• Previous Article
    Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity
  • DCDS Home
  • This Issue
  • Next Article
    Characterization for the existence of bounded solutions to elliptic equations
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 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. CazenaveS. CorreiaF. 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. CazenaveS. CorreiaF. 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.

[1]

Yuya Tanaka, Tomomi Yokota. Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022075

[2]

Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639

[3]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure and Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[4]

Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11

[5]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903

[6]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control and Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

[7]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure and Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

[8]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[9]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085

[10]

Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169

[11]

Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure and Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027

[12]

Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2021, 10 (3) : 599-617. doi: 10.3934/eect.2020082

[13]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032

[14]

Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure and Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243

[15]

Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827

[16]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021

[17]

Xuan Liu, Ting Zhang. Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2721-2757. doi: 10.3934/dcdsb.2021156

[18]

Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134

[19]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006

[20]

Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (341)
  • HTML views (188)
  • Cited by (4)

Other articles
by authors

[Back to Top]