Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation

We carry on the investigation started in [P. Ambrosio and A. Passarelli di Napoli, Regularity results for a class of widely degenerate parabolic equations, preprint (2022), version 3, https://arxiv.org/abs/2204.05966v3] about the regularity of weak solutions to the strongly degenerate parabolic equation u(t) - div[(|Du| - 1) p(- 1) (+) Du | Du|] = f in Omega T = Omega x ( 0, T), where Omega is a bounded domain in R (n) for n >= 2, p >= 2 and ( center dot)(+) stands for the positive part. Here, we weaken the assumption on the right-hand side, by assuming that f is an element of L (p1)(loc) ( 0, T; B-a (p)1 infinity, loc( Omega )), with alpha is an element of ( 0, 1) and p1= p/( p - 1). This leads us to obtain higher fractional differentiability results for a function of the spatial gradient Du of the solutions. Moreover, we establish the higher summability of Du with respect to the spatial variable. The main novelty of the above equation is that the structure function satisfies standard ellipticity and growth conditions only outside the unit ball centered at the origin. We would like to point out that the main result of this paper can be considered, on the one hand, as the parabolic counterpart of an elliptic result contained in [P. Ambrosio, Besov regularity for a class of singular or degenerate elliptic equations, J. Math. Anal. Appl. 505 (2022), no. 2, Paper No. 125636], and on the other hand as the fractional version of some results established in [P. Ambrosio and A. Passarelli di Napoli, Regularity results for a class of widely degenerate parabolic equations, preprint (2022), version 3, https://arxiv.org/abs/2204.05966v3].