824 ISSN 1064–5624, Doklady Mathematics, 2007, Vol. 76, No. 3, pp. 824–827. © Pleiades Publishing, Ltd., 2007. Original Russian Text © S.P. Degtyarev, A.F. Tedeev, 2007, published in Doklady Akademii Nauk, 2007, Vol. 417, No. 2, pp. 156–159. We study the local and global properties of solutions to an anisotropic degenerate doubly nonlinear parabolic equation. Note that much attention has recently been given to anisotropic degenerate equations (see, e.g., [5– 8]). The basic goal of this paper is to derive sharp esti- mates for the qualitative behavior of solutions in the direction of each coordinate axis. Consider the following Cauchy problem for an unknown function u(x, t): (1) (2) Here, N is the dimension of R N , u 0 (x) is a given initial function, and the remaining notation is conventional. To simplify the notation, we assume without loss of generality that p 1 ≤ p 2 ≤ … ≤ p N . Note that Eq. (1) is considered in the case of slow diffusion in all the directions x i . Accordingly, the fol- lowing assumptions are made about the parameters of problem (1), (2): (3) Additionally, we assume that the anisotropy of Eq. (1) is bounded. More specifically, it is assumed that the ∂ u β 1 – u ( ) ∂ t ------------------------ ∂ ∂ x i ------ ∂ u ∂ x i ------ p i 2 – ∂ u ∂ x i ------ ⎝ ⎠ ⎛ ⎞ i 1 = N ∑ – 0, x R N , ∈ = t 0, > u β 1 – ux 0 , ( ) u 0 β 1 – u 0 x () , x R N . ∈ = β 01 ] , p i 1 β , i + > , ( ∈ 12 … N . , , , = scatter of p i from above is not very high, which is expressed as (4) where p is the harmonic mean of p i ; i.e., (5) We also assume that u 0 (x) ≥ 0 and consider only nonnegative solutions to problem (1), (2). The weak solution to this problem in R N × [0, T] is a measurable function u(x, t) on R N × [0, T] such that, for any bounded open subset Ω of R N , it is true that for any t ∈ [0, T) and all t , ∈ (0, T). The function |u| β – 1 u has the property and the corresponding standard integral identity with a sufficiently smooth test function holds true. The initial function u 0 (x) in problem (1), (2) can increase at infinity, but its growth is bounded by the value (6) p i p 1 β N --- + ⎝ ⎠ ⎛ ⎞ , i < 12 … N , , , , = p 1 N --- 1 p i ---- i 1 = N ∑ ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ 1 – . = u x i p i 1 – x τ d d Ω ∫ 0 t ∫ i 1 = N ∑ ∞, u x i p i x τ d d Ω ∫ t t ∫ i 1 = N ∑ ∞ < < t u β 1 – u C 0 T ε – , [ ] L 1 Ω ( ) , ( ) , ∈ u 0 β r ρ k / d – u 0 β x () x d B ρ ∫ ρ r ≥ sup ∞, r 0. > < = MATHEMATICS Bounds for Solutions of the Cauchy Problem for an Anisotropic Degenerate Doubly Nonlinear Parabolic Equation with Growing Initial Data S. P. Degtyarev and A. F. Tedeev Presented by Academician V.A. Il’in June 10, 2007 Received April 3, 2007 DOI: 10.1134/S1064562407060063 Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, ul. R. Lyuksemburg 74, Donetsk, 83114 Ukraine e-mail: spdegt@yahoo.com, tedeev@iam.ac.donetsk.ua