Finite-time blowup for a Schrödinger equation with nonlinear source term

Abstract

We consider the nonlinear Schrödinger equation \[ u_t = i Δu + | u |^αu \quad \mbox{on ${\mathbb R}^N $, $α>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) α\le 4$. Under the additional technical assumptions $α\geq 2$ (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 $U_t=|U|^αU$. In the simplest case, $U(t,x)=(|x|^k-αt)^{-\frac 1α}$ for $t<0$, $x\in {\mathbb R}^N$. For $k$ sufficiently large, $U$ satisfies $|ΔU|\ll U_t$ 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.