Blow up and near soliton dynamics for the \(L^2\) critical gKdV equation. (English) Zbl 1319.35224
Summary: These notes present the main results of [the authors, Acta Math. 212, No. 1, 59–140 (2014; Zbl 1301.35137)] concerning the mass critical (gKdV) equation \(u_t+(u_{xx}+u^5)_x=0\) for initial data in \(H^1\) close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in \(H^1\), construction of various exotic blow up rates in \(H^1\), including grow up in infinite time.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35B44 | Blow-up in context of PDEs |
| 35Q51 | Soliton equations |
| 35C08 | Soliton solutions |
Keywords:
blow upCitations:
Zbl 1301.35137References:
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