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November  2014, 13(6): 2211-2228. doi: 10.3934/cpaa.2014.13.2211

## Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping

 1 School of Mathematics and Computational Science, Xiangtan University, Hunan 411105 2 School of Mathematical Science and Computing Technology, Central South University, Changsha 410075, China

Received  April 2013 Revised  December 2013 Published  July 2014

This paper is concerned with large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. For the nonlinear damping case, i.e. $\beta \neq 0,$ results for the linear damping case are extended to the case of nonlinear damping. Compared with the results obtained by Marcati and Pan, better decay estimates are obtained in this paper.
Citation: Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211
##### References:
 [1] S. Geng and Z. Wang, Convergence rates to nonlinear diffusion waves for solutions to the system of compressible adiabatic flow through porous media,, \emph{Comm. Partial Differential Equations}, 36 (2011), 850.  doi: 10.1080/03605302.2010.520052.  Google Scholar [2] L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping,, \emph{Comm. Math. Phys.}, 143 (1992), 599.  doi: 10.1007/BF02099268.  Google Scholar [3] L. Hsiao and T.-P. Liu, Nonlinear diffusion phenomena of nonlinear hyperbolic system,, \emph{Chin. Ann. Math. Ser. B}, 14 (1993), 465.   Google Scholar [4] L. Hsiao and T. Luo, Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media,, \emph{J. Differential Equations}, 125 (1996), 329.  doi: 10.1006/jdeq.1996.0034.  Google Scholar [5] L. Hsiao and D. Serre, Large-time behavior of solutions for the system of compressible adiabatic flow through porous media,, \emph{Chin. Ann. Math. Ser. B}, 16 (1995), 431.   Google Scholar [6] L. Hsiao and D. Serre, Global existence of solutions for the system of compressible adiabatic flow through porous media,, \emph{SIAM J. Math. Anal.}, 27 (1996), 70.  doi: 10.1137/S0036141094267078.  Google Scholar [7] M. Jiang and C. Zhu, Convergence rates to nonlinear diffusion waves for $p$-system with nonlinear damping on quadrant,, \emph{Discrete Contin. Dyn. Syst. Ser. A}, 23 (2009), 887.  doi: 10.3934/dcds.2009.23.887.  Google Scholar [8] H. Ma and M. Mei, Best asymptotic profile for linear damped p-system with boundary effect,, \emph{J. Differential Equations}, 249 (2010), 446.  doi: 10.1016/j.jde.2010.04.008.  Google Scholar [9] P. Marcati and M. Mei, B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping,, \emph{J. Math. Fluid Mech.}, 7 (2005).  doi: 10.1007/s00021-005-0155-9.  Google Scholar [10] P. Marcati and K. Nishihara, The $L^p-L^q$ estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media,, \emph{J. Differential Equations}, 191 (2003), 445.  doi: 10.1016/S0022-0396(03)00026-3.  Google Scholar [11] P. Marcati and R. Pan, On the diffusive profiles for the system of compressible adiabatice flow through porous media,, \emph{SIAM J. Math. Anal.}, 33 (2001), 790.  doi: 10.1137/S0036141099364401.  Google Scholar [12] M. Mei, Nonlinear diffusion waves for hyperbolic $p$-system with nonlinear damping,, \emph{J. Differential Equations}, 247 (2009), 1275.  doi: 10.1016/j.jde.2009.04.004.  Google Scholar [13] M. Mei, Best asymptotic profile for hyperbolic p-system with damping,, \emph{SIAM J. Math. Anal.}, 42 (2010), 1.  doi: 10.1137/090756594.  Google Scholar [14] K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping,, \emph{J. Differential Equations}, 131 (1996), 171.  doi: 10.1006/jdeq.1996.0159.  Google Scholar [15] K. Nishihara, Asymptotic toward the diffusion wave for a one-dimensional compressible flow through porous media,, \emph{Proceedings of the Royal Society of Edinburgh}, 133A (2003), 177.  doi: 10.1017/S0308210500002341.  Google Scholar [16] K. Nishihara and M. Nishikawa, Asymptotic behavior of solutions to the system of compressible adiabatic flow through porous media,, \emph{SIAM J. Math. Anal.}, 33 (2001), 216.  doi: 10.1137/S003614109936467X.  Google Scholar [17] K. Nishihara, W. Wang and T. Yang, $L_p$ -convergence rate to nonlinear diffusion waves for p-system with damping,, \emph{J. Differential Equations}, 161 (1999), 191.  doi: 10.1006/jdeq.1999.3703.  Google Scholar [18] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws,, \emph{Funkcial. Ekvac.}, 41 (1998), 107.   Google Scholar [19] R. Pan, Darcy's law as long-time limit of adiabatic porous media flow,, \emph{J. Differential Equations}, 220 (2006), 121.  doi: 10.1016/j.jde.2004.10.013.  Google Scholar [20] H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of p-system with damping,, \emph{J. Differential Equations}, 174 (2001), 200.  doi: 10.1006/jdeq.2000.3936.  Google Scholar [21] C. Zhu, Convergence rates to nonlinear diffusion waves for weak solutions to $p$-system with damping,, \emph{Sci. Chin. Ser. A}, 46 (2003), 562.  doi: 10.1360/03ys9057.  Google Scholar [22] C. Zhu and M. Jiang, $L^p$-decay rates to nonlinear diffusion waves for $p$-system with nonlinear damping,, \emph{Sciences in China, 49 (2006), 721.  doi: 10.1007/s11425-006-0721-5.  Google Scholar

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##### References:
 [1] S. Geng and Z. Wang, Convergence rates to nonlinear diffusion waves for solutions to the system of compressible adiabatic flow through porous media,, \emph{Comm. Partial Differential Equations}, 36 (2011), 850.  doi: 10.1080/03605302.2010.520052.  Google Scholar [2] L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping,, \emph{Comm. Math. Phys.}, 143 (1992), 599.  doi: 10.1007/BF02099268.  Google Scholar [3] L. Hsiao and T.-P. Liu, Nonlinear diffusion phenomena of nonlinear hyperbolic system,, \emph{Chin. Ann. Math. Ser. B}, 14 (1993), 465.   Google Scholar [4] L. Hsiao and T. Luo, Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media,, \emph{J. Differential Equations}, 125 (1996), 329.  doi: 10.1006/jdeq.1996.0034.  Google Scholar [5] L. Hsiao and D. Serre, Large-time behavior of solutions for the system of compressible adiabatic flow through porous media,, \emph{Chin. Ann. Math. Ser. B}, 16 (1995), 431.   Google Scholar [6] L. Hsiao and D. Serre, Global existence of solutions for the system of compressible adiabatic flow through porous media,, \emph{SIAM J. Math. Anal.}, 27 (1996), 70.  doi: 10.1137/S0036141094267078.  Google Scholar [7] M. Jiang and C. Zhu, Convergence rates to nonlinear diffusion waves for $p$-system with nonlinear damping on quadrant,, \emph{Discrete Contin. Dyn. Syst. Ser. A}, 23 (2009), 887.  doi: 10.3934/dcds.2009.23.887.  Google Scholar [8] H. Ma and M. Mei, Best asymptotic profile for linear damped p-system with boundary effect,, \emph{J. Differential Equations}, 249 (2010), 446.  doi: 10.1016/j.jde.2010.04.008.  Google Scholar [9] P. Marcati and M. Mei, B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping,, \emph{J. Math. Fluid Mech.}, 7 (2005).  doi: 10.1007/s00021-005-0155-9.  Google Scholar [10] P. Marcati and K. Nishihara, The $L^p-L^q$ estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media,, \emph{J. Differential Equations}, 191 (2003), 445.  doi: 10.1016/S0022-0396(03)00026-3.  Google Scholar [11] P. Marcati and R. Pan, On the diffusive profiles for the system of compressible adiabatice flow through porous media,, \emph{SIAM J. Math. Anal.}, 33 (2001), 790.  doi: 10.1137/S0036141099364401.  Google Scholar [12] M. Mei, Nonlinear diffusion waves for hyperbolic $p$-system with nonlinear damping,, \emph{J. Differential Equations}, 247 (2009), 1275.  doi: 10.1016/j.jde.2009.04.004.  Google Scholar [13] M. Mei, Best asymptotic profile for hyperbolic p-system with damping,, \emph{SIAM J. Math. Anal.}, 42 (2010), 1.  doi: 10.1137/090756594.  Google Scholar [14] K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping,, \emph{J. Differential Equations}, 131 (1996), 171.  doi: 10.1006/jdeq.1996.0159.  Google Scholar [15] K. Nishihara, Asymptotic toward the diffusion wave for a one-dimensional compressible flow through porous media,, \emph{Proceedings of the Royal Society of Edinburgh}, 133A (2003), 177.  doi: 10.1017/S0308210500002341.  Google Scholar [16] K. Nishihara and M. Nishikawa, Asymptotic behavior of solutions to the system of compressible adiabatic flow through porous media,, \emph{SIAM J. Math. Anal.}, 33 (2001), 216.  doi: 10.1137/S003614109936467X.  Google Scholar [17] K. Nishihara, W. Wang and T. Yang, $L_p$ -convergence rate to nonlinear diffusion waves for p-system with damping,, \emph{J. Differential Equations}, 161 (1999), 191.  doi: 10.1006/jdeq.1999.3703.  Google Scholar [18] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws,, \emph{Funkcial. Ekvac.}, 41 (1998), 107.   Google Scholar [19] R. Pan, Darcy's law as long-time limit of adiabatic porous media flow,, \emph{J. Differential Equations}, 220 (2006), 121.  doi: 10.1016/j.jde.2004.10.013.  Google Scholar [20] H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of p-system with damping,, \emph{J. Differential Equations}, 174 (2001), 200.  doi: 10.1006/jdeq.2000.3936.  Google Scholar [21] C. Zhu, Convergence rates to nonlinear diffusion waves for weak solutions to $p$-system with damping,, \emph{Sci. Chin. Ser. A}, 46 (2003), 562.  doi: 10.1360/03ys9057.  Google Scholar [22] C. Zhu and M. Jiang, $L^p$-decay rates to nonlinear diffusion waves for $p$-system with nonlinear damping,, \emph{Sciences in China, 49 (2006), 721.  doi: 10.1007/s11425-006-0721-5.  Google Scholar
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