Blow up for the critical gKdV equation III: exotic regimes

We consider the blow up problem in the energy space for the critical (gKdV) equation in the continuation of part I and part II. We know from part I that the unique and stable blow up rate for solutions close to the solitons with strong decay on the right is $1/t$. In this paper, we construct non-generic blow up regimes in the energy space by considering initial data with explicit slow decay on the right in space. We obtain finite time blow up solutions with speed $t^{-\nu}$ where $\nu>11/13,$ as well as global in time growing up solutions with both exponential growth or power growth. These solutions can be taken with initial data arbitrarily close to the ground state solitary wave

On collapsing ring blow up solutions to the mass supercritical NLS

We consider the nonlinear Schr\"odinger equation i\pa_tu+\Delta u+u|u|^{p-1}=0 in dimension $N\geq 2$ and in the mass super critical and energy subcritical range $1+\frac 4N<p<\min\{\frac{N+2}{N-2},5\}.$ For initial data $u_0\in H^1$ with radial symmetry, we prove a universal upper bound on the blow up speed. We then prove that this bound is sharp and attained on a family of collapsing ring blow up solutions first formally predicted by Gavish, Fibich and Wang.Comment: 48 page

Blow up dynamics for smooth equivariant solutions to the energy critical Schr\"odinger map

We consider the energy critical Schr\"odinger map problem with the 2-sphere target for equivariant initial data of homotopy index $k=1$. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blow up solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy