An example of nonlinear wave equation whose solutions decay faster than exponentially (Q1107740)

The author studies the mixed problem: \[ u_{tt}-u_{xx}+(\partial /\partial t)f(x,u)+g(x,u) = 0 text{\quad in\quad} I\times R+, \tag{1} \] where \(I\) is a bounded interval in \(R\), with the conditions \[ u(x,0) = u_ 0,\quad u_ t(x,0)=u_ 1\quad and\quad u |_{\partial_ I}=0. \tag{2} \] Under certain assumptions for the growth of (\(\partial /\partial t)f(x,u)\) and g(x,u) the author proves the existence of a generalized solution \(u\in L^{\infty}(R\) \(+;\overset\circ H_ 1)\cap W^{1,\infty}(R\) \(+,L\) 2) of the problem (1), (2) satisfying the inequality E(u(t))\(\leq C_ 0\exp \{-\exp \lambda t\}\) for any \(t>0\). Here \(C_ 0\) is a positive constant depending on initial functions \(u_ 0\), \(u_ 1\) and E(u(t)) denotes the energy operator generated by the equation (1) and \(\lambda\) is a parameter depending of the growth conditions.