Asymptotic Behavior of Solution to Nonlinear Damped P-system with Boundary Effect

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@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 × 2 damped p-system with nonlinear source,
\begin{equation*}
\begin{cases}
&v_t-u_x=0, \\
&u_t+p(v)_x=-α u-β |u|^{q-1}u,q\;\geq\;2 \\
&(v, u)|_t=0(v_0, u_0)(x) → (v_+, u_+)\;\;as x → +∞,\\
&u_{x=0}=0, u_+\neq0,
\end{cases}
\quad (x,t)∈\mathbb{R}_+×\mathbb{R}_+,
\end{equation*}
when β › 0, or β ‹ 0 but |β| ‹ \frac{α}{|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=-α\bar{u}, \\
&\bar{v}|_t=0\bar{v}0(x) → v_+\;\;as x →+∞,\\
&\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 β ‹ 0 and |β| ›
\frac{α}{|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}
}

TY - JOUR
T1 - Asymptotic Behavior of Solution to Nonlinear Damped P-system with Boundary Effect
AU - C. Lin, C. Lin & M. Mei
JO - International Journal of Numerical Analysis Modeling Series B
VL - 1
SP - 70
EP - 92
PY - 2010
DA - 2010/01
SN - 1
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnamb/326.html
KW - p-system of hyperbolic conservation laws
KW - nonlinear damping
KW - IBVP
KW - porous equations
KW - diffusion waves
KW - asymptotic behavior
KW - convergence rates
KW - blow-up
AB - For the initial-boundary value problem to the 2 × 2 damped p-system with nonlinear source,
\begin{equation*}
\begin{cases}
&v_t-u_x=0, \\
&u_t+p(v)_x=-α u-β |u|^{q-1}u,q\;\geq\;2 \\
&(v, u)|_t=0(v_0, u_0)(x) → (v_+, u_+)\;\;as x → +∞,\\
&u_{x=0}=0, u_+\neq0,
\end{cases}
\quad (x,t)∈\mathbb{R}_+×\mathbb{R}_+,
\end{equation*}
when β › 0, or β ‹ 0 but |β| ‹ \frac{α}{|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=-α\bar{u}, \\
&\bar{v}|_t=0\bar{v}0(x) → v_+\;\;as x →+∞,\\
&\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 β ‹ 0 and |β| ›
\frac{α}{|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.

C. Lin, C. Lin & M. Mei. (1970). Asymptotic Behavior of Solution to Nonlinear Damped P-system with Boundary Effect .

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*International Journal of Numerical Analysis Modeling Series B*.*1*(1). 70-92. doi: