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June  2020, 28(2): 777-794. doi: 10.3934/era.2020039

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

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

* Corresponding author: Ting Zhang

Received  February 2020 Revised  April 2020 Published  May 2020

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.
Citation: Xuan Liu, Ting Zhang. $ H^2 $ blowup result for a Schrödinger equation with nonlinear source term. Electronic Research Archive, 2020, 28 (2) : 777-794. doi: 10.3934/era.2020039
References:
[1]

S. Alinhac, Blowup for Nonlinear Hyperbolic Equations. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., 17. Boston, MA, 1995. doi: 10.1007/978-1-4612-2578-2.  Google Scholar

[2]

T. CazenaveS. CorreiaF. Dicksteinand 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.  Google Scholar

[3]

T. CazenaveD. Y. Fang and Z. Han, Continuous dependence for NLS in fractional order spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 135-147.  doi: 10.1016/j.anihpc.2010.11.005.  Google Scholar

[4]

T. CazenaveD. Y. Fang and Z. Han, Local well-posedness for the $H^2$-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 368 (2016), 7911-7934.  doi: 10.1090/tran6683.  Google Scholar

[5]

T. Cazenave, Z. Han and Y. Martel, Blowup on an arbitrary compact set for a Schrödinger equation with nonlinear source term, (2019), arXiv: 1906.02983. Google Scholar

[6]

T. CazenaveY. Martel and L. Zhao, Finite-time blowup for a Schröding equation with nonlinear source term, Discrete Contin. Dynam. Systems., 39 (2019), 1171-1183.  doi: 10.3934/dcds.2019050.  Google Scholar

[7]

T. CazenaveY. Martel and L. F. Zhao, Solutions blowing up on any given compact set for the energy subcritical wave equation, J. Differential Equations, 268 (2020), 680-706.  doi: 10.1016/j.jde.2019.08.030.  Google Scholar

[8]

T. CazenaveY. Martel and L. F. Zhao, Solutions with prescribed local blow-up surface for the nonlinear wave equation, Adv. Nonlinear Stud., 19 (2019), 639-675.  doi: 10.1515/ans-2019-2059.  Google Scholar

[9]

T. CazenaveY. Martel and L. F. Zhao, Finite-time blowup for a Schrödinger equation with nonlinear source term, Discrete Contin. Dyn. Syst., 39 (2019), 1171-1183.  doi: 10.3934/dcds.2019050.  Google Scholar

[10]

C. Collot, T. E. Ghouland N. Masmoudi, Singularity formation for Burgers equation with transverse viscosity, (2018), arXiv: 1803.07826. Google Scholar

[11]

G. M. Constantine and T. H. Savits, A multivariate Faa di Bruno formula with applications, Trans. Amer. Math. Soc., 348 (1996), 503-520.  doi: 10.1090/S0002-9947-96-01501-2.  Google Scholar

[12]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.   Google Scholar

[13]

S. Kawakami and S. Machihara, Blowup solutions for the nonlinear Schrödinger equation with complex coefficient, (2019), arXiv: 1905.13037. Google Scholar

[14]

R. KillipS. MasakiJ. Murphy and M. Visan, The radial mass-subcritical NLS in negative order Sobolev spaces, Discrete Contin. Dyn. Syst., 39 (2019), 553-583.  doi: 10.3934/dcds.2019023.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

F. Merle and H. Zaag, O.D.E. type behavior of blow-up solutions of nonlinear heat equations, Discrete Contin. Dyn., 8 (2002), 435-450.  doi: 10.3934/dcds.2002.8.435.  Google Scholar

[18]

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.  Google Scholar

[19]

I. Moerdijk and G. Reyes, Models for Smooth Infinitesimal Analysis, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4757-4143-8.  Google Scholar

[20]

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.  Google Scholar

[21]

H. Pecher, Solutions of semilinear Schrödinger equations in $H^s$, Ann. Inst. H. Poincareé Phys. Théor., 67 (1997), 259-296.   Google Scholar

[22]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[23]

J. Speck, Stable ODE-type blowup for some quasilinear wave equations with derivative-quadratic nonlinearity, Analysis and PDE, 13 (2020), 93–146, arXiv: 1709.04778. doi: 10.2140/apde.2020.13.93.  Google Scholar

[24]

R. Z. XuY. X. ChenY. B. YangS. H. ChenJ. H. ShenT. Yu and Z. S. Xu, Global well-posedness of semilinear hyperbolic equations, parabolic equations and Schrödinger equations, Electron. J. Differential Equations, 2018 (2018), 1-52.   Google Scholar

[25]

M. Zhang and M. Ahmed, Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential, Adv. Nonlinear Anal., 9 (2020), 882-894.   Google Scholar

show all references

References:
[1]

S. Alinhac, Blowup for Nonlinear Hyperbolic Equations. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., 17. Boston, MA, 1995. doi: 10.1007/978-1-4612-2578-2.  Google Scholar

[2]

T. CazenaveS. CorreiaF. Dicksteinand 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.  Google Scholar

[3]

T. CazenaveD. Y. Fang and Z. Han, Continuous dependence for NLS in fractional order spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 135-147.  doi: 10.1016/j.anihpc.2010.11.005.  Google Scholar

[4]

T. CazenaveD. Y. Fang and Z. Han, Local well-posedness for the $H^2$-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 368 (2016), 7911-7934.  doi: 10.1090/tran6683.  Google Scholar

[5]

T. Cazenave, Z. Han and Y. Martel, Blowup on an arbitrary compact set for a Schrödinger equation with nonlinear source term, (2019), arXiv: 1906.02983. Google Scholar

[6]

T. CazenaveY. Martel and L. Zhao, Finite-time blowup for a Schröding equation with nonlinear source term, Discrete Contin. Dynam. Systems., 39 (2019), 1171-1183.  doi: 10.3934/dcds.2019050.  Google Scholar

[7]

T. CazenaveY. Martel and L. F. Zhao, Solutions blowing up on any given compact set for the energy subcritical wave equation, J. Differential Equations, 268 (2020), 680-706.  doi: 10.1016/j.jde.2019.08.030.  Google Scholar

[8]

T. CazenaveY. Martel and L. F. Zhao, Solutions with prescribed local blow-up surface for the nonlinear wave equation, Adv. Nonlinear Stud., 19 (2019), 639-675.  doi: 10.1515/ans-2019-2059.  Google Scholar

[9]

T. CazenaveY. Martel and L. F. Zhao, Finite-time blowup for a Schrödinger equation with nonlinear source term, Discrete Contin. Dyn. Syst., 39 (2019), 1171-1183.  doi: 10.3934/dcds.2019050.  Google Scholar

[10]

C. Collot, T. E. Ghouland N. Masmoudi, Singularity formation for Burgers equation with transverse viscosity, (2018), arXiv: 1803.07826. Google Scholar

[11]

G. M. Constantine and T. H. Savits, A multivariate Faa di Bruno formula with applications, Trans. Amer. Math. Soc., 348 (1996), 503-520.  doi: 10.1090/S0002-9947-96-01501-2.  Google Scholar

[12]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.   Google Scholar

[13]

S. Kawakami and S. Machihara, Blowup solutions for the nonlinear Schrödinger equation with complex coefficient, (2019), arXiv: 1905.13037. Google Scholar

[14]

R. KillipS. MasakiJ. Murphy and M. Visan, The radial mass-subcritical NLS in negative order Sobolev spaces, Discrete Contin. Dyn. Syst., 39 (2019), 553-583.  doi: 10.3934/dcds.2019023.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

F. Merle and H. Zaag, O.D.E. type behavior of blow-up solutions of nonlinear heat equations, Discrete Contin. Dyn., 8 (2002), 435-450.  doi: 10.3934/dcds.2002.8.435.  Google Scholar

[18]

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.  Google Scholar

[19]

I. Moerdijk and G. Reyes, Models for Smooth Infinitesimal Analysis, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4757-4143-8.  Google Scholar

[20]

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.  Google Scholar

[21]

H. Pecher, Solutions of semilinear Schrödinger equations in $H^s$, Ann. Inst. H. Poincareé Phys. Théor., 67 (1997), 259-296.   Google Scholar

[22]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[23]

J. Speck, Stable ODE-type blowup for some quasilinear wave equations with derivative-quadratic nonlinearity, Analysis and PDE, 13 (2020), 93–146, arXiv: 1709.04778. doi: 10.2140/apde.2020.13.93.  Google Scholar

[24]

R. Z. XuY. X. ChenY. B. YangS. H. ChenJ. H. ShenT. Yu and Z. S. Xu, Global well-posedness of semilinear hyperbolic equations, parabolic equations and Schrödinger equations, Electron. J. Differential Equations, 2018 (2018), 1-52.   Google Scholar

[25]

M. Zhang and M. Ahmed, Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential, Adv. Nonlinear Anal., 9 (2020), 882-894.   Google Scholar

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