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Hölder estimates for solutions of uniformly degenerate quasilinear parabolic equations. (English) Zbl 0562.35051
The author deals with the quasilinear parabolic equation \(u_ t- (a_{ij}(x,t,u)u_{x_ j};)_{x_ i}+b_ i(x,t,u)\partial u/\partial x_ i+c(x,t,u)\quad with\quad \nu (| u|)| \xi |^ 2\leq a_{ij}(x,t,u)\xi_ i\xi_ j\leq \Lambda \nu (| u|)| \xi |^ 2\) for any \(\xi \in {\mathbb{R}}^ n\) where \(\Lambda\) is a constant and \(\nu (0)=0\) and \(\nu (s)>0\) if \(s>0\). Applying the DeGiorgi iteration procedure the author establishes interior and global Hölder estimates for positive solutions u(x,t) of the first boundary value problem for the above equation with Hölder exponents and coefficients independent of the lower bound of u(x,t) under some assumptions.

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K65 Degenerate parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)