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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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[11] 
Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term, Adv. Math. Sci. Appl., 18 (2008), 329343. 
[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. 
[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. 
[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. 
[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. 
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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[11] 
Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term, Adv. Math. Sci. Appl., 18 (2008), 329343. 
[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. 
[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. 
[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. 
[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. 
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