Sharp Sobolev regularity for widely degenerate parabolic equations

We consider local weak solutions to the widely degenerate parabolic PDE ∂tu − div (|Du| − λ)p−1+Du|Du|= f in ΩT = Ω × (0, T), where p ≥ 2, Ω is a bounded domain in Rn for n ≥ 2, λ is a non-negative constant and (·)+ stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue-Besov parabolic space when p > 2 and that f ∈ L2loc(ΩT ) if p = 2, we prove theSobolev spatial regularity of a novel nonlinear function of the spatial gradient of the weak solutions. This result, in turn, implies the existence of the weak time derivative for the solutions of the evolutionary p-Poisson equation. The main novelty here is that f only has a Besov or Lebesgue spatial regularity, unlike the previous work [7], where f was assumed to possess a Sobolev spatial regularity of integer order. We emphasize that the results obtained here can be considered, on the one hand, as the parabolic analog of some elliptic results established in [6], and on the other hand as the extension to a strongly degenerate setting of some known results for less degenerate parabolic equations.