JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 133, 14-26 (1988) Strong Ergodic Properties of a First-Order Partial Differential Equation RYSZARD RUDNICKI institute of Mathematics, Silesian University, 40-007 Katowice, Poland Submitted by Avner Friedman Received June 16, 1986 1. INTRODUCTION The purpose of this article is to study the first-order differential equation u, + c(x) ux =f(x, u) for (t, x)E [0, co) x [0, l] (1) with the initial condition u(0, x) = u(x) for XE [0, 11. (2) Equation (1) arises in a natural way as a model of nonhomogeneous population of cells [6]. Under certain regularity conditions (I), (2) generate a semiflow (S,},,, on C[O, l] defined by the formula S,4x) = 44 xl, where u is the solution of (l), (2). It is known [3,6] that such semiflow may exhibit irregular “chaotic” behaviour. This property of the semiflow is closely related with the fact that for such semiflow it is possible to construct an invariant measure having nontrivial ergodic properties. For example, Brunovsky and Komornik 163 proved the existence of an invariant measure such that {S,} is exact. We recall that {S, > is exact on a measure space (X, Z; 11) with a probabilistic invariant measure p if the a-algebra fl I, 0 S,‘(Z) contains only sets of measure zero or one (Rochlin [S]). Exactness is a strong property, implying ergodicity and mixing. Our aim is to show the existence of an exact invariant measure p having some additional properties. Namely, ~1 is positive on every nonempty open subset and the second moment of p is finite. From these properties of p follow additional features of {S,} : chaos in the sense of Auslander and 14 0022-247X/88 $3.00 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved