Gradient regularity for a singular parabolic equation in non-divergence form

Abstract

In this paper we consider viscosity solutions of a class of nonhomogeneous singular parabolic equations ∂tu − |Du| γ ∆ N p u = f, where −1 < γ < 0, 1 < p < ∞, and f is a given bounded function. We establish interior Hölder regularity of the gradient by studying two alternatives: The first alternative uses an iteration which is based on an approximation lemma. In the second alternative we use a small perturbation argument.

On the second-order regularity of solutions to the parabolic p-Laplace equation

“…In [24], Parviainen and Vázquez established Harnack's inequality and asymptotic behaviour by using the fact that for radial solutions equation (1.1) is equivalent to a divergence form equation but in fictitious dimension. Attouchi [2] in the degenerate case and Attouchi-Ruosteenoja [4] in the singular case established spatial C 1,α loc -regularity for an equation of type (1.1) but with a source term. The elliptic Harnack's inequality in the singular range was obtained in [17].…”

Regularity for quasi-linear parabolic equations with nonhomogeneous degeneracy or singularity

“…Imbert-Jin-Silvestre [34] showed the interior C 1,α -regularity of viscosity solutions u to (1.5) in Q 1 , which states that Du C α (Q 1/2 ) ≤ C and sup Later, for the nonhomogeneous analogue, ∂ t u − |Du| q ∆ N p u = f (x, t), the local C 1,α -regularity of solutions was completed under the assumption that f is continuous and bounded; see [2] for the degenerate case q ≥ 0 and [6] for the singular case −1 < q < 0. Additionally, several extra aspects of such equations have already been explored as well, such as existence and uniqueness of solutions [15,27], the comparison principles [32,50], Aleksandrov-Bakelman-Pucci type estimate [1], parabolic Harnack's inequality [51].…”