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

We consider the Cauchy problem ut = u +eu, x∈ RN, t ∈ (0,T ), u(x, 0) = u0, x∈ RN, where u0 ∈ C(RN) 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. © 2005 Elsevier Inc. All rights reserved