Finite-time blowup for a Schr\"odinger equation with nonlinear source term

TL;DR

This paper constructs explicit finite-time blowup solutions for a nonlinear Schr"odinger equation with specific nonlinearities, demonstrating blowup at arbitrary points and providing detailed profiles and rates.

Contribution

It introduces a method to explicitly construct finite-time blowup solutions with prescribed profiles and blow-up points for certain nonlinear Schr"odinger equations.

Findings

Explicit blowup solutions with known profiles and rates.

Blowup can occur at any finite set of points in space.

Construction uses approximate solutions and energy estimates.

Abstract

We consider the nonlinear Schr\"odinger equation \[ u_t = i \Delta u + | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) \alpha \le 4$. Under the additional technical assumptions $\alpha\geq 2$ (and thus $N\leq 4$), we construct $H^1$ solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finite set of points of ${\mathbb R}^N$. The construction involves explicit functions $U$, solutions of the ordinary differential equation $U_t=|U|^\alpha U$. In the simplest case, $U(t,x)=(|x|^k-\alpha t)^{-\frac 1\alpha}$ for $t<0$, $x\in {\mathbb R}^N$. For $k$ sufficiently large, $U$ satisfies $|\Delta U|\ll U_t$ close to the blow-up point $(t,x)=(0,0)$, so that it is a suitable approximate solution of the problem. To construct an actual solution $u$ close to $U$, we use energy estimates and a compactness argument.