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

$v_t$-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 $v_0(0)= v_+$.