Singular solutions for fractional parabolic boundary value problems

The standard problem for the classical heat equation posed in a bounded domain ¿ 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 non-zero boundary data must be singular, i.e., the solution u(t, x) blows up as x approaches ¿¿ 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 one-sided Weyl-type law holds in the general class, which was previously known for the restricted and spectral fractional Laplacians, but is new for the censored (or regional) fractional Laplacian. This yields bounds on the fractional heat kernel.HC has received funding from the European Research Council under the Grant Agreement No 721675, and the Spanish Government under Grant CEX2019-000904-S funded by MCIN/AEI/10.13039/501100011033 and PID2020-113596GB-I00. The research of DGC was supported by the Advanced Grant Nonlocal-CPD of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 883363). The research of DGC and JLV was partially supported by grant PGC2018-098440-B-I00 from the Ministerio de Ciencia, Innovación y Universidades (MICINN) of the Spanish Government and Grant RED2018-102650-T funded by MCIN/AEI/10.13039/501100011033. JLV is an Honorary Professor at Univ. Complutense. We would like to thank Gerd Grubb for her valuable comments on the first draft of the manuscript