Singular perturbation of a nonlinear time-dependent problem in hilbert space

Abstract

The convergence of the solution of the nonlinear singular perturbation problem Ud'J + aU´ { + e2AnUe £ + g(AUe e) + h(t, A)]f(A)Ue e = 0, C/e e(0) = x0, f/´ €'(0) = zi for ϵ > 0 to the solution of the degenerate problem (ϵ - 0) is studied. Here a [math001] 0 is a constant and A is a (generally unbounded) self- adjoint linear operator in Hilbert space. The problem is nonlocal, nonlinear, and time dependent because of g(‖AU‖)f(A)U and h(t,A). The existence of solutions is proven simultaneously with asymptotic behaviour.