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

Abstract

We consider the Cauchy problem { u t = Δ u + e u , x ∈ R N , t ∈ ( 0 , T ) , u ( x , 0 ) = u 0 , x ∈ R N , where u 0 ∈ 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.