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

Thierry Cazenave; Yvan Martel; Lifeng Zhao

doi:10.3934/dcds.2019050

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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

Thierry Cazenave ^1,, , Yvan Martel ^2, and Lifeng Zhao ^3,

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.

Keywords: Nonlinear Schrödinger equation, finite-time blowup, blow-up profile.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B44, 35B40.

Citation: Thierry Cazenave, Yvan Martel, Lifeng Zhao. Finite-time blowup for a Schrödinger equation with nonlinear source term. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1171-1183. doi: 10.3934/dcds.2019050

References:

[1]	S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Progress in Nonlinear Differential Equations and their Applications, 17. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-2578-2.
[2]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.
[3]	T. Cazenave, S. Correia, F. Dickstein and F. B. Weissler, A Fujita-type blowup result and low energy scattering for a nonlinear Schrödinger equation, São Paulo J. Math. Sci., 9 (2015), 146-161. doi: 10.1007/s40863-015-0020-6.
[4]	C. Collot, T. -E. Ghoul and N. Masmoudi, Singularity formation for Burgers equation with transverse viscosity, preprint, arXiv: 1803.07826.
[5]	Y. Martel, Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math., 127 (2005), 1103-1140. doi: 10.1353/ajm.2005.0033.
[6]	F. Merle, Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., 129 (1990), 223-240. doi: 10.1007/BF02096981.
[7]	F. Merle and H. Zaag, O.D.E. type behavior of blow-up solutions of nonlinear heat equations, Current developments in partial differential equations (Temuco, 1999), Discrete Contin. Dyn. Syst., 8 (2002), 435-450. doi: 10.3934/dcds.2002.8.435.
[8]	F. Merle and H. Zaag, Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension, Séminaire: Équations aux Dérivées Partielles. 2009-2010, Exp. No. Ⅺ, 10 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2012, Available from http://sedp.cedram.org/sedp-bin/fitem?id=SEDP_2009-2010____A11_0
[9]	F. Merle and H. Zaag, On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations, Comm. Math. Phys., 333 (2015), 1529-1562. doi: 10.1007/s00220-014-2132-8.
[10]	N. Nouaili and H. Zaag, Construction of a blow-up solution for the complex Ginzburg-Landau equation in a critical case, Arch. Ration. Mech. Anal., 228 (2018), 995-1058. doi: 10.1007/s00205-017-1211-3.
[11]	T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 25 (2006), 403-408. doi: 10.1007/s00526-005-0349-2.
[12]	P. Raphaël and J. Szeftel, Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc., 24 (2011), 471-546. doi: 10.1090/S0894-0347-2010-00688-1.
[13]	J. Speck, Stable ODE-type blowup for some quasilinear wave equations with derivative-quadratic nonlinearity, preprint, arXiv: 1709.04778.

show all references

References:

[1]	S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Progress in Nonlinear Differential Equations and their Applications, 17. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-2578-2.
[2]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.
[3]	T. Cazenave, S. Correia, F. Dickstein and F. B. Weissler, A Fujita-type blowup result and low energy scattering for a nonlinear Schrödinger equation, São Paulo J. Math. Sci., 9 (2015), 146-161. doi: 10.1007/s40863-015-0020-6.
[4]	C. Collot, T. -E. Ghoul and N. Masmoudi, Singularity formation for Burgers equation with transverse viscosity, preprint, arXiv: 1803.07826.
[5]	Y. Martel, Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math., 127 (2005), 1103-1140. doi: 10.1353/ajm.2005.0033.
[6]	F. Merle, Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., 129 (1990), 223-240. doi: 10.1007/BF02096981.
[7]	F. Merle and H. Zaag, O.D.E. type behavior of blow-up solutions of nonlinear heat equations, Current developments in partial differential equations (Temuco, 1999), Discrete Contin. Dyn. Syst., 8 (2002), 435-450. doi: 10.3934/dcds.2002.8.435.
[8]	F. Merle and H. Zaag, Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension, Séminaire: Équations aux Dérivées Partielles. 2009-2010, Exp. No. Ⅺ, 10 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2012, Available from http://sedp.cedram.org/sedp-bin/fitem?id=SEDP_2009-2010____A11_0
[9]	F. Merle and H. Zaag, On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations, Comm. Math. Phys., 333 (2015), 1529-1562. doi: 10.1007/s00220-014-2132-8.
[10]	N. Nouaili and H. Zaag, Construction of a blow-up solution for the complex Ginzburg-Landau equation in a critical case, Arch. Ration. Mech. Anal., 228 (2018), 995-1058. doi: 10.1007/s00205-017-1211-3.
[11]	T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 25 (2006), 403-408. doi: 10.1007/s00526-005-0349-2.
[12]	P. Raphaël and J. Szeftel, Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc., 24 (2011), 471-546. doi: 10.1090/S0894-0347-2010-00688-1.
[13]	J. Speck, Stable ODE-type blowup for some quasilinear wave equations with derivative-quadratic nonlinearity, preprint, arXiv: 1709.04778.

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