Describing the singular behaviour of parabolic equations on cones in fractional Sobolev spaces

Abstract

In this paper, the Dirichlet problem for parabolic equations in a wedge is considered. In particular, we study the smoothness of the solutions in the fractional Sobolev scale $$H^s$$ , $$s\in \mathbb {R}$$ . The regularity in these spaces is related with the approximation order that can be achieved by numerical schemes based on uniform grid refinements. Our results provide a first attempt to generalize the well-known $$H^{3/2}$$ -Theorem of Jerison and Kenig (J Funct Anal 130:161–219, 1995) to parabolic PDEs. As a special case the heat equation on radial-symmetric cones is investigated.