The approximate weak inertial manifolds of a class of nonlinear hyperbolic dynamical systems (Q1814840)

The author considers the infinite-dimensional dynamical system described by the nonlinear hyperbolic evolution equation \[ {d^2y\over dt^2}(t)+ aA^\alpha{dy\over dt}(t)+ Ay(t)+ g(y(t))=0,\;y(0)=y_0,\;{dy\over dt}(0)= y_1,\tag{1} \] where \(a>0\), \(0<\alpha<1\), \(A\) is a positive selfadjoint operator on a Hilbert space, and \(g\) is a suitably restricted nonlinear term. Homeomorphic mappings are used to transform (1) into equivalent first-order systems which are analyzed to show the existence and uniqueness of solutions, and the existence of a global attractor and semi-approximate weak inertial manifolds. The existence of approximate weak inertial manifolds for the case \(\alpha=0\) is also established.