Fractional Cauchy Problems on Bounded Domains: Survey of Recent Results

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

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Fujita exponents for a space–time weighted parabolic problem in bounded domains

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2021.103358 ◽

2021 ◽

Vol 62 ◽

pp. 103358

Author(s):

Xizheng Sun ◽

Bingchen Liu ◽

Fengjie Li

Keyword(s):

Parabolic Problem ◽

Space Time ◽

Bounded Domains

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Least energy sign-changing solutions for a class of Schrödinger–Poisson system on bounded domains

Journal of Mathematical Physics ◽

10.1063/5.0040741 ◽

2021 ◽

Vol 62 (3) ◽

pp. 031509

Author(s):

Sofiane Khoutir

Keyword(s):

Bounded Domains ◽

Poisson System ◽

Sign Changing Solutions ◽

Least Energy

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Fractional abstract Cauchy problem on complex interpolation scales

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0057 ◽

2020 ◽

Vol 23 (4) ◽

pp. 1125-1140

Author(s):

Andriy Lopushansky ◽

Oleh Lopushansky ◽

Anna Szpila

Keyword(s):

Cauchy Problem ◽

Time Derivative ◽

Abstract Cauchy Problem ◽

Initial Boundary ◽

Sectorial Operator ◽

Complex Interpolation ◽

Bounded Domains ◽

Initial Boundary Value Problems ◽

Fractional Time Derivative ◽

Initial Boundary Value

AbstractAn fractional abstract Cauchy problem generated by a sectorial operator is investigated. An inequality of coercivity type for its solution with respect to a complex interpolation scale generated by a sectorial operator with the same parameters is established. An application to differential parabolic initial-boundary value problems in bounded domains with a fractional time derivative is shown.

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