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- W1968172806 abstract "In this paper, we study the asymptotic behavior and the convergencerates of solutions to the so-called $p$-system with nonlineardamping 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 Neumannboundary 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 ofnull-Dirichlet boundary condition on $u$, we show that thecorresponding problem admits a unique global solution $(v(x,t),u(x,t))$ and such a solution tends time-asymptotically to thecorresponding nonlinear diffusion wave $(bar{v}(x,t),bar{u}(x,t))$ governed by the classical Darcy's law providedthat the corresponding prescribed initial error function$(w_0(x), z_0(x))$ lies in $(H^3timesH^2)(mathbb{R}^+)$ and$||v_0(x)-v_+||_{L^1}+||w_0||_3+||z_0||_2+||V_0||_5+||Z_0||_4$ issufficiently small. Its optimal $L^infty$ convergence rate is alsoobtained by using the Green function of the diffusion equation. Inthe case of null-Neumann boundary condition on $u$, the globalexistence of smooth solution with small initial data is obtained inboth 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 $v_0(0)= v_+$." @default.
- W1968172806 created "2016-06-24" @default.
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- W1968172806 date "2009-01-01" @default.
- W1968172806 modified "2023-10-01" @default.
- W1968172806 title "Convergence rates to nonlinear diffusion waves for $p$-system with nonlinear damping on quadrant" @default.
- W1968172806 doi "https://doi.org/10.3934/dcds.2009.23.887" @default.
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