On solutions of quasilinear wave equations with nonlinear damping terms

Park, Jong Yeoul, and Bae, Jeong Ja. "On solutions of quasilinear wave equations with nonlinear damping terms." Czechoslovak Mathematical Journal 50.3 (2000): 565-585. <http://eudml.org/doc/30585>.

@article{Park2000,
abstract = {In this paper we consider the existence and asymptotic behavior of solutions of the following problem: \[ u\_\{tt\}(t,x)-(\alpha +\beta \Vert \nabla u(t,x)\Vert \_2^2 +\beta \Vert \nabla v(t,x)\Vert \_2^2)\Delta u(t,x) +\delta |u\_t(t,x)|^\{p-1\}u\_t(t,x) \quad =\mu |u(t,x)|^\{q-1\}u(t,x), \quad x \in \Omega ,\quad t \ge 0, v\_\{tt\}(t,x)-(\alpha +\beta \Vert \nabla u(t,x)\Vert \_2^2+ \beta \Vert \nabla v(t,x)\Vert \_2^2) \Delta v(t,x) +\delta |v\_t(t,x)|^\{p-1\}v\_t(t,x) \quad =\mu |v(t,x)|^\{q-1\}v(t,x), \quad x \in \Omega ,\quad t \ge 0, u(0,x)=u\_0(x),\quad u\_t(0,x)=u\_1(x), \quad x \in \Omega , v(0,x)=v\_0(x),\quad v\_t(0,x)=v\_1(x), \quad x \in \Omega , u|\_\{\_\{\partial \Omega \}\}=v|\_\{\_\{\partial \Omega \}\}=0 \] where $q > 1$, $ p \ge 1$, $ \delta >0$, $ \alpha > 0$, $ \beta \ge 0 $, $\mu \in \mathbb \{R\} $ and $\Delta $ is the Laplacian in $\mathbb \{R\}^N$.},
author = {Park, Jong Yeoul, Bae, Jeong Ja},
journal = {Czechoslovak Mathematical Journal},
keywords = {quasilinear wave equation; existence and uniqueness; asymptotic behavior; Galerkin method; quasilinear wave equation; existence and uniqueness; asymptotic behavior; Galerkin method},
language = {eng},
number = {3},
pages = {565-585},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On solutions of quasilinear wave equations with nonlinear damping terms},
url = {http://eudml.org/doc/30585},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Park, Jong Yeoul
AU - Bae, Jeong Ja
TI - On solutions of quasilinear wave equations with nonlinear damping terms
JO - Czechoslovak Mathematical Journal
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 3
SP - 565
EP - 585
AB - In this paper we consider the existence and asymptotic behavior of solutions of the following problem: \[ u_{tt}(t,x)-(\alpha +\beta \Vert \nabla u(t,x)\Vert _2^2 +\beta \Vert \nabla v(t,x)\Vert _2^2)\Delta u(t,x) +\delta |u_t(t,x)|^{p-1}u_t(t,x) \quad =\mu |u(t,x)|^{q-1}u(t,x), \quad x \in \Omega ,\quad t \ge 0, v_{tt}(t,x)-(\alpha +\beta \Vert \nabla u(t,x)\Vert _2^2+ \beta \Vert \nabla v(t,x)\Vert _2^2) \Delta v(t,x) +\delta |v_t(t,x)|^{p-1}v_t(t,x) \quad =\mu |v(t,x)|^{q-1}v(t,x), \quad x \in \Omega ,\quad t \ge 0, u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x), \quad x \in \Omega , v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x), \quad x \in \Omega , u|_{_{\partial \Omega }}=v|_{_{\partial \Omega }}=0 \] where $q > 1$, $ p \ge 1$, $ \delta >0$, $ \alpha > 0$, $ \beta \ge 0 $, $\mu \in \mathbb {R} $ and $\Delta $ is the Laplacian in $\mathbb {R}^N$.
LA - eng
KW - quasilinear wave equation; existence and uniqueness; asymptotic behavior; Galerkin method; quasilinear wave equation; existence and uniqueness; asymptotic behavior; Galerkin method
UR - http://eudml.org/doc/30585
ER -