Singular solutions for fractional parabolic boundary value problems

Abstract The standard problem for the classical heat equation posed in a bounded domain $$\Omega $$ Ω of $${\mathbb {R}}^n$$ R n is initial and boundary value problem. If Laplace operator replaced by version fractional Laplacian, can still be solved on condition that non-zero data must singular, i.e., solution u ( t , x ) blows up as approaches $$\partial \Omega ∂ definite way. In this paper we construct theory existence uniqueness solutions parabolic with singular taken very precise sense, also admitting forcing term. When are zero recover semigroup. A general class integro-differential operators may replace Laplacian operators, thus enlarging scope work. As further results spectral semigroup, show one-sided Weyl-type law holds class, which was previously known restricted Laplacians, but new censored (or regional) Laplacian. This yields bounds kernel.