Ukrainian Mathematical Journal, Vol. 48, No. 7, 1996 TWO-SIDED ESTIMATES OF A SOLUTION OF THE NEUMANN PROBLEM AS t --> o. FOR A SECOND-ORDER QUASI-LINEAR PARABOLIC EQUATION A. F. Tedeev UDC 517.956 We establish exact upper and lower bounds as t --+ ~ for the norm [I u(., t) IlL(n) of a solution of the Neumann problem for a second-order quasilinear parabolic equation in the region D = f2 x { t > 0 }, where f2 is a region with noncompact boundary. 1. Introduction Let f2 C R ~, n > 2, be an unbounded domain with sufficiently smooth noncompact boundary. Consider the Neumann initial boundary-value problem O~ m-1 ~, = ~x (IVul Uxi)' i=1 (i) n IVulm-luxvilaa• = O. i=1 (2) u(x,O) = Uo(X ), x~ f2, (3) in the domain D = f2 x { t > 0}, where V i are cosines of the outward unit normal to Of 2, m > (n - 1 )/(n + 1 ), 2 )1/2, =--~U/OXi, and ut=Ou/Ot. I Vul= (Ux2, +--. + u.n % This paper is a continuation of [1], where the case m > 1 was studied. In the case where m = 1, problem (1)-(3) was investigated in [2, 3] (see also references in [3]). For the Cauchy problem (1), (3) for (n - 1 )/(n + 1 ) < m < 1 (if2 = Rn), local estimates in the classes of increasing initial data and other qualitative properties of solutions were obtained in [4]. The survey of results on qualitative properties of degenerate parabolic equations can be found in [5]. 2. Several Auxiliary Assertions Following [2], we introduce the following class of domains: Consider a volume function v > 0, l(u) = infmesn_ 1 (~Q A f2), where Q is an arbitrary open set f2 and the infimum is taken over all Q such that mesnQ = v. Assume that g(v), v > 0, is a nonnegative continuous function with the following property: For all v > 0, v(n-1)/~/g(v) is monotonically nondecreasing. We say that f2 ~l(g) if l(v)>g(v) forall v>0. Denote Zp/q+p-1 E~.- f lu(x)lX~, Jp- ~ IVulP~. Gp, q(Z) = (g(z)) p , f2 f2 Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk. Translated from Ukrainskii Matematicheskii Zhumal, Vol. 48, No. 7, pp. 989-998, July, 1996. Original article submitted April 10, 1995. 0041-5995/96/4807-1119 $15.00 9 1997 Plenum Publishing Corporation 1119