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
