A potential well argument for a semilinear parabolic equation with exponential nonlinearity.

"We consider the Cauchy problem for a two space dimensional parabolic equation with square exponential nonlinearity. More precisely, {∂tu=Δu−u+λf(u)u(0,x)=u0(x)in (0,T)×R2,in R2, where λ>0, and f(u):=2α0ueα0u2, for some α0>0. We take into account initial data in the energy space H1(R2), i.e. u0∈H1(R2), and in view of the Trudinger-Moser inequality, the nonlinearity f (which has square exponential growth at infinity) is in the energy critical regime. "We look for sufficient conditions in order to predict from the initial data whether the solution blows up in finite time or the solution exists globally in time. Our main tools are energy methods, and the so-called potential well argument. If 0<12α0, we prove that for energies below the ground state level, the dichotomy between blow-up and global existence is determined by the sign of a suitable functional.''