Energy decay for a damped nonlinear hyperbolic equation (Q1808524)

This paper concentrates on the energy decay of the solution to the initial boundary value problem for a damped nonlinear hyperbolic equation \[ u''+ k_1\Delta^2u+ k_2\Delta^2 u'+\Delta g(\Delta u)= 0\quad\text{in }\Omega\times \mathbb{R}^+,\tag{1} \] \[ u= 0,\quad{\partial u\over\partial\nu}= 0\quad\text{on }\Gamma\times \mathbb{R}^+,\tag{2} \] \[ u(x,0)= u_0(x),\quad u'(x,0)= u_1(x)\quad\text{on }\Omega,\tag{3} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundary \(\Gamma\), \(k_1\) and \(k_2\) are two positive constants, \(g\) is a \(C^2\) class real-valued function, \(u_0\) and \(u_1\) are given functions on \(\Omega\). The global existence of weak solutions by using a variational method and the semigroup formulation have been proven, so the present note deals with the decay properties of solutions of equations (1), (2), (3) and establishes the uniform stability of solutions. The main theorem of the paper states that under certain conditions the relation \(E(t)\leq E(0)\exp(1-\omega t)\) for all \(t\in\mathbb{R}^+\) holds, where \[ E(t)={1\over 2} \int_\Omega (|u'|^2+ k_1|\Delta u|^2) dx+ \int_\Omega G(\Delta u) dx,\quad G(x)= \int^x_0 g(t) dt. \]