CONVERGENCE ESTIMATES FOR SOME ABSTRACT SECOND ORDER DIFFERENTIAL EQUATIONS IN HILBERT SPACES

Abstract

n a real Hilbert space H we consider the following perturbed Cauchy problem ( " u′′ " (t) + u′ " (t) + Au " (t) + B(u " (t)) = f(t), t ∈ (0, T ), u " (0) = u0, u′ " (0) = u1, (P " ) where u0, u1 ∈ H, f : [0, T ] 7→ H and ", are two small parameters, A is a linear self-adjoint operator, B is a locally Lipschitz and monotone operator. We study the behavior of solutions u " to the problem (P " ) in two different cases: (i) when " → 0 and ≥ 0 > 0; (ii) when " → 0 and → 0. We establish that the solution to the unperturbed problem has a singular behavior, relative to the parameters, in the neighborhood of t = 0. We show the boundary layer and boundary layer function in both cases.