JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 128, 45469 (1987) On the Behaviour as t + co of the Solutions of the Cauchy Problem for Certain Nonlinear Parabolic Equations S. KAMIN School of Mathematical Sciences, Raymond and Beverly Sackler Faculry of Exact Sciences, Tel Aviv University, Tel Aviv, Israel AND M. UGHI Istitufo Matematico “U. Dini,” Universlta Degli Studi, Firenze, Italy Suhmirted by A. Friedman Received July 7, 1986 1. INTR~OUCTI~N We are interested in the large time behaviour of nonnegative solutions of the following two Cauchy problems: (PM) i u, = Au” in R”x (0, ~3) u(x, 0)= q(x) in R” (1.1) (1.2) and (PMA) ! v, =Av”-19’ in R” x (0, 03) (1.3) v(x, 0)= q(x) in IR” (1.4) where n > 1, m > 1, p > m + 2/n and q(x) is a given bounded nonnegative function. The existence and uniqueness of nonnegative bounded solutions of (PM) and (PMA) defined in a certain weak sense, is well established (see [3,4, 13)). It was shown in some recent papers [ 1, 6-111 that the large time 456 0022-247X/87 $3.00 Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.