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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2412.04049 |
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- This paper investigates the asymptotic behavior of solutions to $u_t=Δu+|u|^{p-1}u$ in the Sobolev critical case. Our main result is a classification of the dynamics near the ground states in the six dimensional case. It is shown that if the initial data $u_0\in H^1(\mathbb{R}^6)$ satisfies $\|u_0-{\sf Q}\|_{\dot H^1(\mathbb{R}^6)}\ll1$, then the solution falls into one of the following three scenarios: 1) It is globally defined and converge to one of the ground states as $t\to\infty$. 2) It is globally defined and converge to $0$ in $\dot H^1(\mathbb{R}^6)$ as $t\to\infty$. 3) It exhibits finite time blowup with a type I rate. This paper extends the classification result in the case $n\geq7$, previously obtained by Collot-Merle-Raphaël, to the borderline case $n=6$.