Person:
Gómez Castro, David

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First Name
David
Last Name
Gómez Castro
URI
https://hdl.handle.net/20.500.14352/75645
Affiliation
Universidad Complutense de Madrid
Faculty / Institute
Ciencias Matemáticas
Department
Análisis Matemático Matemática Aplicada
Area
Matemática Aplicada
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Author: Chan, Hardy ×

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    Singular solutions for space-time fractional equations in a bounded domain
    (2023) Chan, Hardy; Gómez Castro, David; Vázquez, Juan Luis
    This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann-Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense.
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    Singular solutions for fractional parabolic boundary value problems
    (2020) Chan, Hardy; Gómez Castro, David; Vázquez, Juan Luis
    The standard problem for the classical heat equation posed in a bounded domain Ω of Rn is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial and boundary value problem can still be solved on the condition that the nonzero boundary data must be singular, i.e., the solution u(t, x) blows up as x approaches ∂Ω in a definite way. In this paper we construct a theory of existence and uniqueness of solutions of the parabolic problem with singular data taken in a very precise sense, and also admitting initial data and a forcing term. When the boundary data are zero we recover the standard fractional heat semigroup. A general class of integro-differential operators may replace the classical fractional Laplacian operators, thus enlarging the scope of the work. As further results on the spectral theory of the fractional heat semigroup, we show that a Weyl-type law holds in the general class, which was previously known for the restricted and spectral fractionalLaplacians, but is new for the censored (or regional) fractional Laplacian. This yields bounds on the fractional heat kerne.