On Harnack inequality and Holder continuity to the degenerate parabolic equations

In this paper, the Harnack inequality and Holder continuity results are established for a new class of degenerate parabolic equations partial derivative(xi) (a(ij) (t, x) partial derivative(xj) u) - partial derivative(t)u = 0 on a bounded domain D subset of Rn+1. The coefficients matrix A = parallel to a(ij) (t, x)parallel to, i, j = 1, 2, ... n is real, symmetric, positive and such that there exist positive constants c(1), c(2) with c(1)omega (t, x)vertical bar xi vertical bar(2) <= a(ij) (t, x)xi(i)xi(j) <= c(2)omega (t, x)vertical bar xi vertical bar(2) for for all xi is an element of R-n and arbitrary (t, x) is an element of D. The exclusive Muckenhoupt condition omega(p) is an element of A(1+p/r): (integral(QT) omega(p) dxdt)(1/p) (integral(QT) sigma(r) dxdt)(1/r) <= c(0) is assumed for a p > (n + 2)/2, an r > n/2 such that n/2r + (n + 2)/2p < 1; the cylinders {Q(T) = K-R(x0) x (t(0) - T, t(0)): (t(0), x(0)) is an element of D, T = C R-2/(integral(QT) omega(P) dxdt)(1/p)} sigma = 1/omega, K-r(x) = {y is an element of R-n: vertical bar y - x vertical bar < r}. The cylinders are of T -> 0 as R -> 0. (C) 2022 Elsevier Inc. All rights reserved.