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

Abstract

We consider the nonlinear Schrödinger equation \[ u_t = i Δu + | u |^αu \quad \mbox{on ${\mathbb{R}}^N$, $\alpha >0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2)\alpha \le 4$. Under the additional technical assumptions $\alpha \ge 2$ (and thus $N\le 4$), 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.

References in corpus (2)

• Construction of a blow-up solution for the Complex Ginzburg-Landau equation in some critical case
• Singularity formation for Burgers equation with transverse viscosity

Cited by in corpus (5)

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• Blowup on an arbitrary compact set for a Schödinger equation with nonlinear source term
• Solutions blowing up on any given compact set for the energy subcritical wave equation
• Solutions with prescribed local blow-up surface for the nonlinear wave equation
• Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient