Description

• Title:
  Convergence estimates for abstract second order differential equations with two small parameters and monotone nonlinearities
• Creator:
  Perjan, Andrei;
  Rusu, Galina
• Description:
  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.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2019 Topological Methods in Nonlinear Analysis
• Date:
  2019.11.14
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 54, No 2B (December 2019); 1093 - 1110;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.089/23690 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2019.089/23691
• Subject:
  Singular perturbation;
  abstract second order Cauchy problem;
  boundary layer function;
  a priori estimate;
  35B25;
  35K15;
  35L15;
  34G10
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.089
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other