Slow blowup solutions for the focusing energy supercritical nonlinear Schr\"odinger equation

Abstract

We consider the focusing energy supercritical nonlinear Schr\"odinger equation \begin{equation*} i\partial_t\psi+\Delta\psi+|\psi|^{2p}\psi=0 \end{equation*} in dimensions $d\geq11$, with $p\in\mathbb N^\ast$, under the Joseph--Lundgren condition $2p+1>p_{JL}(d)$. For every $\nu>\nu_0(d,p)$, there exists a radial finite-time type II blow-up solution concentrating the ground state $Q$ in the form \begin{equation*} \psi(t,x)=\lambda^{\frac{1}{p}}(t)Q(\lambda(t)x)+\zeta(t,x),\quad \lambda(t)=t^{-\frac12-\nu}. \end{equation*} All homogeneous Sobolev norms $\|\psi(t)\|_{\dot H^s}$, $1\leq s<s_c$, remain bounded. The radiation converges to a limiting profile \begin{equation*} \zeta^*\in \dot H^s(\mathbb R^d), \qquad s_c<s<s_c+2\nu|\lambda_+|, \end{equation*} where $\lambda_+$ is determined by the leading correction in the asymptotic expansion of $Q$ at infinity. Our proof combines matched asymptotic expansions across three spatial scales with modified energy estimates. The key ingredient is the systematic treatment of the singular nonlinear background and coupled oscillations generated by the slow algebraic tail of $Q$.