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

We consider the Cauchy problem{ut=Δu+eu,x∈RN,t∈(0,T),u(x,0)=u0,x∈RN, where u0∈C(RN)u0∈C(RN) and T>0T>0. We first study the radial steady states of the equation and the number of intersections distinguishing four different cases: N=1N=1, N=2N=2, 3⩽N⩽93⩽N⩽9 and N⩾10N⩾10, writing explicitly every steady state for N=1N=1 and N=2N=2. Then we study the large time behavior of solutions of the parabolic problem.