@Article{IJNAMB-1-70,
author = {C. Lin, C. Lin and M. Mei},
title = {Asymptotic Behavior of Solution to Nonlinear Damped P-system with Boundary Effect },
journal = {International Journal of Numerical Analysis Modeling Series B},
year = {2010},
volume = {1},
number = {1},
pages = {70--92},
abstract = {For the initial-boundary value problem to the 2 &#215 2 damped p-system with nonlinear source,

\begin{equation*}
  \begin{cases}
    &v_t-u_x=0,          \\
    &u_t+p(v)_x=-&#945 u-&#946 |u|^{q-1}u,q\;\geq\;2 \\
    &(v, u)|_t=0(v_0, u_0)(x) &#8594 (v_+, u_+)\;\;as x &#8594 +&#8734,\\
    &u_{x=0}=0, u_+\neq0,
  \end{cases}
  \quad (x,t)&#8712\mathbb{R}_+&#215\mathbb{R}_+,
\end{equation*}

when &#946 &#8250 0, or &#946 &#8249 0 but |&#946| &#8249 \frac{&#945}{|u_+|^{q-1}}, the solution (v, u)(x, t) is proved to globally exist and
converge to the solution of the corresponding porous media equations

\begin{equation*}
  \begin{cases}
    &\bar{v}_t-\bar{u}_x=0,          \\
    &p_(\bar{v})_x=-&#945\bar{u}, \\
    &\bar{v}|_t=0\bar{v}0(x) &#8594 v_+\;\;as x &#8594+&#8734,\\
    &\bar{v}|_x=0,
  \end{cases}
  \quad (x,t)\in\mathbb{R}_+\times\mathbb{R}_+,
\end{equation*}

with a specially selected initial data \bar{v}0(x). The optimal convergence rates 
\|\partial^k_x(v-\bar{v},u-\bar{u})(t)\|_L^2=0(1)(t^{-\frac{2k+3}{4}}, t^{-\frac{2k+5}{4}}), 
k = 0, 1, are also obtained, as the initial perturbation is in L^1(\mathbb{R}_+)\cap H^3(\mathbb{R}_+).
 If the initial perturbation is in the weighted space L^{1,\gamma}(\mathbb{R}_+)\cap H^3(\mathbb{R}_+) with the best
choice of 
\gamma=\frac{1}
{4} , some new and much better decay rates are further obtained: 
\|\partial^k_x(v-\bar{v})(t)\|_L^2=0(1+t)^{-\frac{2k+3}{4}-\frac{\gamma}{2}}, 
k = 0, 1. The proof is based on the technical weighted energy method
combining with the Green function method. However, when &#946 &#8249 0 and |&#946| &#8250 
\frac{&#945}{|u_+|^{q-1}}, then
the solution will blow up at a finite time. Finally, numerical simulations are carried out to
confirm the theoretical results by using the central-upwind scheme. In particular, the interest
phenomenon of coexistence of the global solution v(x, t) and the blow-up solution u(x, t) is observed
and numerically demonstrated.},
issn = {},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/ijnamb/326.html}
}