Geodesic
On the existence of solutions for some nondegenerate nonlinear wave equations of Kirchhoff type
Czechoslovak Mathematical Journal, Tome 52 (2002) no. 4, pp. 781-795 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $\alpha \ge 0$.
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 $\alpha \ge 0$.
Classification : 35D05, 35L15, 35L20, 35L70, 35L75, 35L80, 65M60
Keywords: existence and uniqueness; Galerkin method; nondegenerate wave equation 
@article{CMJ_2002_52_4_a10,
     author = {Park, Jong Yeoul and Bae, Jeong Ja},
     title = {On the existence of solutions for some nondegenerate nonlinear wave equations of {Kirchhoff} type},
     journal = {Czechoslovak Mathematical Journal},
     pages = {781--795},
     year = {2002},
     volume = {52},
     number = {4},
     mrnumber = {1940059},
     zbl = {1011.35096},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a10/}
}
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
SP  - 781
EP  - 795
VL  - 52
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a10/
LA  - en
ID  - CMJ_2002_52_4_a10
ER  - 
%0 Journal Article
%A Park, Jong Yeoul
%A Bae, Jeong Ja
%T On the existence of solutions for some nondegenerate nonlinear wave equations of Kirchhoff type
%J Czechoslovak Mathematical Journal
%D 2002
%P 781-795
%V 52
%N 4
%U http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a10/
%G en
%F CMJ_2002_52_4_a10
Park, Jong Yeoul; Bae, Jeong Ja. On the existence of solutions for some nondegenerate nonlinear wave equations of Kirchhoff type. Czechoslovak Mathematical Journal, Tome 52 (2002) no. 4, pp. 781-795. http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a10/

[1] M. M.  Cavalcanti, V. N.  Domingos Cavalcanti, J. S.  Prates Filho and J. A.  Soriano: Existence and exponential decay for a Kirchhoff-Carrier model with viscosity. J.  Math. Anal. Appl. 226 (1998), 40–60. | DOI | MR

[2] R. Ikehata: On the existence of global solutions for some nonlinear hyperbolic equations with Neumann conditions. TRU Math. 24 (1988), 1–17. | MR | Zbl

[3] J. L.  Lions: Quelques méthode de résolution des probléme aux limites nonlinéaire. Dunod Gauthier–Villars, Paris (1969). | MR

[4] T.  Matsuyama and R.  Ikehata: On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms. J.  Math. Anal. Appl. 204 (1996), 729–753. | DOI | MR

[5] M.  Nakao: Asymptotic stability of the bounded or almost periodic solutions of the wave equations with nonlinear damping terms. J.  Math. Anal. Appl. 58 (1977), 336–343. | DOI | MR

[6] K. Narasimha: Nonlinear vibration of an elastic string. J.  Sound Vibration 8 (1968), 134–146. | DOI

[7] K.  Nishihara and Y.  Yamada: On global solutions of some degenerate quasilinear hyperbolic equation with dissipative damping terms. Funkcial. Ekvac. 33 (1990), 151–159. | MR

[8] K.  Ono: Global existence, decay and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings. J.  Differential Equations 137 (1997), 273–301. | DOI | MR | Zbl