On the existence of solutions for some nondegenerate nonlinear wave equations of Kirchhoff type

Park, Jong Yeoul, and Bae, Jeong Ja. "On the existence of solutions for some nondegenerate nonlinear wave equations of Kirchhoff type." Czechoslovak Mathematical Journal 52.4 (2002): 781-795. <http://eudml.org/doc/30744>.

@article{Park2002,
abstract = {Let $\Omega $ be a bounded domain in $\{\mathbb \{R\}\}^n$ with a smooth boundary $\Gamma $. In this work we study the existence of solutions for the following boundary value problem: \[ \frac\{\partial ^2 y\}\{\partial t^2\}-M\biggl (\int \_\Omega |\nabla y|^2\mathrm \{d\}x\biggr ) \Delta y -\frac\{\partial \}\{\partial t\}\Delta y=f(y) \quad \text\{in\} Q=\Omega \times (0,\infty ),.1 y=0 \quad \text\{in\} \Sigma \_1=\Gamma \_\{\!1\} \times (0,\infty ), M\biggl (\int \_\Omega |\nabla y|^2\mathrm \{d\}x\biggr ) \frac\{\partial y\}\{\partial \nu \} +\frac\{\partial \}\{\partial t\}\Bigl (\frac\{\partial y\}\{\partial \nu \}\Bigr )=g \quad \text\{in\} \Sigma \_0=\Gamma \_\{\!0\} \times (0,\infty ), y(0)=y\_0,\quad \frac\{\partial y\}\{\partial t\}\,(0)=y\_1 \quad \text\{in\} \quad \Omega , \qquad \mathrm \{(1)\}\] where $M$ is a $C^1$-function such that $M(\lambda ) \ge \lambda _0 >0$ for every $\lambda \ge 0$ and $f(y)=|y|^\alpha y$ for $\alpha \ge 0$.},
author = {Park, Jong Yeoul, Bae, Jeong Ja},
journal = {Czechoslovak Mathematical Journal},
keywords = {existence and uniqueness; Galerkin method; nondegenerate wave equation; existence and uniqueness; Galerkin method},
language = {eng},
number = {4},
pages = {781-795},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the existence of solutions for some nondegenerate nonlinear wave equations of Kirchhoff type},
url = {http://eudml.org/doc/30744},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Park, Jong Yeoul
AU - Bae, Jeong Ja
TI - On the existence of solutions for some nondegenerate nonlinear wave equations of Kirchhoff type
JO - Czechoslovak Mathematical Journal
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 4
SP - 781
EP - 795
AB - Let $\Omega $ be a bounded domain in ${\mathbb {R}}^n$ with a smooth boundary $\Gamma $. In this work we study the existence of solutions for the following boundary value problem: \[ \frac{\partial ^2 y}{\partial t^2}-M\biggl (\int _\Omega |\nabla y|^2\mathrm {d}x\biggr ) \Delta y -\frac{\partial }{\partial t}\Delta y=f(y) \quad \text{in} Q=\Omega \times (0,\infty ),.1 y=0 \quad \text{in} \Sigma _1=\Gamma _{\!1} \times (0,\infty ), M\biggl (\int _\Omega |\nabla y|^2\mathrm {d}x\biggr ) \frac{\partial y}{\partial \nu } +\frac{\partial }{\partial t}\Bigl (\frac{\partial y}{\partial \nu }\Bigr )=g \quad \text{in} \Sigma _0=\Gamma _{\!0} \times (0,\infty ), y(0)=y_0,\quad \frac{\partial y}{\partial t}\,(0)=y_1 \quad \text{in} \quad \Omega , \qquad \mathrm {(1)}\] where $M$ is a $C^1$-function such that $M(\lambda ) \ge \lambda _0 >0$ for every $\lambda \ge 0$ and $f(y)=|y|^\alpha y$ for $\alpha \ge 0$.
LA - eng
KW - existence and uniqueness; Galerkin method; nondegenerate wave equation; existence and uniqueness; Galerkin method
UR - http://eudml.org/doc/30744
ER -