manuscripta math. 75, 65- 80 (1992) manuscripta mathematica Sprmger-Verlag 1992 ON THE LOCAL BEHAVIOUR OF SOLUTIONS OF A CERTAIN CLASS OF DOUBLY NONLINEAR PARABOLIC EQUATIONS Vincenzo Vespri Here we prove HSlder regularity for bounded weak solutions of nonhnear parabolic equations with measurable coefficients. The prototype of this class of equations is ut = Div(lulZlDuF-2Du) p > 1, ~ > 1 -p 1 Introduction Consider quasilinear parabolic equations in divergence form of the type: ut - Div(d(]u])a(x,u, Du)) = b(x,t,u,Du) in Dt(~'~T) (l) where ~ is a bounded set of RN,0 < T < +or and FtT= f~ x [0, T]. The functions a : R 2N+1 ~ R N, b : R 2N+2 ~ R, r : R + ---+ R + are assumed only to be measurable and to satisfy the structure conditions a(x,z,q)~ > ~0r ~ - ~0(x) (2) la(x, z,~)l <_ c,r '-~ + r162 (3) (a(x, z, 71) - a(x, z, ~2))(~, - ~2) > 0 (4) Ib(x,t,z,,7)l < c~r ' + r (5) .71[zi~t < r < ,~2lz]~2 foreach0<lz]<l whereZ2>fll>l-p (6) 0<h~(M)_<r <A~(M) < +oo for eachl <_ Izi_<M (7) Here c+, i = 1,2, 3 ; flj, 7#, A j, j = 1,2; M are given positive constants;(x, t) 6 flT,Z 6 R, 7, qi 6 R N, i = 1, 2, are arbitrary. The functions r = 0,1,2 are nonnegative,defined in fit and subject to the condition 65