Singular perturbation for an abstract non-densely defined Cauchy problem
type: Item
instance of: scholarly article
title: Singular perturbation for an abstract non-densely defined Cauchy problem
zbMATH Open document ID: 1387.34089
author: Pierre Magal
author: Ousmane Seydi
author: Arnaud Ducrot
published in: Journal of Evolution Equations
publication date: 2018-01-05T00:00:00Z
review text: This work deals with a class of singularly perturbed non-densely defined abstract Cauchy problems of the form: \[ \varepsilon\frac{du_{\varepsilon}(t)}{dt}=Au_{\varepsilon}(t)+F(u_{\varepsilon}(t),Lv_{\varepsilon}(t)), \] \[ \frac{dv_{\varepsilon}(t)}{dt}=Bv_{\varepsilon}(t)+G(u_{\varepsilon}(t),v_{\varepsilon}(t)) \] for \(t\geq 0\) and the initial data \(u_{\varepsilon}(0)=x_{\varepsilon}\in\overline{D(A)}\), \(v_{\varepsilon}(0)=y_{\varepsilon}\in\overline{D(B)}\). Here, \(\varepsilon>0\) is a small parameter and the initial data satisfy \[ \lim_{\varepsilon\to 0^+}x_{\varepsilon}=x_0\in\overline{D(A)}, \lim_{\varepsilon\to 0^+}y_{\varepsilon}=y_0\in\overline{D(B)}. \] \(A:D(A)\subset X\to X, B:D(B)\subset Y\to Y\) are two given Hille-Yosida linear operators acting respectively on the Banach spaces \(X,Y\). \(L\in{\mathcal L}(Y,Z)\) is a bounded linear operator defined on the Banach space \(Y\) taking values in a Banach space \(Z\). \(F(\cdot,\cdot)\) and \(G(\cdot,\cdot)\) are Lipschitz continuous in bounded sets functions. The paper extends the classical Tikhonov theorem for ordinary differential equations to the case of abstract Cauchy problems. It is proved that the solutions rapidly evolve and stay in some neighbourhood of the slow manifold. One obtains that the solutions of the given Cauchy problem converge on each compact time interval, as the singular parameter goes to zero, towards the solutions of the so-called reduced problem. These results are applied to an example of age-structured model as well as to a class of functional differential equations.
reviewed by: Vasile Dragan
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34K26
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 47D62
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 6822632
zbMATH Keywords: singular perturbation
zbMATH Keywords: Tikhonov's theorem
zbMATH Keywords: age-structured model
zbMATH Keywords: functional differential equations
MaRDI profile type: Publication
full work available at URL: s00028-016-0374-y
OpenAlex ID: W2553837460
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DOI: 10.1007/S00028-016-0374-Y
description: scientific article; zbMATH DE number 6822632