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The Super-Diffusive Singular Perturbation Problem

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Bibliographic Details
Title:	The Super-Diffusive Singular Perturbation Problem
Authors:	Edgardo Alvarez, Carlos Lizama
Source:	Mathematics, Vol 8, Iss 3, p 403 (2020)
Publisher Information:	MDPI AG
Publication Year:	2020
Collection:	Directory of Open Access Journals: DOAJ Articles
Subject Terms:	singular perturbation, fractional partial differential equations, analytic semigroup, super-diffusive processes, Mathematics, QA1-939
Description:	In this paper we study a class of singularly perturbed defined abstract Cauchy problems. We investigate the singular perturbation problem ( P ϵ ) ϵ α D t α u ϵ ( t ) + u ϵ ′ ( t ) = A u ϵ ( t ) , t ∈ [ 0 , T ] , 1 < α < 2 , ϵ > 0 , for the parabolic equation ( P ) u 0 ′ ( t ) = A u 0 ( t ) , t ∈ [ 0 , T ] , in a Banach space, as the singular parameter goes to zero. Under the assumption that A is the generator of a bounded analytic semigroup and under some regularity conditions we show that problem ( P ϵ ) has a unique solution u ϵ ( t ) for each small ϵ > 0 . Moreover u ϵ ( t ) converges to u 0 ( t ) as ϵ → 0 + , the unique solution of equation ( P ) .
Document Type:	article in journal/newspaper
Language:	English
Relation:	https://www.mdpi.com/2227-7390/8/3/403; https://doaj.org/toc/2227-7390; https://doaj.org/article/f9a548d2fde64e06af4dbda162bfa089
DOI:	10.3390/math8030403
Availability:	https://doi.org/10.3390/math8030403 https://doaj.org/article/f9a548d2fde64e06af4dbda162bfa089
Accession Number:	edsbas.92FD6F20
Database:	BASE

Description
Abstract:	In this paper we study a class of singularly perturbed defined abstract Cauchy problems. We investigate the singular perturbation problem <semantics> ( P ϵ ) ϵ α D t α u ϵ ( t ) + u ϵ ′ ( t ) = A u ϵ ( t ) , t ∈ [ 0 , T ] </semantics> , <semantics> 1 < α < 2 , ϵ > 0 , </semantics> for the parabolic equation <semantics> ( P ) u 0 ′ ( t ) = A u 0 ( t ) , t ∈ [ 0 , T ] , </semantics> in a Banach space, as the singular parameter goes to zero. Under the assumption that A is the generator of a bounded analytic semigroup and under some regularity conditions we show that problem <semantics> ( P ϵ ) </semantics> has a unique solution <semantics> u ϵ ( t ) </semantics> for each small <semantics> ϵ > 0 . </semantics> Moreover <semantics> u ϵ ( t ) </semantics> converges to <semantics> u 0 ( t ) </semantics> as <semantics> ϵ → 0 + , </semantics> the unique solution of equation <semantics> ( P ) </semantics> .
DOI:	10.3390/math8030403