Mh. Math. 122, 157-170 (1996) N o ~ hit 9 Springer-Verlag 1996 Printed in Austria Division Problem and Partial Differential Equations with Constant Coefficients in Colombeau's Space of New Generalized Functions By Marko Nedeljkov, Stevan Pilipovi6, Novi Sad, and Dimitris Scarpal6zos, Paris (Received 20 July 1994; in revised form 2 October 1995) Abstract. We study the problem of division in Colombeau's space of new generalized functions in the associated sense: more precisely we solve the equation of the form F" G ~pH with adequate assumptions on F. By using the Fourier transformation we construct, by simple methods, the solution in the p-associated sense of a partial differential equation with constant coefficients. 1. Introduction Colombeau's generalized functions as well as generalized func- tions of Egorov and Rosinger have a lot of advantages in studying linear and, especially, non-linear problems; see Ill, [-2], [3], [4], [8], [9], [10] and references there. The problem of division, which is the subject of the paper, is closely connected with such investigations. The division of a distribution by a polynomial or by a real analytic function has been studied in [5] and [6]. Now we consider such a problem in the more general case of new generalized functions and give explicit p-associated solutions. The basic notions in this paper are COI~OMBEAU'S spaces ~ and ~f, from [2] (see also [7]) and the t-Fourier transformation ([7]). We consider the division problem in the associated sense and, in Theorem 3.1, give sufficient conditions on F~q such that G is the solution of F'G,,~H and F'G~vH (see also Remark 2). In 1991 Mathematics Subject Classification: 35E05, 26C10, 46F05. Key words: Generalized functions, division problem, Fourier transformation, algebraic variety.