Asymptotics for a parabolic equation with critical exponential nonlinearity
Asymptotics for a parabolic equation with critical exponential nonlinearity
复制标题
DOI：
10.1007/s00028-020-00649-z
复制
发表时间：
2020-11
影响因子：
1.4
通讯作者：
M. Ishiwata;B. Ruf;Federica Sani;E. Terraneo
中科院分区：
文献类型：
--
作者：
M. Ishiwata;B. Ruf;Federica Sani;E. Terraneo
我们考虑柯西问题：∂ tu= Δ u-u+ λ f (u) in (0, T)× R 2, u (0, x)= u 0 (x) in R 2, 其中 λ> 0 λ> 0, f (u):= 2 α _0 ue^ α _ 0 u^ 2,\quad 对于某些 α _ 0> 0, f (u):= 2 α 0 ue α 0 u 2，对于某些 α 0> 0，初始数据 u_0 ∈ H^ 1 (R^ 2) u 0∈ H 1 (R 2)。鉴于 Trudinger-Moser 嵌入，非线性项 f 在能量空间 H^ 1 (R^ 2) H 1 (R 2) 中具有无穷大临界增长。我们的目标是从初始数据 u_0 ∈ H^ 1 (R^ 2) u 0∈ H 1 (R 2) 中调查解是否在有限时间内爆炸或解在时间上是全局的。对于 0< λ< 1 2 α _0 0< λ< 1 2 α 0，我们证明对于能量低于或等于基态水平的初始数据，有限时间爆炸和全局存在之间的二分法可以通过势阱论证来确定。
We consider the Cauchy problem:∂ tu= Δ u-u+ λ f (u) in (0, T)× R 2, u (0, x)= u 0 (x) in R 2, where λ> 0 λ> 0, f (u):= 2 α _0 ue^ α _ 0 u^ 2,\quad for some α _ 0> 0, f (u):= 2 α 0 ue α 0 u 2, for some α 0> 0, with initial data u_0 ∈ H^ 1 (R^ 2) u 0∈ H 1 (R 2). The nonlinear term f has a critical growth at infinity in the energy space H^ 1 (R^ 2) H 1 (R 2) in view of the Trudinger-Moser embedding. Our goal is to investigate from the initial data u_0 ∈ H^ 1 (R^ 2) u 0∈ H 1 (R 2) whether the solution blows up in finite time or the solution is global in time. For 0< λ< 1 2 α _0 0< λ< 1 2 α 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.