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Let $S$ be a minimal mass blow up solution of the critical generalized KdVequation as constructed by Martel, Merle and Rapha\"el in arXiv:1204.4624. Weprove both time and space sharp asymptotics for $S$ close to the blow up time.Let $Q$ be the unique ground state of (gKdV), satisfying $Q"+Q^5=Q$.  First, we show that there exist universal smooth profiles$Q_k\in\mathcal{S}(\mathbb{R})$ (with $Q_0=Q$) and a constant$c_0\in\mathbb{R}$ such that, fixing the blow up time at $t=0$ and appropriatescaling and translation parameters, $S$ satisfies, for any $m\geqslant 0$, \[\partial_x^m S(t) - \sum_{k=0}^{[m/2]} \frac 1{t^{\frac 12+m-2k}}Q_k^{(m-k)}\left(\frac{\cdot+ \frac1t}{t}+c_0\right)\to 0\quad \mbox{in}\ L^2\mbox{as}\ t\downarrow 0. \] Second, we prove that, for $0<t\ll 1$, $x\leqslant-\frac 1t -1$, \[ S(t,x) \sim - \frac 12 \|Q\|_{L^1} |x|^{-3/2}, \] and relatedbounds for the derivatives of $S(t)$ of any order. We also prove$\int_{\mathbb{R}} S(t,x)\,dx=0$.

Sharp asymptotics for the minimal mass blow up solution of critical gKdV equation