Volume 1, Issue 1
Asymptotic Behavior of Solution to Nonlinear Damped P-system with Boundary Effect

C. Lin, C. Lin & M. Mei

DOI:

Int. J. Numer. Anal. Mod. B, 1 (2010), pp. 70-92

Published online: 2010-01

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  • 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.

  • Keywords

p-system of hyperbolic conservation laws nonlinear damping IBVP porous equations diffusion waves asymptotic behavior convergence rates blow-up

  • AMS Subject Headings

35L50 35L60 35L65 76R50

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COPYRIGHT: © Global Science Press

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