Continuum of finite point blowup rates for the critical generalized Korteweg-de Vries equation
Didier Pilod, Professor @ Department of Mathematics, UiB
Abstract
For any $\nu\in(\frac 37,\frac12)$, we prove the existence of a finite energy solution $u$ of the mass critical generalized Korteweg-de Vries equation on the time interval $(0,T_0]$, for some $T_0>0$, which blows up at the time $t=0$ and at the point $x=0$ with the rate $\|\partial_x u (t)\|_{L^2} \sim t^{-\nu}$.
By construction, this blowup rate is associated to an $H^1$ blowup residue, obtained by passing to the limit in the solution $u(t)$ as $t\downarrow 0$ after subtracting the singular bubble, of the form $r_\alpha(x)= C x^{\alpha -\frac 12}$ for $x>0$ close to the blowup point, where $\alpha=\frac{3\nu-1}{2-4\nu}$.
Note that $\nu=\frac25$ is equivalent to $\alpha=\frac12$, while the condition $\nu\in(\frac37,\frac12)$ is equivalent to $\alpha>1$. Thus, this range of $\nu$ corresponds exactly to the range of $\alpha$ for which the residue $r_\alpha$ belongs to $H^1$.For this reason, these are the only possible finite point blowup rates $\nu$ accessible by our construction method.
Finally, we present some open problems regarding the blowup phenomenon for the mass critical
gKdV equation.
The talk is based on a joint work with Yvan Martel.