Abstract
We consider the Cauchy problem:
where \(\lambda >0\),
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.
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Ishiwata, M., Ruf, B., Sani, F. et al. Asymptotics for a parabolic equation with critical exponential nonlinearity. J. Evol. Equ. 21, 1677–1716 (2021). https://doi.org/10.1007/s00028-020-00649-z
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DOI: https://doi.org/10.1007/s00028-020-00649-z