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oai:docta.ucm.es:20.500.14352/72302023-08-26T18:49:30Zcom_20.500.14352_14col_20.500.14352_15
Docta Complutense author Chan, Hardy author Gómez Castro, David author Vázquez, Juan Luis 2023-06-17T08:28:32Z 2023-06-17T08:28:32Z 2020-07-28 https://hdl.handle.net/20.500.14352/7230 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. eng Singular solutions for fractional parabolic boundary value problems journal article URL https://docta.ucm.es/bitstreams/5602a526-c248-4965-a951-94fddc052b29/download File MD5 1f1b03c6a5b95d286a30815ac7cff72a 325987 application/pdf gomezcastro_singularsolutions.pdf URL https://docta.ucm.es/bitstreams/c8d9db69-4759-4962-9999-cc7b4b7faad3/download File MD5 be924cd1bbaa6c13b18ff4e519a2659a 78332 text/plain gomezcastro_singularsolutions.pdf.txt