Sharp upper and lower bounds on the blow-up rate for nonlinear Schrödinger equation with potential

Abstract

We consider the blow-up solutions of the Cauchy problem for critical nonlinear Schrödinger equation with a repulsive harmonic potential. In terms of Merle and Raphaël’s recent arguments as well as Carles’ transform, the sharp upper and lower bounds of the blow-up rate are obtained.

Introduction

In this paper, we study the Cauchy problem of the following nonlinear Schrödinger equation with a repulsive harmonic potential:

$$\mathrm{i}{u}_t+\frac{1}{2}\mathrm{\Delta }u=-\frac{{\omega }^2}{2}|x{|}^{2}u-|u{|}^{4/n}u\text{,}t\in [0\text{,}T)\text{,}x\in R^n\text{,}$$

$$u(0\text{,}x)=u_0\text{,}x\in R^n\text{,}$$

where ω is a positive parameter; n is the space dimension; u = u(t, x): [0, T) × Rⁿ → C and 0 < T ⩽ ∞; $\mathrm{i}=\sqrt{-1}$; Δ is the Laplace operator on Rⁿ. Carles[1] established the local well-posedness in the corresponding energy field. At the same time, Carles [1] also proved for some initial data, the solutions of the Cauchy problem (1.1), (1.2) blow up in a finite time T < ∞.

In Eq. (1.1), if we replace the repulsive harmonic potential with $\frac{{\omega }^2}{2}|x{|}^{2}u$, it models the remarkable Bose–Einstein condensate with attractive interactions under magnetic trap (see for example [14]). In this case, Oh [11] established the local well-posedness; Cazenave [3], Zhang [14], [15], [16] and Carles [2] showed the existence of blow-up solutions for some initial data, Li et al. [4], [5] showed some dynamical properties of the blow-up solutions.

We recall the Cauchy problem of the classical nonlinear Schrödinger equation

$$\mathrm{i}{v}_t+\frac{1}{2}\mathrm{\Delta }v=-|v{|}^{4/n}v\text{,}t\in [0\text{,}T)\text{,}x\in R^n\text{,}$$

$$v(0\text{,}x)=u_0\text{,}x\in R^n\text{.}$$

Using the spectral property, Merle and Raphaël [8], [9], [10] established the sharp upper and lower bounds on the blow-up rate for the blow-up solutions: there are two constants C₁ > 0, C₂ > 0 such that

$$C_1{\left(\frac{\ln |\ln (T-t)|}{T-t}\right)}^{\frac{1}{2}}⩽\Vert \nabla v(t){\Vert }_{L^2}⩽C_2{\left(\frac{\ln |\ln (T-t)|}{T-t}\right)}^{\frac{1}{2}}\text{,}$$

which is according with the numerical computations [6].

In this paper, in terms of Merle and Raphaël’s arguments [8], [9], [10], by applying the transform provided by Carles [1], we obtain the sharp upper and lower bounds on the blow-up rate for the blow-up solution of (1.1), (1.2), as follows:

$$C_{\ast }{\left(\frac{\ln |\ln \omega (T-t)|}{\omega (T-t)}\right)}^{\frac{1}{2}}⩽\Vert \nabla u(t){\Vert }_{L^2}⩽C^{\ast }{\left(\frac{\ln |\ln \omega (T-t)|}{\omega (T-t)}\right)}^{\frac{1}{2}}\text{,}$$

where C_∗ > 0, C^∗ > 0 are constants.

We conclude this section with several notations. We abbreviate L^q(Rⁿ), $\Vert ·{\Vert }_{L^q(R^n)}$, H^s(Rⁿ) and ${\int }_{R^n}·\mathrm{d}x$ by L^q, $\Vert ·{\Vert }_{L^q}$, H^s and $\int ·\mathrm{d}x$.