
Previous Article
Stability of a VlasovBoltzmann binary mixture at the phase transition on an interval
 KRM Home
 This Issue

Next Article
A remark on the ultraanalytic smoothing properties of the spatially homogeneous Landau equation
Global stability of stationary waves for damped wave equations
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 
References:
[1] 
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New YorkLondon, 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 multidimensional half space,[Decay estimates of planar stationary waves for damped wave equations with nonlinear convection in multidimensional half space] Acta Mathematica Scientia Ser. B Engl. Ed., 31 (2011), 13891410. doi: 10.1016/S02529602(11)603263. Google Scholar 
[3] 
L.L. Fan, H.X. Liu and H.J. Zhao, Onedimensional damped wave equation with large initial perturbation, Analysis and Applications, 11 (2013), 1350013, 40 pp 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), 545590. 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), 141172. Google Scholar 
[6] 
S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for onedimensional gas motion, Comm. Math. Phys., 101 (1985), 97127. doi: 10.1007/BF01212358. Google Scholar 
[7] 
S. Kawashima, S. Nishibata and M. Nishikawa, $L^p$ energy method for multidimensional viscous coonservation laws and applications to the stability of planar waves, J. Hyperbolic Differential Equations, 1 (2004), 581603. 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), 293308. 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), 296320. 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), 795823. doi: 10.1002/(SICI)10970312(199608)49:8<795::AIDCPA2>3.0.CO;23. 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), 329343. 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), 147179. Google Scholar 
[13] 
Y. Ueda, T. Nakamura and S. Kawashima, Stability of planar stationary waves for damped wave equations with nonlinear convection in multidimensional half space, Kinetic and Related Models, 1 (2008), 4964. 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), 735762. doi: 10.1007/s0020501003698. 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), 11691199. doi: 10.1016/j.jde.2010.10.003. Google Scholar 
show all references
References:
[1] 
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New YorkLondon, 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 multidimensional half space,[Decay estimates of planar stationary waves for damped wave equations with nonlinear convection in multidimensional half space] Acta Mathematica Scientia Ser. B Engl. Ed., 31 (2011), 13891410. doi: 10.1016/S02529602(11)603263. Google Scholar 
[3] 
L.L. Fan, H.X. Liu and H.J. Zhao, Onedimensional damped wave equation with large initial perturbation, Analysis and Applications, 11 (2013), 1350013, 40 pp 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), 545590. 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), 141172. Google Scholar 
[6] 
S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for onedimensional gas motion, Comm. Math. Phys., 101 (1985), 97127. doi: 10.1007/BF01212358. Google Scholar 
[7] 
S. Kawashima, S. Nishibata and M. Nishikawa, $L^p$ energy method for multidimensional viscous coonservation laws and applications to the stability of planar waves, J. Hyperbolic Differential Equations, 1 (2004), 581603. 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), 293308. 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), 296320. 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), 795823. doi: 10.1002/(SICI)10970312(199608)49:8<795::AIDCPA2>3.0.CO;23. 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), 329343. 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), 147179. Google Scholar 
[13] 
Y. Ueda, T. Nakamura and S. Kawashima, Stability of planar stationary waves for damped wave equations with nonlinear convection in multidimensional half space, Kinetic and Related Models, 1 (2008), 4964. 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), 735762. doi: 10.1007/s0020501003698. 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), 11691199. 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 multidimensional half space. Kinetic & Related Models, 2008, 1 (1) : 4964. 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) : 4155. doi: 10.3934/cpaa.2016.15.41 
[3] 
Wei Wang, Yan Lv. Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete & Continuous Dynamical Systems  B, 2010, 13 (1) : 175193. doi: 10.3934/dcdsb.2010.13.175 
[4] 
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) : 13511358. doi: 10.3934/cpaa.2019065 
[5] 
Ge Zu, Bin Guo. Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy. Evolution Equations & Control Theory, 2021, 10 (2) : 259270. doi: 10.3934/eect.2020065 
[6] 
Hiroshi Takeda. Global existence of solutions for higher order nonlinear damped wave equations. Conference Publications, 2011, 2011 (Special) : 13581367. doi: 10.3934/proc.2011.2011.1358 
[7] 
Linglong Du, Caixuan Ren. Pointwise wave behavior of the initialboundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems  B, 2019, 24 (7) : 32653280. doi: 10.3934/dcdsb.2018319 
[8] 
Junichi 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) : 843859. doi: 10.3934/cpaa.2015.14.843 
[9] 
John Boyd. Strongly nonlinear perturbation theory for solitary waves and bions. Evolution Equations & Control Theory, 2019, 8 (1) : 129. doi: 10.3934/eect.2019001 
[10] 
Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021019 
[11] 
Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 17891811. doi: 10.3934/dcds.2016.36.1789 
[12] 
Varga K. Kalantarov, Edriss S. Titi. Global stabilization of the NavierStokesVoight and the damped nonlinear wave equations by finite number of feedback controllers. Discrete & Continuous Dynamical Systems  B, 2018, 23 (3) : 13251345. doi: 10.3934/dcdsb.2018153 
[13] 
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, 2011, 31 (1) : 119138. doi: 10.3934/dcds.2011.31.119 
[14] 
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) : 100117. doi: 10.3934/proc.1998.1998.100 
[15] 
Soichiro Katayama. Global existence for systems of nonlinear wave and kleingordon equations with compactly supported initial data. Communications on Pure & Applied Analysis, 2018, 17 (4) : 14791497. doi: 10.3934/cpaa.2018071 
[16] 
Bilgesu A. Bilgin, Varga K. Kalantarov. Nonexistence of global solutions to nonlinear wave equations with positive initial energy. Communications on Pure & Applied Analysis, 2018, 17 (3) : 987999. doi: 10.3934/cpaa.2018048 
[17] 
Salim A. Messaoudi, Ala A. Talahmeh. Blow up of negative initialenergy solutions of a system of nonlinear wave equations with variableexponent nonlinearities. Discrete & Continuous Dynamical Systems  S, 2021 doi: 10.3934/dcdss.2021107 
[18] 
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) : 10751100. doi: 10.3934/dcdsb.2012.17.1075 
[19] 
Genni Fragnelli, Dimitri Mugnai. Stability of solutions for nonlinear wave equations with a positivenegative damping. Discrete & Continuous Dynamical Systems  S, 2011, 4 (3) : 615622. doi: 10.3934/dcdss.2011.4.615 
[20] 
Fabrício Cristófani, Ademir Pastor. Nonlinear stability of periodicwave solutions for systems of dispersive equations. Communications on Pure & Applied Analysis, 2020, 19 (10) : 50155032. doi: 10.3934/cpaa.2020225 
2020 Impact Factor: 1.432
Tools
Metrics
Other articles
by authors
[Back to Top]