Differential and Integral Equations

On solutions of nondegenerate wave equations with nonlinear damping terms

Jeong Ja Bae and Jong Yeoul Park

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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 $$ \|\nabla u\|^2_2 =\sum_{i=1}^N \int_\Omega|\frac{\partial u}{\partial x_i}(t,x)|^2dx, \quad u_t=\frac{\partial u}{\partial t}\quad\hbox{and}\quad \Delta u=\sum_{i=1}^N \frac{\partial^2u}{\partial x_i^2}. $$

Article information

Source
Differential Integral Equations, Volume 14, Number 12 (2001), 1421-1440.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356123004

Mathematical Reviews number (MathSciNet)
MR1859915

Zentralblatt MATH identifier
1161.35448

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 35B35: Stability 35L15: Initial value problems for second-order hyperbolic equations 35L20: Initial-boundary value problems for second-order hyperbolic equations 35L55: Higher-order hyperbolic systems

Citation

Park, Jong Yeoul; Bae, Jeong Ja. On solutions of nondegenerate wave equations with nonlinear damping terms. Differential Integral Equations 14 (2001), no. 12, 1421--1440. https://projecteuclid.org/euclid.die/1356123004


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