A threshold result for a non-local parabolic equation

In this paper, we consider the Cauchy problem: [GRAPHICS] The stationary problem for (ECP) is the famous Choquard-Pekar problem, and it has a unique positive solution (u) over bar(x) as long as p(x) is radial, continuous in R-3, p(x) greater than or equal to (a) over bar > 0, and lim(\x\-->infinity)p(x) = (p) over bar > 0. In this paper, we prove that if the initial data 0 less than or equal to u(0)(x) less than or equal to (not equal) (u) over bar(x), then the corresponding solution u(x,t) exists globally and it tends to the zero steady-state solution as t --> infinity, if u(0)(x) greater than or equal to (not equal) (u) over bar(x), then the solution u(x,t) blows up in finite time. (C) 1997 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.