Stability of steady states of the Cauchy problem for the exponential reaction-diffusion equation

We consider the Cauchy problem ut = Delta u + e(u), X is an element of R-N, t is an element of (0, T), u(x, 0) = u(0), X is an element of R-N, where u(0) is an element of C(R-N) and T > 0. We first study the radial steady states of the equation and the number of intersections distinguishing four different cases: N = 1, N = 2, 3 <= N <= 9 and N >= 10, writing explicitly every steady state for N = 1 and N = 2. Then we study the large time behavior of solutions of the parabolic problem. (c) 2005 Elsevier Inc. All rights reserved.