Convergence estimates for abstract second order differential equations with two small parameters and monotone nonlinearities

Perjan Andrei; Rusu Galina

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PERJAN, Andrei, RUSU, Galina. Convergence estimates for abstract second order differential equations with two small parameters and monotone nonlinearities. In: Topological Methods in Nonlinear Analysis, 2019, nr. 2(54), pp. 1093-1110. ISSN 1230-3429. DOI: https://doi.org/10.12775/TMNA.2019.089

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Topological Methods in Nonlinear Analysis

Numărul 2(54) / 2019 / ISSN 1230-3429

Convergence estimates for abstract second order differential equations with two small parameters and monotone nonlinearities

DOI:https://doi.org/10.12775/TMNA.2019.089

Pag. 1093-1110

Perjan Andrei, Rusu Galina

Moldova State University

Disponibil în IBN: 6 aprilie 2020

Rezumat

In a real Hilbert space H we consider the following perturbed Cauchy problem [formula presented], where u₀, u₁ ∈ H, f: [0, T] ↦ H and ε, δ 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_εδ to the problem (P_εδ) in two different cases: (i) when ε → 0 and δ ≥ δ₀ > 0; (ii) when ε → 0 and δ → 0. 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 neighborhood of t = 0. We show the boundary layer and boundary layer function in both cases.

Cuvinte-cheie
a priori estimate, Abstract second order, Cauchy problem, boundary layer function, singular perturbation

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