Journal of Evolution Equations ( IF 1.1 ) Pub Date : 2020-11-26 , DOI: 10.1007/s00028-020-00649-z Michinori Ishiwata , Bernhard Ruf , Federica Sani , Elide Terraneo
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.
中文翻译：
具有临界指数非线性的抛物型方程的渐近性。
我们考虑柯西问题：
$$ \ begin {aligned} {\ left \ {\ begin {array} {ll} \ partial _ {t} u = \ Delta u-u + \ lambda f（u）＆{} \ text {in}（0， T）\ times {\ mathbb {R}} ^ {2}，\\ u（0，x）= u_ {0}（x）＆{} \ text {in} {\ mathbb {R}} ^ {2 }，\ end {array} \ right。} \ end {aligned} $$其中\（\ lambda> 0 \），
$$ \ begin {aligned} f（u）{：=} 2 \ alpha _0 ue ^ {\ alpha _ {0} u ^ {2}}，\ quad \ text {对于某些} \ alpha _ {0}> 0，\ end {aligned} $$带有初始数据\（u_0 \ in H ^ {1}（{\ mathbb {R}} ^ {2}）\）。考虑到Trudinger-Moser嵌入，非线性项f在能量空间\（H ^ {1}（{\ mathbb {R}} ^ {2}）\）中的无穷大处具有临界增长。我们的目标是从初始数据\（u ^ \ in H ^ {1}（{\ mathbb {R}} ^ {2}）\）中调查解决方案是否在有限时间内爆炸或该解决方案在时间上是全局的。对于\（0 <\ lambda <\ frac {1} {2 \ alpha _0} \），我们证明对于能量低于或等于基态能级的初始数据，有限时间爆炸和全局存在之间的二分法可以通过一个潜在的井参数来确定。
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