# American Institute of Mathematical Sciences

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February  2019, 39(2): 1171-1183. doi: 10.3934/dcds.2019050

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

 1 Sorbonne Université & CNRS, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France 2 CMLS, École Polytechnique, CNRS, 91128 Palaiseau Cedex, France 3 Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China

* Corresponding author

Received  May 2018 Published  November 2018

Fund Project: The third author thanks the hospitality of Professor F. Merle when he visited IHES and Professor Y. Martel when he visited CMLS, École Polytechnique, where part of the work was done. He is partly supported by grant NSFC of China no. 11771415.

We consider the nonlinear Schrödinger equation
 ${u_t} = i\Delta u + {\left| u \right|^\alpha }u\;\;\;\;{\rm{on}}\;\;\;{\mathbb{R}^N},\;\;\alpha >0$
for
 $H^1$
-subcritical or critical nonlinearities:
 $(N-2) α ≤ 4$
. Under the additional technical assumptions
 $α≥ 2$
(and thus
 $N≤4$
), 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
 $U_t=|U|^α U$
. In the simplest case,
 $U(t,x)=(|x|^k-α t)^{-\frac 1α}$
for
 $t<0$
,
 $x∈ \mathbb{R}^N$
. For
 $k$
sufficiently large,
 $U$
satisfies
 $|Δ U|\ll U_t$
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.
Citation: Thierry Cazenave, Yvan Martel, Lifeng Zhao. Finite-time blowup for a Schrödinger equation with nonlinear source term. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1171-1183. doi: 10.3934/dcds.2019050
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