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Open AccessArticle

The Super-Diffusive Singular Perturbation Problem

by Edgardo Alvarez 1,*,† and Carlos Lizama 2,†
1
Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla, Colombia
2
Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencia, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2020, 8(3), 403; https://doi.org/10.3390/math8030403
Received: 26 January 2020 / Revised: 24 February 2020 / Accepted: 28 February 2020 / Published: 12 March 2020
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 ) . View Full-Text
Keywords: singular perturbation; fractional partial differential equations; analytic semigroup; super-diffusive processes singular perturbation; fractional partial differential equations; analytic semigroup; super-diffusive processes
MDPI and ACS Style

Alvarez, E.; Lizama, C. The Super-Diffusive Singular Perturbation Problem. Mathematics 2020, 8, 403.

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