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 partial derivative(t)u = [vertical bar Du vertical bar(q) + a(x, t) vertical bar Du vertical bar(s)](Delta u + (p - 2)< D(2)u Du/vertical bar Du vertical bar, Du/vertical bar Du vertical bar >), where 1<p<infinity, -1<q <= s<infinity 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-infinity to C-1,C-alpha regularity theory for the aforementioned problem. To be precise, under some proper assumptions, we use geometrical methods to establish the local Holder regularity of spatial gradients of viscosity solutions.