On the limit behavior of the trajectory attractor of a nonlinear hyperbolic equation containing a small parameter at the highest derivative (Q1033984)

The considered hyperbolic equation contains a small parameter at the second derivative with respect to time such that the limit equation, for parameter equals zero, is a parabolic one. An initial boundary value problem is associated to the hyperbolic equation. Under a special assumption regarding external forces in both equations, one proves that the trajectory attractor of the hyperbolic equation converges to the trajectory attractor of the limit parabolic equation, in the corresponding topology. The result applies even if the solution for the Cauchy problem is not unique.