Asymptotics for a parabolic equation with critical exponential nonlinearity

We consider the Cauchy problem: {∂tu=Δu-u+λf(u)in(0,T)×R2,u(0,x)=u0(x)inR2,where λ> 0 , f(u):=2α0ueα0u2,for someα0>0,with initial data u∈ H1(R2). The nonlinear term f has a critical growth at infinity in the energy space H1(R2) in view of the Trudinger-Moser embedding. Our goal is to investigate from the initial data u∈ H1(R2) whether the solution blows up in finite time or the solution is global in time. For 0<12α0, we prove that for initial data with energies below or equal to the ground state level, the dichotomy between finite time blow-up and global existence can be determined by means of a potential well argument.