Abstract
In a real Hilbert space $H$ we consider the following perturbed Cauchy problem \begin{equation} \begin{cases} \varepsilon\,u''_{\varepsilon\delta}(t)+ \delta\,u'_{\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} \tag{${\rm P}_{\varepsilon\delta}$} \end{equation} 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 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.
Citation
Andrei Perjan. Galina Rusu. "Convergence estimates for abstract second order differential equations with two small parameters and monotone nonlinearities." Topol. Methods Nonlinear Anal. 54 (2B) 1093 - 1110, 2019. https://doi.org/10.12775/TMNA.2019.089
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