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