Global stability of stationary waves for damped wave equations

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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

Abstract
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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 and 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, Vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 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 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), 1389-1410. doi: 10.1016/S0252-9602(11)60326-3.
[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), 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), 545-590. 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), 141-172.
[6]	S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.
[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), 581-603. 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), 293-308. 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), 296-320. 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), 795-823. doi: 10.1002/(SICI)1097-0312(199608)49:8<795::AID-CPA2>3.0.CO;2-3.
[11]	Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term, Adv. Math. Sci. Appl., 18 (2008), 329-343.
[12]	Y. Ueda and S. Kawashima, Large time behavior of solutions to a semilinear hyperbolic system with relaxation, J. Hyperbolic Differ. Equ., 4 (2007), 147-179.
[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), 49-64. 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), 735-762. doi: 10.1007/s00205-010-0369-8.
[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), 1169-1199. doi: 10.1016/j.jde.2010.10.003.

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References:

[1]	R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 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 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), 1389-1410. doi: 10.1016/S0252-9602(11)60326-3.
[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), 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), 545-590. 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), 141-172.
[6]	S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.
[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), 581-603. 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), 293-308. 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), 296-320. 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), 795-823. doi: 10.1002/(SICI)1097-0312(199608)49:8<795::AID-CPA2>3.0.CO;2-3.
[11]	Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term, Adv. Math. Sci. Appl., 18 (2008), 329-343.
[12]	Y. Ueda and S. Kawashima, Large time behavior of solutions to a semilinear hyperbolic system with relaxation, J. Hyperbolic Differ. Equ., 4 (2007), 147-179.
[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), 49-64. 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), 735-762. doi: 10.1007/s00205-010-0369-8.
[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), 1169-1199. doi: 10.1016/j.jde.2010.10.003.

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