Ukrainian Mathematical Journal, Vol. 64, No. 11, April, 2013 (Ukrainian Original Vol. 64, No. 11, November, 2012) ON THE BEHAVIOR OF SOLUTIONS OF THE CAUCHY PROBLEM FOR A DEGENERATE PARABOLIC EQUATION WITH SOURCE IN THE CASE WHERE THE INITIAL FUNCTION SLOWLY VANISHES A. V. Martynenko, 1 A. F. Tedeev, 2 and V. N. Shramenko 3 UDC 517.946 We study the Cauchy problem for a degenerate parabolic equation with source and inhomogeneous density of the form u t D div..x/u m1 jDuj 1 Du/ C u p in the case where the initial function slowly vanishes as jxj!1: We establish conditions for the exis- tence and nonexistence of a global (in time) solution. These conditions strongly depend on the behavior of the initial data as jxj!1: In the case of global solvability, we establish a sharp estimate of the solution for large times. 1. Introduction In the present paper, we study the Cauchy problem of the form u t D div..x/u m1 jDuj 1 Du/ C u p ; (1) .x;t/ 2 Q T D R N  .0; T /; T > 0; N  1; u.x; 0/ D u 0 .x/; x 2 R N : (2) In the case .x/  1; m D 1;  D 1; this problem was considered in [1]. It was shown that, for p>p  D 1 C 2=N; there exist initial data for which the global (in time) solution of the problem exists and, at the same time, for 1<p<p  ; all nonzero solutions become unbounded for a finite period of time (“blow up”). The number p  is called the critical exponent. Later, the results of this kind (called Fujita-type theorems) were extended to more general equations. In the case of homogeneous density ..x/  1/; the Fujita-type theorems for the equations of porous media . D 1; m > 1/ were obtained in [2–4]. For the equation of non-Newtonian filtration . > 1; m D 1/; these theorems were established in [5] and, for the equation with double inequality . > 1; m > 1/; in [6, 7]. In [7], for problem 1 Shevchenko Lugansk National University, Lugansk, Ukraine. 2 Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk, Ukraine. 3 “Kiev Polytechnic Institute” National Technical University, Kiev, Ukraine. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 11, pp. 1500–1515, November, 2012. Original article submitted Febru- ary 27, 2012. 1698 0041-5995/13/6411–1698 c  2013 Springer Science+Business Media New York