# American Institute of Mathematical Sciences

March  2014, 13(2): 835-858. doi: 10.3934/cpaa.2014.13.835

## Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux

 1 Department of Mathematics, University of Iowa, Iowa City, IA 52242 2 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received  June 2013 Revised  August 2013 Published  October 2013

This paper is concerned with the initial-boundary value problem for the generalized Benjamin-Bona-Mahony-Burgers equation in the half space $R_+$ \begin{eqnarray} u_t-u_{txx}-u_{xx}+f(u)_{x}=0,\quad t>0, x\in R_+,\\ u(0,x)=u_0(x)\to u_+, \quad as \ \ x\to +\infty,\\ u(t,0)=u_b. \end{eqnarray} Here $u(t,x)$ is an unknown function of $t>0$ and $x\in R_+$, $u_+\not=u_b$ are two given constant states and the nonlinear function $f(u)$ is assumed to be a non-convex function which has one or finitely many inflection points. In this paper, we consider $u_b Citation: Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure and Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835 ##### References:  [1] H. P. Cui, The asymptotic behavior of solutions of an initial boundary value problem for the generalized Benjamin-Bona-Mahony equation, Indian J. Pure Appl. Math., 43 (2012), 323-342. doi: 10.1007/s13226-012-0020-5. [2] E. Grenier and F. Rousset, Stability of one-dimensional boundary layers by using Green's functions, Communications on Pure and Applied Mathematics, 54 (2001), 1343-1385. doi: DOI: 10.1002/cpa.10006. [3] Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space, Commun. Math. Phys., 266 (2006), 401-430. doi: 10.1007/s00220-006-0017-1. [4] S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358. [5] S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Commun. Pure Appl. Math., 47 (1994), 1547-1569. doi: 10.1002/cpa.3160471202. [6] S. Kawashima, S. Nishibata and M. Nishikawa, Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane, Discrete and Continuous Dynamical Systems, Supplement (2003), 469-476. [7] S. Kawashima, S. Nishibata and M. Nishikawa,$L^p$energy method for multi-dimensional viscous coonservation laws and applications to the stability of planar waves, J. Hyperbolic Differential Equations, 1 (2004), 581-603. doi: 10.1142/S0219891604000196. [8] T. P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect, J. Differential Equations, 133 (1997), 296-320. doi: 10.1006/jdeq.1996.3217. [9] A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Commun. Math. Phys., 165 (1994), 83-96. doi: 10.1007/BF02099739. [10] T. Nakamura, S. Nishibata and, T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line, J. Differential Equations, 241 (2007), 94-111. doi: 10.1016/j.jde.2007.06.016. [11] M. Nishikawa, Convergence rates to the travelling wave for viscous conservation laws, Funk. Ekvac., 41 (1998), 107-132. [12] Y. Ueda, Asymptotic convergence toward stationary waves to the wave equation with a convection term in the half space, Advances in Mathematical Sciences and Applications, 18 (2008), 329-343. [13] H. Yin and H. J. Zhao, Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equations in the half-space, Kinetic and Related Models, 2 (2009), 3144-3216. [14] H. Yin, H. J. Zhao and J. S. Kim, Convergence rates of solutions toward boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equations in the half-space, J. Differential Equations, 245 (2008), 3144-3216. doi: 10.1016/j.jde.2007.12.012. [15] C. J. Zhu, Asymptotic behavior of solutions for$p$-system with relaxation, J. Differential Equations, 180 (2002), 273-306. doi: 10.1006/jdeq.2001.4063. [16] P. C. Zhu, Nonlinear Waves for the Compressible Navier-Stokes Equations in the Half Space, the report for JSPS postdoctoral research at Kyushu University, Fukuoka, Japan, August 2001. show all references ##### References:  [1] H. P. Cui, The asymptotic behavior of solutions of an initial boundary value problem for the generalized Benjamin-Bona-Mahony equation, Indian J. Pure Appl. Math., 43 (2012), 323-342. doi: 10.1007/s13226-012-0020-5. [2] E. Grenier and F. Rousset, Stability of one-dimensional boundary layers by using Green's functions, Communications on Pure and Applied Mathematics, 54 (2001), 1343-1385. doi: DOI: 10.1002/cpa.10006. [3] Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space, Commun. Math. Phys., 266 (2006), 401-430. doi: 10.1007/s00220-006-0017-1. [4] S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358. [5] S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Commun. Pure Appl. Math., 47 (1994), 1547-1569. doi: 10.1002/cpa.3160471202. [6] S. Kawashima, S. Nishibata and M. Nishikawa, Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane, Discrete and Continuous Dynamical Systems, Supplement (2003), 469-476. [7] S. Kawashima, S. Nishibata and M. Nishikawa,$L^p$energy method for multi-dimensional viscous coonservation laws and applications to the stability of planar waves, J. Hyperbolic Differential Equations, 1 (2004), 581-603. doi: 10.1142/S0219891604000196. [8] T. P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect, J. Differential Equations, 133 (1997), 296-320. doi: 10.1006/jdeq.1996.3217. [9] A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Commun. Math. Phys., 165 (1994), 83-96. doi: 10.1007/BF02099739. [10] T. Nakamura, S. Nishibata and, T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line, J. Differential Equations, 241 (2007), 94-111. doi: 10.1016/j.jde.2007.06.016. [11] M. Nishikawa, Convergence rates to the travelling wave for viscous conservation laws, Funk. Ekvac., 41 (1998), 107-132. [12] Y. Ueda, Asymptotic convergence toward stationary waves to the wave equation with a convection term in the half space, Advances in Mathematical Sciences and Applications, 18 (2008), 329-343. [13] H. Yin and H. J. Zhao, Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equations in the half-space, Kinetic and Related Models, 2 (2009), 3144-3216. [14] H. Yin, H. J. Zhao and J. S. Kim, Convergence rates of solutions toward boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equations in the half-space, J. Differential Equations, 245 (2008), 3144-3216. doi: 10.1016/j.jde.2007.12.012. [15] C. J. Zhu, Asymptotic behavior of solutions for$p\$-system with relaxation, J. Differential Equations, 180 (2002), 273-306. doi: 10.1006/jdeq.2001.4063. [16] P. C. Zhu, Nonlinear Waves for the Compressible Navier-Stokes Equations in the Half Space, the report for JSPS postdoctoral research at Kyushu University, Fukuoka, Japan, August 2001.
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