Sharp Sobolev regularity for widely degenerate parabolic equations: Sharp Sobolev regularity

We consider local weak solutions to the widely degenerate parabolic PDE ∂ t u - div ( | D u | - λ ) + p - 1 Du | D u | = f in Ω T = Ω × ( 0 , T ) , where p ≥ 2 , Ω is a bounded domain in R n 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 ∈ L loc 2 ( Ω T ) if p = 2 , we prove the Sobolev 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.