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Construction of multi-bubble solutions for the critical gKdV equation. (arXiv:1706.09870v1 [math.AP])
来源于:arXiv
We prove the existence of solutions of the mass critical generalized
Korteweg-de Vries equation $\partial_t u + \partial_x(\partial_{xx} u + u^5) =
0$ containing an arbitrary number $K\geq 2$ of blow up bubbles, for any choice
of sign and scaling parameters: for any $\ell_1>\ell_2>\cdots>\ell_K>0$ and
$\epsilon_1,\ldots,\epsilon_K\in\{\pm1\}$, there exists an $H^1$ solution $u$
of the equation such that \[ u(t) - \sum_{k=1}^K \frac
{\epsilon_k}{\lambda_k^\frac12(t)} Q\left( \frac {\cdot - x_k(t)}{\lambda_k(t)}
\right) \longrightarrow 0 \quad\mbox{ in }\ H^1 \mbox{ as }\ t\downarrow 0, \]
with $\lambda_k(t)\sim \ell_k t$ and $x_k(t)\sim -\ell_k^{-2}t^{-1}$ as
$t\downarrow 0$. The construction uses and extends techniques developed mainly
by Martel, Merle and Rapha\"el. Due to strong interactions between the bubbles,
it also relies decisively on the sharp properties of the minimal mass blow up
solution (single bubble case) proved by the authors in arXiv:1602.03519. 查看全文>>