$ H^2 $ blowup result for a Schrödinger equation with nonlinear source term

Xuan Liu; Ting Zhang; Xuan Liu; Ting Zhang

doi:10.3934/era.2020039

Electronic Research Archive

2020, Volume 28, Issue 2: 777-794. doi: 10.3934/era.2020039

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$ H^2 $ blowup result for a Schrödinger equation with nonlinear source term

Xuan Liu ,
Ting Zhang ^,

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

Received: 01 February 2020 Revised: 01 April 2020
Primary: 35L71; Secondary: 35B30, 35B44

In this paper, we consider the nonlinear Schrödinger equation on $ \mathbb{R}^N, N\ge1 $,
$ \partial_tu = i\Delta u+\lambda|u|^\alpha u , $
with $ H^2 $-subcritical nonlinearities: $ \alpha>0, (N-4)\alpha<4 $ and Re$ \lambda>0 $. For any given compact set $ K\subset\mathbb{R}^N $, we construct $ H^2 $ solutions that are defined on $ (-T, 0) $ for some $ T>0 $, and blow up exactly on $ K $ at $ t = 0 $. We generalize the range of the power $ \alpha $ in the result of Cazenave, Han and Martel [5]. The proof is based on the energy estimates and compactness arguments.
- Schrödinger equations,
- finite-time blowup
Citation: Xuan Liu, Ting Zhang. $ H^2 $ blowup result for a Schrödinger equation with nonlinear source term[J]. Electronic Research Archive, 2020, 28(2): 777-794. doi: 10.3934/era.2020039

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Abstract

In this paper, we consider the nonlinear Schrödinger equation on $ \mathbb{R}^N, N\ge1 $,

$ \partial_tu = i\Delta u+\lambda|u|^\alpha u , $

with $ H^2 $-subcritical nonlinearities: $ \alpha>0, (N-4)\alpha<4 $ and Re$ \lambda>0 $. For any given compact set $ K\subset\mathbb{R}^N $, we construct $ H^2 $ solutions that are defined on $ (-T, 0) $ for some $ T>0 $, and blow up exactly on $ K $ at $ t = 0 $. We generalize the range of the power $ \alpha $ in the result of Cazenave, Han and Martel [5]. The proof is based on the energy estimates and compactness arguments.

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