doi: 10.3934/dcdsb.2021156

Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient

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

* Corresponding author: Ting Zhang

Received  December 2020 Early access  June 2021

We consider the fourth-order Schrödinger equation
$ i\partial_tu+\Delta^2 u+\mu\Delta u+\lambda|u|^\alpha u = 0, $
where
$ \alpha>0, \mu = \pm1 $
or
$ 0 $
and
$ \lambda\in\mathbb{C} $
. Firstly, we prove local well-posedness in
$ H^4\left( {\mathbb R}^N\right) $
in both
$ H^4 $
subcritical and critical case:
$ \alpha>0 $
,
$ (N-8)\alpha\leq8 $
. Then, for any given compact set
$ K\subset\mathbb{R}^N $
, we construct
$ H^4( {\mathbb R}^N) $
solutions that are defined on
$ (-T, 0) $
for some
$ T>0 $
, and blow up exactly on
$ K $
at
$ t = 0 $
.
Citation: Xuan Liu, Ting Zhang. Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021156
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]

G. BaruchG. Fibich and E. Mandelbaum, Singular solutions of the biharmonic nonlinear Schrödinger equation, SIAM J. Appl. Math., 70 (2010), 3319-3341.  doi: 10.1137/100784199.  Google Scholar

[3]

G. BaruchG. Fibich and N. Gavish, Singular standing ring solutions of nonlinear partial differential equations, Physica D, 239 (2010), 1968-1983.  doi: 10.1016/j.physd.2010.07.009.  Google Scholar

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G. Baruch and G. Fibic, Singular solutions of the $L^2$-supercritical biharmonic nonlinear Schrödinger equation, Physica D, 24 (2011), 1843-1859.  doi: 10.1088/0951-7715/24/6/009.  Google Scholar

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M. Ben-ArtziH. Koch and J. C. Saut, Disperion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 87-92.  doi: 10.1016/S0764-4442(00)00120-8.  Google Scholar

[6]

T. Boulenger and E. Lenzmann, Blowup for Biharmonic NLS, Ann. Sci. Ecole Norm. Sup., 50 (2017), 503-544.  doi: 10.24033/asens.2326.  Google Scholar

[7]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, American Mathematical Society, New York; Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

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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.  Google Scholar

[9]

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

[10]

T. CazenaveZ. Han and Y. Martel, Blowup on an arbitrary compact set for a Schrödinger equation with nonlinear source term, J. Dyn. Diff. Equat., 33 (2021), 941-960.  doi: 10.1007/s10884-020-09841-8.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

Y. ChoT. Ozawa and C. Wang, Finite time blowup for the fourth-order NLS, Bull. Korean Math. Soc., 53 (2016), 615-640.  doi: 10.4134/BKMS.2016.53.2.615.  Google Scholar

[15]

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

[16]

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

[17]

M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112.   Google Scholar

[18]

G. FibichB. Ilan and G. Papanicolaou, Self-focusing with forth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.  Google Scholar

[19]

S. B. Cui and C. H. Guo, Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces $H^s({\mathbb R}^N)$ and applications, Nonlinear Anal., 67 (2007), 687-707.  doi: 10.1016/j.na.2006.06.020.  Google Scholar

[20]

V. D. Dinh, On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation, J. Dynam. Differential Equations, 31 (2019), 1793-1823.  doi: 10.1007/s10884-018-9690-y.  Google Scholar

[21]

V. I. Karpman, Lyapunov approach to the soliton stability in highly dispersive systems I. Fourth order nonlinear Schrödinger equations, Phys. Lett. A, 215 (1996), 254-256.  doi: 10.1016/0375-9601(96)00231-9.  Google Scholar

[22]

V. I. Karpman and A. G. Shagalov, Influence of high-order dispersion on self-focusing. II. Numerical investigation, Phys. Lett. A, 160 (1991), 538-540.   Google Scholar

[23]

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

[24]

T. Kato, Nonlinear Schrödinger equations, Schrödinger operators (Sønderborg, 1988), Lecture Notes in Phys, 345 (1989), 218-263.  doi: 10.1007/3-540-51783-9_22.  Google Scholar

[25]

S. Kawakami and S. Machihara, Blowup solutions for the nonlinear Schrödinger equation with complex coefficient, Differential Integral Equations, 33 (2020), 445-464.   Google Scholar

[26]

M. Keel and T. Tao, Endpoint Strichartz inequalities, Amer. J. Math., 120 (1998), 955-980.   Google Scholar

[27]

X. Liu and T. Zhang, $H^2$ blowup result for a Schrödinger equation with nonlinear source term, Electron. Res. Arch., 28 (2020), 777-794.  doi: 10.3934/era.2020039.  Google Scholar

[28]

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.   Google Scholar

[29]

Y. Martel, Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math., 127 (2005), 1103-1140.   Google Scholar

[30]

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

[31]

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

[32]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Differential Equations, 246 (2009), 3715-3749.  doi: 10.1016/j.jde.2008.11.011.  Google Scholar

[33]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy critical nonlinear Schrödinger equations of fourth order in dimensions $d \geq9$, J. Differential Equations, 251 (2011), 3381-3402.  doi: 10.1016/j.jde.2011.08.009.  Google Scholar

[34]

A. Mielke, The Ginzburg-Landau equation in its role as a modulation equation, North-Holland, Amsterdam, Handbook of Dynamical Systems, 2 (2002), 759-834.  doi: 10.1016/S1874-575X(02)80036-4.  Google Scholar

[35]

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

[36]

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

[37]

B. Pausader, Global well-posedness for energy-critical fourth-order Schrödinger equation in the radial case, Dyn. Partial Differ. Equ., 4 (2007), 197-225.  doi: 10.4310/DPDE.2007.v4.n3.a1.  Google Scholar

[38]

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

[39]

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

[40]

K. Stewartson and J. T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, J. Fluid Mech., 48 (1971), 529-545.  doi: 10.1017/S0022112071001733.  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]

G. BaruchG. Fibich and E. Mandelbaum, Singular solutions of the biharmonic nonlinear Schrödinger equation, SIAM J. Appl. Math., 70 (2010), 3319-3341.  doi: 10.1137/100784199.  Google Scholar

[3]

G. BaruchG. Fibich and N. Gavish, Singular standing ring solutions of nonlinear partial differential equations, Physica D, 239 (2010), 1968-1983.  doi: 10.1016/j.physd.2010.07.009.  Google Scholar

[4]

G. Baruch and G. Fibic, Singular solutions of the $L^2$-supercritical biharmonic nonlinear Schrödinger equation, Physica D, 24 (2011), 1843-1859.  doi: 10.1088/0951-7715/24/6/009.  Google Scholar

[5]

M. Ben-ArtziH. Koch and J. C. Saut, Disperion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 87-92.  doi: 10.1016/S0764-4442(00)00120-8.  Google Scholar

[6]

T. Boulenger and E. Lenzmann, Blowup for Biharmonic NLS, Ann. Sci. Ecole Norm. Sup., 50 (2017), 503-544.  doi: 10.24033/asens.2326.  Google Scholar

[7]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, American Mathematical Society, New York; Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

T. CazenaveZ. Han and Y. Martel, Blowup on an arbitrary compact set for a Schrödinger equation with nonlinear source term, J. Dyn. Diff. Equat., 33 (2021), 941-960.  doi: 10.1007/s10884-020-09841-8.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

Y. ChoT. Ozawa and C. Wang, Finite time blowup for the fourth-order NLS, Bull. Korean Math. Soc., 53 (2016), 615-640.  doi: 10.4134/BKMS.2016.53.2.615.  Google Scholar

[15]

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

[16]

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

[17]

M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112.   Google Scholar

[18]

G. FibichB. Ilan and G. Papanicolaou, Self-focusing with forth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.  Google Scholar

[19]

S. B. Cui and C. H. Guo, Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces $H^s({\mathbb R}^N)$ and applications, Nonlinear Anal., 67 (2007), 687-707.  doi: 10.1016/j.na.2006.06.020.  Google Scholar

[20]

V. D. Dinh, On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation, J. Dynam. Differential Equations, 31 (2019), 1793-1823.  doi: 10.1007/s10884-018-9690-y.  Google Scholar

[21]

V. I. Karpman, Lyapunov approach to the soliton stability in highly dispersive systems I. Fourth order nonlinear Schrödinger equations, Phys. Lett. A, 215 (1996), 254-256.  doi: 10.1016/0375-9601(96)00231-9.  Google Scholar

[22]

V. I. Karpman and A. G. Shagalov, Influence of high-order dispersion on self-focusing. II. Numerical investigation, Phys. Lett. A, 160 (1991), 538-540.   Google Scholar

[23]

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

[24]

T. Kato, Nonlinear Schrödinger equations, Schrödinger operators (Sønderborg, 1988), Lecture Notes in Phys, 345 (1989), 218-263.  doi: 10.1007/3-540-51783-9_22.  Google Scholar

[25]

S. Kawakami and S. Machihara, Blowup solutions for the nonlinear Schrödinger equation with complex coefficient, Differential Integral Equations, 33 (2020), 445-464.   Google Scholar

[26]

M. Keel and T. Tao, Endpoint Strichartz inequalities, Amer. J. Math., 120 (1998), 955-980.   Google Scholar

[27]

X. Liu and T. Zhang, $H^2$ blowup result for a Schrödinger equation with nonlinear source term, Electron. Res. Arch., 28 (2020), 777-794.  doi: 10.3934/era.2020039.  Google Scholar

[28]

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.   Google Scholar

[29]

Y. Martel, Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math., 127 (2005), 1103-1140.   Google Scholar

[30]

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

[31]

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

[32]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Differential Equations, 246 (2009), 3715-3749.  doi: 10.1016/j.jde.2008.11.011.  Google Scholar

[33]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy critical nonlinear Schrödinger equations of fourth order in dimensions $d \geq9$, J. Differential Equations, 251 (2011), 3381-3402.  doi: 10.1016/j.jde.2011.08.009.  Google Scholar

[34]

A. Mielke, The Ginzburg-Landau equation in its role as a modulation equation, North-Holland, Amsterdam, Handbook of Dynamical Systems, 2 (2002), 759-834.  doi: 10.1016/S1874-575X(02)80036-4.  Google Scholar

[35]

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

[36]

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

[37]

B. Pausader, Global well-posedness for energy-critical fourth-order Schrödinger equation in the radial case, Dyn. Partial Differ. Equ., 4 (2007), 197-225.  doi: 10.4310/DPDE.2007.v4.n3.a1.  Google Scholar

[38]

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

[39]

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

[40]

K. Stewartson and J. T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, J. Fluid Mech., 48 (1971), 529-545.  doi: 10.1017/S0022112071001733.  Google Scholar

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