On quasilinear parabolic higher order equations with Hölder continuous solutions

It is introduced a class of quasilinear parabolic higher order equations whose generalized solutions are Höolder continuous functions. A model equation of this class is the equation 1 https://www.w3.org/1998/Math/MathML"> ∂ u ∂ t + Σ | α | = m ( - 1 ) | α | D α [ | D m u | p - 2 D α u ] - Σ i = 1 n ∂ ∂ x i [ | ∂ u ∂ x | q - 2 ∂ u ∂ x i ] = f ( x , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math1o.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> if p ≥ 2, q > mp. For quasilinear parabolic equation with Caratheodory's coefficients they are obtained interior regularity of solutions, regularity of solutions near the boundary for Dirichlet or Neumann boundary conditions. It is established that the solutions belong to a class of functions B_q,s which generalizes the corresponding classes of O.A. Ladyzhenskaya and N.N.Uraltzeva (q = 2, s = 1) [3] and of E. di Benedetto (s = 1) [2], The imbeddings of classes B_q,s in spaces of Holder continuous functions are established. Corresponding results about interior regularity were published in papers [4,5], regularity near the boundary the author proved together with prof. F.Nicolosi.