Higher regularity for weak solutions to degenerate parabolic problems

Abstract

AbstractIn this paper, we study two related features of the regularity of the weak solutions to the following strongly degenerate parabolic equation $$\begin{aligned} u_t-\textrm{div}\left( \left( \left| Du\right| -1\right) _+^{p-1}\frac{Du}{\left| Du\right| }\right) =f\qquad \text{ in } \Omega _T =\Omega \times (0,T), \end{aligned}$$ u t - div D u - 1 + p - 1 Du D u = f in Ω T = Ω × ( 0 , T ) , where $$\Omega $$ Ω is a bounded domain in $$\mathbb {R}^{n}$$ R n for $$n\ge 2$$ n ≥ 2 , $$p\ge \text {and}\, T>0$$ p ≥ and T > 0 . We prove the higher differentiability of a nonlinear function of the spatial gradient of the weak solutions, assuming only that $$f\in L^{2}_{\textrm{loc}}\left( \Omega _T\right) $$ f ∈ L loc 2 Ω T . This allows us to establish the higher integrability of the spatial gradient under the same minimal requirement on the datum f.