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,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u = \Delta u - u+ \lambda f(u) &{} \text {in } (0,T) \times {\mathbb {R}}^{2}, \\ u(0,x)=u_{0}(x) &{} \text {in } {\mathbb {R}}^{2}, \end{array}\right. } \end{aligned}$$\end{document}where λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document}, f(u):=2α0ueα0u2,for someα0>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(u){:=}2 \alpha _0 u e^{\alpha _{0}u^{2}}, \quad \text {for some } \alpha _{0}>0, \end{aligned}$$\end{document}with initial data u0∈H1(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0\in H^{1}({\mathbb {R}}^{2})$$\end{document}. The nonlinear term f has a critical growth at infinity in the energy space H1(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H^{1}({\mathbb {R}}^{2})$$\end{document} in view of the Trudinger-Moser embedding. Our goal is to investigate from the initial data u0∈H1(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0\in H^{1}({\mathbb {R}}^{2})$$\end{document} whether the solution blows up in finite time or the solution is global in time. For 0<λ<12α0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\lambda <\frac{1}{2\alpha _0}$$\end{document}, 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.