Asymptotics for a parabolic equation with critical exponential nonlinearity

Abstract

We consider the Cauchy problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u = \Delta u - u+ \lambda f(u) &{} \text {in } (0,T) \times {\mathbb {R}}^{2}, \\ u(0,x)=u_{0}(x) &{} \text {in } {\mathbb {R}}^{2}, \end{array}\right. } \end{aligned}$$

where $\lambda >0$,

$$\begin{aligned} f(u){:=}2 \alpha _0 u e^{\alpha _{0}u^{2}}, \quad \text {for some } \alpha _{0}>0, \end{aligned}$$

with initial data $u_0\in H^{1}({\mathbb {R}}^{2})$. The nonlinear term f has a critical growth at infinity in the energy space $H^{1}({\mathbb {R}}^{2})$ in view of the Trudinger-Moser embedding. Our goal is to investigate from the initial data $u_0\in H^{1}({\mathbb {R}}^{2})$ whether the solution blows up in finite time or the solution is global in time. For $0<\lambda <\frac{1}{2\alpha _0}$, 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.