Regularity results for a class of widely degenerate parabolic equations

Abstract Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE u t - div ⁡ ( (| D ⁢ u | - ν) + p - 1 ⁢ D ⁢ u | D ⁢ u |) = f in ⁢ Ω T = Ω × (0, T), uₓ-div ( (Du-) +^p-1Du% Du) =f ₓ= (0, T), where Ω is a bounded domain in ℝ n R^{n} for n ≥ 2 n 2, p ≥ 2 p 2, ν is a positive constant and (⋅) + (\, \, ) + stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue–Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative u t {u_{