CONVERGENCE ESTIMATES FOR ABSTRACT SECOND ORDER DIFFERENTIAL EQUATIONS WITH TWO SMALL PARAMETERS AND LIPSCHITZIAN NONLINEARITIES

Abstract

In a real Hilbert space H we consider the following singularly perturbed Cauchy problem { εu′′ εδ (t) + δ u′ εδ (t) + Auεδ (t) + B(uεδ (t)) = f (t), t ∈ (0, T ), uεδ (0) = u0, u′ εδ (0) = u1, where u0, u1 ∈ H, f : [0, T ] 7 → H, ε, δ are two small parameters, A is a linear self-adjoint operator and B is a nonlinear A1/2 Lipschitzian operator. We study the behavior of solutions uεδ in two different cases: ε → 0 and δ ≥ δ0 > 0; ε → 0 and δ → 0, relative to solution to the corresponding unperturbed problem. We obtain some a priori estimates of solutions to the perturbed problem, which are uniform with respect to parameters, and a relationship between solutions to both problems. We establish that the solution to the unperturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of t = 0