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

We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity ∂ t u = [ | D u | q + a ( x , t ) | D u | s ] Δ u + ( p - 2 ) D 2 u Du | D u | , Du | D u | , where 1 < p < ∞ , - 1 < q ≤ s < ∞ and a ( x , t ) ≥ 0 . The motivation to investigate this model stems not only from the connections to tug-of-war like stochastic games with noise, but also from the non-standard growth problems of double phase type. According to different values of q , s , such equations include nonhomogeneous degeneracy or singularity, and may involve these two features simultaneously. In particular, when q = p - 2 and q < s , it will encompass the parabolic p -Laplacian both in divergence form and in non-divergence form. We aim to explore the L ∞ to C 1 , α regularity theory for the aforementioned problem. To be precise, under some proper assumptions, we use geometrical methods to establish the local Hölder regularity of spatial gradients of viscosity solutions.