Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation

Abstract

Abstract 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 ⁡ [ ( | D ⁢ u | - 1 ) + p - 1 ⁢ D ⁢ u | D ⁢ u | ] = f   in  ⁢ Ω T = Ω × ( 0 , T ) , u_{t}-\operatorname{div}\Bigl{[}(\lvert Du\rvert-1)_{+}^{p-1}\frac{Du}{\lvert Du% \rvert}\Bigr{]}=f\quad\text{in }\Omega_{T}=\Omega\times(0,T), where Ω is a bounded domain in ℝ n {\mathbb{R}^{n}} for n ≥ 2 {n\geq 2} , p ≥ 2 {p\geq 2} and ( ⋅ ) + {(\,\cdot\,)_{+}} stands for the positive part. Here, we weaken the assumption on the right-hand side, by assuming that f ∈ L loc p ′ ⁢ ( 0 , T ; B p ′ , ∞ , loc α ⁢ ( Ω ) ) , f\in L_{\mathrm{loc}}^{p^{\prime}}(0,T;B_{p^{\prime},\infty,\mathrm{loc}}^{% \alpha}(\Omega)), with α ∈ ( 0 , 1 ) {\alpha\in(0,1)} and p ′ = p / ( p - 1 ) {p^{\prime}=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, 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].