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The Super-Diffusive Singular Perturbation Problem

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Titel: The Super-Diffusive Singular Perturbation Problem
Autoren: Edgardo Alvarez, Carlos Lizama
Quelle: Mathematics, Vol 8, Iss 3, p 403 (2020)
Verlagsinformationen: MDPI AG
Publikationsjahr: 2020
Bestand: Directory of Open Access Journals: DOAJ Articles
Schlagwörter: singular perturbation, fractional partial differential equations, analytic semigroup, super-diffusive processes, Mathematics, QA1-939
Beschreibung: 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 ) .
Publikationsart: article in journal/newspaper
Sprache: 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
Verfügbarkeit: https://doi.org/10.3390/math8030403
https://doaj.org/article/f9a548d2fde64e06af4dbda162bfa089
Dokumentencode: edsbas.92FD6F20
Datenbank: BASE
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  Data: The Super-Diffusive Singular Perturbation Problem
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  Data: &lt;searchLink fieldCode=&quot;AR&quot; term=&quot;%22Edgardo+Alvarez%22&quot;&gt;Edgardo Alvarez&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;AR&quot; term=&quot;%22Carlos+Lizama%22&quot;&gt;Carlos Lizama&lt;/searchLink&gt;
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  Data: Mathematics, Vol 8, Iss 3, p 403 (2020)
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  Data: MDPI AG
– Name: DatePubCY
  Label: Publication Year
  Group: Date
  Data: 2020
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  Data: Directory of Open Access Journals: DOAJ Articles
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  Data: &lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22singular+perturbation%22&quot;&gt;singular perturbation&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22fractional+partial+differential+equations%22&quot;&gt;fractional partial differential equations&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22analytic+semigroup%22&quot;&gt;analytic semigroup&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22super-diffusive+processes%22&quot;&gt;super-diffusive processes&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22Mathematics%22&quot;&gt;Mathematics&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22QA1-939%22&quot;&gt;QA1-939&lt;/searchLink&gt;
– Name: Abstract
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  Data: In this paper we study a class of singularly perturbed defined abstract Cauchy problems. We investigate the singular perturbation problem &lt;semantics&gt; ( P ϵ ) ϵ α D t α u ϵ ( t ) + u ϵ ′ ( t ) = A u ϵ ( t ) , t ∈ [ 0 , T ] &lt;/semantics&gt; , &lt;semantics&gt; 1 &lt; α &lt; 2 , ϵ &gt; 0 , &lt;/semantics&gt; for the parabolic equation &lt;semantics&gt; ( P ) u 0 ′ ( t ) = A u 0 ( t ) , t ∈ [ 0 , T ] , &lt;/semantics&gt; 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 &lt;semantics&gt; ( P ϵ ) &lt;/semantics&gt; has a unique solution &lt;semantics&gt; u ϵ ( t ) &lt;/semantics&gt; for each small &lt;semantics&gt; ϵ &gt; 0 . &lt;/semantics&gt; Moreover &lt;semantics&gt; u ϵ ( t ) &lt;/semantics&gt; converges to &lt;semantics&gt; u 0 ( t ) &lt;/semantics&gt; as &lt;semantics&gt; ϵ → 0 + , &lt;/semantics&gt; the unique solution of equation &lt;semantics&gt; ( P ) &lt;/semantics&gt; .
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– Name: DOI
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  Data: 10.3390/math8030403
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  Data: https://doi.org/10.3390/math8030403&lt;br /&gt;https://doaj.org/article/f9a548d2fde64e06af4dbda162bfa089
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        Value: 10.3390/math8030403
    Languages:
      – Text: English
    Subjects:
      – SubjectFull: singular perturbation
        Type: general
      – SubjectFull: fractional partial differential equations
        Type: general
      – SubjectFull: analytic semigroup
        Type: general
      – SubjectFull: super-diffusive processes
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      – SubjectFull: Mathematics
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      – SubjectFull: QA1-939
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      – TitleFull: The Super-Diffusive Singular Perturbation Problem
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            NameFull: Carlos Lizama
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              Y: 2020
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            – TitleFull: Mathematics, Vol 8, Iss 3, p 403 (2020
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