

L.-L. Fan, H.-X. Liu and H. Yin, Decay estimates of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space,[Decay estimates of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space], Acta Mathematica Scientia Ser. B Engl. Ed., 31 (2011), 1389.  doi: 10.1016/S0252-9602(11)60326-3.  Google Scholar

[3]

L.-L. Fan, H.-X. Liu and H.-J. Zhao, One-dimensional damped wave equation with large initial perturbation,, Analysis and Applications, 11 (2013).  doi: 10.1142/S0219530513500139.  Google Scholar

[4]

L.-L. Fan, H.-X. Liu and H.-J. Zhao, Nonlinear stability of planar boundary layer solutions for damped wave equation,, J. Hyperbolic Differ. Equ., 8 (2011), 545.  doi: 10.1142/S0219891611002494.  Google Scholar

[5]

L.-L. Fan, H. Yin and H.-J. Zhao, Decay rates toward stationary waves of solutions for damped wave equations,, J. Partial Differential Equations, 21 (2008), 141.   Google Scholar

[6]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion,, Comm. Math. Phys., 101 (1985), 97.  doi: 10.1007/BF01212358.  Google Scholar

[7]

S. Kawashima, S. Nishibata and M. Nishikawa, $L^p$ energy method for multi-dimensional viscous coonservation laws and applications to the stability of planar waves,, J. Hyperbolic Differential Equations, 1 (2004), 581.  doi: 10.1142/S0219891604000196.  Google Scholar

[8]

T.-P. Liu, A. Matsumura and K. Nishihara, Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves,, SIAM J. Math. Anal., 29 (1998), 293.  doi: 10.1137/S0036141096306005.  Google Scholar

[9]

T.-P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect,, J. Differential Equations, 133 (1997), 296.  doi: 10.1006/jdeq.1996.3217.  Google Scholar

[10]

R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws,, Comm. Pure Appl. Math., 49 (1996), 795.  doi: 10.1002/(SICI)1097-0312(199608)49:8<795::AID-CPA2>3.0.CO;2-3.  Google Scholar

[11]

Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term,, Adv. Math. Sci. Appl., 18 (2008), 329.   Google Scholar

[12]

Y. Ueda and S. Kawashima, Large time behavior of solutions to a semilinear hyperbolic system with relaxation,, J. Hyperbolic Differ. Equ., 4 (2007), 147.   Google Scholar

[13]

Y. Ueda, T. Nakamura and S. Kawashima, Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space,, Kinetic and Related Models, 1 (2008), 49.  doi: 10.3934/krm.2008.1.49.  Google Scholar

[14]

Y. Ueda, T. Nakamura and S. Kawashima, Stability of degenerate stationary waves for viscous gases,, Arch. Ration. Mech. Anal., 198 (2010), 735.  doi: 10.1007/s00205-010-0369-8.  Google Scholar

[15]

Y. Ueda, T. Nakamura and S. Kawashima, Energy method in the partial Fourier space and application to stability problems in the half space,, J. Differential Equations, 250 (2011), 1169.  doi: 10.1016/j.jde.2010.10.003.  Google Scholar

[1]

Yoshihiro Ueda, Tohru Nakamura, Shuichi Kawashima. Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space. Kinetic & Related Models, 2008, 1 (1) : 49-64. doi: 10.3934/krm.2008.1.49
[2]

Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure & Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41
[3]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065
[4]

Wei Wang, Yan Lv. Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 175-193. doi: 10.3934/dcdsb.2010.13.175
[5]

Hiroshi Takeda. Global existence of solutions for higher order nonlinear damped wave equations. Conference Publications, 2011, 2011 (Special) : 1358-1367. doi: 10.3934/proc.2011.2011.1358
[6]

Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319
[7]

Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
[8]

John Boyd. Strongly nonlinear perturbation theory for solitary waves and bions. Evolution Equations & Control Theory, 2019, 8 (1) : 1-29. doi: 10.3934/eect.2019001
[9]

Varga K. Kalantarov, Edriss S. Titi. Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1325-1345. doi: 10.3934/dcdsb.2018153
[10]

A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119
[11]

Björn Birnir, Kenneth Nelson. The existence of smooth attractors of damped and driven nonlinear wave equations with critical exponent , s = 5. Conference Publications, 1998, 1998 (Special) : 100-117. doi: 10.3934/proc.1998.1998.100
[12]

Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789
[13]

Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071
[14]

Bilgesu A. Bilgin, Varga K. Kalantarov. Non-existence of global solutions to nonlinear wave equations with positive initial energy. Communications on Pure & Applied Analysis, 2018, 17 (3) : 987-999. doi: 10.3934/cpaa.2018048
[15]

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
[16]

Feng Xie. Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1075-1100. doi: 10.3934/dcdsb.2012.17.1075
[17]

John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31
[18]

P. Fabrie, C. Galusinski, A. Miranville. Uniform inertial sets for damped wave equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 393-418. doi: 10.3934/dcds.2000.6.393
[19]

Alexandre Nolasco de Carvalho, Jan W. Cholewa, Tomasz Dlotko. Damped wave equations with fast growing dissipative nonlinearities. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1147-1165. doi: 10.3934/dcds.2009.24.1147
[20]

Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences