    July  2009, 23(3): 887-918. doi: 10.3934/dcds.2009.23.887

## Convergence rates to nonlinear diffusion waves for $p$-system with nonlinear damping on quadrant

 1 Laboratory of Nonlinear Analysis, Department of Mathematics, Huazhong Normal University, Wuhan 430079, China

Received  November 2007 Revised  June 2008 Published  November 2008

In this paper, we study the asymptotic behavior and the convergence rates of solutions to the so-called $p$-system with nonlinear damping on quadrant $\mathbb{R^+}\times \mathbb{R^+}=(0,\infty)\times (0,\infty)$,

$v_t$-u_x=0, $u_t$+p(v)_x=-αu-g(u)

with the Dirichlet boundary condition $u|_{x=0}=0$ or the Neumann boundary condition $u_x|_{x=0}=0$. The initial data $(v_0,u_0)(x)$ has the constant states $(v_+,u_+)$ at $x=\infty$. In the case of null-Dirichlet boundary condition on $u$, we show that the corresponding problem admits a unique global solution $(v(x,t), u(x,t))$ and such a solution tends time-asymptotically to the corresponding nonlinear diffusion wave $(\bar{v}(x,t), \bar{u}(x,t))$ governed by the classical Darcy's law provided that the corresponding prescribed initial error function $(w_0(x), z_0(x))$ lies in $(H^3\times H^2)(\mathbb{R}^+)$ and $||v_0(x)-v_+||_{L^1}+||w_0||_3+||z_0||_2+||V_0||_5+||Z_0||_4$ is sufficiently small. Its optimal $L^\infty$ convergence rate is also obtained by using the Green function of the diffusion equation. In the case of null-Neumann boundary condition on $u$, the global existence of smooth solution with small initial data is obtained in both of the case of $v_0(0)= v_+$ and $v_0(0)\neq v_+$. The solution $(v(x,t), u(x,t))$ is proved to tend to $(\bar v(x,t), 0)$ as $t$ tends to infinity, and we also get the optimal $L^\infty$ convergence rate in the case of $v_0(0)= v_+$.

Citation: Mina Jiang, Changjiang Zhu. Convergence rates to nonlinear diffusion waves for $p$-system with nonlinear damping on quadrant. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 887-918. doi: 10.3934/dcds.2009.23.887
  José A. Carrillo, Jean Dolbeault, Ivan Gentil, Ansgar Jüngel. Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1027-1050. doi: 10.3934/dcdsb.2006.6.1027  Lizhi Ruan, Changjiang Zhu. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 331-352. doi: 10.3934/dcds.2012.32.331  Narcisa Apreutesei, Vitaly Volpert. Reaction-diffusion waves with nonlinear boundary conditions. Networks & Heterogeneous Media, 2013, 8 (1) : 23-35. doi: 10.3934/nhm.2013.8.23  Walter Allegretto, Yanping Lin, Zhiyong Zhang. Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations. Conference Publications, 2009, 2009 (Special) : 11-23. doi: 10.3934/proc.2009.2009.11  Margaret Beck. Stability of nonlinear waves: Pointwise estimates. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 191-211. doi: 10.3934/dcdss.2017010  Huafei Di, Yadong Shang, Jiali Yu. Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source. Electronic Research Archive, 2020, 28 (1) : 221-261. doi: 10.3934/era.2020015  Giuseppina Autuori, Patrizia Pucci. Kirchhoff systems with nonlinear source and boundary damping terms. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1161-1188. doi: 10.3934/cpaa.2010.9.1161  Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361  Jun-Ren Luo, Ti-Jun Xiao. Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping. Evolution Equations & Control Theory, 2020, 9 (2) : 359-373. doi: 10.3934/eect.2020009  Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583  Tae Gab Ha. On viscoelastic wave equation with nonlinear boundary damping and source term. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1543-1576. doi: 10.3934/cpaa.2010.9.1543  Takeshi Taniguchi. Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1571-1585. doi: 10.3934/cpaa.2017075  Ryo Ikehata, Shingo Kitazaki. Optimal energy decay rates for some wave equations with double damping terms. Evolution Equations & Control Theory, 2019, 8 (4) : 825-846. doi: 10.3934/eect.2019040  Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37  Ruiying Wei, Yin Li, Zheng-an Yao. Global existence and convergence rates of solutions for the compressible Euler equations with damping. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2949-2967. doi: 10.3934/dcdsb.2020047  Yeping Li, Jie Liao. Stability and $L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1281-1302. doi: 10.3934/cpaa.2019062  Linghai Zhang. Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2181-2200. doi: 10.3934/dcdss.2016091  Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285  R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339  Ruy Coimbra Charão, Jáuber Cavalcante Oliveira, Gustavo Alberto Perla Menzala. Energy decay rates of magnetoelastic waves in a bounded conductive medium. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 797-821. doi: 10.3934/dcds.2009.25.797

2020 Impact Factor: 1.392