2001 On solutions of nondegenerate wave equations with nonlinear damping terms
Jeong Ja Bae, Jong Yeoul Park
Differential Integral Equations 14(12): 1421-1440 (2001). DOI: 10.57262/die/1356123004

Abstract

Let $\Omega $ be a bounded domain in $\Bbb R^N$ with smooth boundary $\partial \Omega$. In this paper, we consider the existence and energy decay of solutions of the following problem: \begin{align} & u_{tt}(t,x)-(a+b (\|\nabla u(t,x)\|_2^2+\|\nabla v(t,x)\|_2^2)^\gamma) \Delta u(t,x)+\beta \Delta^2 u(t,x) \nonumber \\ & \quad+\delta|u_t(t,x)|^{\rho}u_t(t,x) =\mu|u(t,x)|^{\alpha}u(t,x), \quad x \in \Omega,\ t \in [0, T], \nonumber \\ & v_{tt}(t,x)-(a+b (\|\nabla u(t,x)\|_2^2+\|\nabla v(t,x)\|_2^2)^\gamma ) \Delta v(t,x)+\beta \Delta^2 v(t,x) \nonumber \\ & \quad+\delta|v_t(t,x)|^{\rho}v_t(t,x) =\mu|v(t,x)|^{\alpha}v(t,x), \quad x \in \Omega,\ t \in [0, T], \tag*{(1.1)} \\ & u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x), \quad x \in \Omega, \nonumber \\ & v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x), \quad x \in \Omega, \nonumber \\ & u|_{\partial \Omega}=v|_{\partial \Omega}=0, \nonumber \end{align} where $T >0$, $\alpha > 0$, $\rho \geq 0$, $\delta >0$, $\mu \in \Bbb R$, $a+b \geq 0$, $b \geq 0$, $\gamma \geq 1 $ and $$ \|\nabla u\|^2_2 =\sum_{i=1}^N \int_\Omega|\frac{\partial u}{\partial x_i}(t,x)|^2dx, \quad u_t=\frac{\partial u}{\partial t}\quad\hbox{and}\quad \Delta u=\sum_{i=1}^N \frac{\partial^2u}{\partial x_i^2}. $$

Citation

Download Citation

Jeong Ja Bae. Jong Yeoul Park. "On solutions of nondegenerate wave equations with nonlinear damping terms." Differential Integral Equations 14 (12) 1421 - 1440, 2001. https://doi.org/10.57262/die/1356123004

Information

Published: 2001
First available in Project Euclid: 21 December 2012

zbMATH: 1161.35448
MathSciNet: MR1859915
Digital Object Identifier: 10.57262/die/1356123004

Subjects:
Primary: 35L70
Secondary: 35B35 , 35L15 , 35L20 , 35L55

Rights: Copyright © 2001 Khayyam Publishing, Inc.

Vol.14 • No. 12 • 2001
Back to Top