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

Abstract

In a real Hilbert space $H$ we consider the following perturbed Cauchy problem $$ \begin{cases} \varepsilonu''_{\varepsilon\delta}(t)+ \deltau'_{\varepsilon\delta}(t)+Au_{\varepsilon\delta}(t)+B(u_{\varepsilon\delta}(t))= f(t),\quad t\in(0,T),\\ u_{\varepsilon\delta}(0)=u_0,\quad u'_{\varepsilon\delta}(0)=u_1, \end{cases} \leqno(\rom{P}_{\varepsilon\delta}) $$% where $u_0, u_1\in H$, $f\colon [0,T]\mapsto H$ and $\varepsilon,$ $\delta$ 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 $u_{\varepsilon\delta}$ to the problem (P$_{\varepsilon\delta}$) in two different cases: \begin{enumerate} \item[(i)] when $\varepsilon\to 0$ and $\delta \geq \delta_0> 0 ;$ \item[(ii)] when $\varepsilon\to 0$ and $\delta \to 0.$ \end{enumerate} We obtain some {\it 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.