June  2020, 28(2): 599-625. doi: 10.3934/era.2020032

Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Jun Zhou

Received  March 2020 Revised  March 2020 Published  April 2020

In this paper, the initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity is invsitgated. First, we establish the local well-posedness of solutions by means of the semigroup theory. Then by using ordinary differential inequalities, potential well theory and energy estimate, we study the conditions on global existence and finite time blow-up. Moreover, the lifespan (i.e., the upper bound of the blow-up time) of the finite time blow-up solution is estimated.

Citation: Xu Liu, Jun Zhou. Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28 (2) : 599-625. doi: 10.3934/era.2020032
References:
[1]

B. AbdellaouiI. Peral and A. Primo, Influence of the Hardy potential in a semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 897-926.  doi: 10.1017/S0308210508000152.  Google Scholar

[2]

B. AbdellaouiI. Peral and A. Primo, Strong regularizing effect of a gradient term in the heat equation with the Hardy potential, J. Funct. Anal., 258 (2010), 1247-1272.  doi: 10.1016/j.jfa.2009.11.008.  Google Scholar

[3]

A. AttarS. Merchán and I. Peral, A remark on the existence properties of a semilinear heat equation involving a Hardy-Leray potential, J. Evol. Equ., 15 (2015), 239-250.  doi: 10.1007/s00028-014-0259-x.  Google Scholar

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M. Bawin, Electron-bound states in the field of dipolar molecules, Phys. Rev. A, 70 (2004), 022505. doi: 10.1103/PhysRevA.70.022505.  Google Scholar

[5]

M. Bawin and S. A. Coon, Neutral atom and a charged wire: From elastic scattering to absorption, Phys. Rev. A, 63 (2001), 034701. doi: 10.1103/PhysRevA.63.034701.  Google Scholar

[6]

M. Bawin and S. A. Coon, Singular inverse square potential, limit cycles, and self-adjoint extensions, Phys. Rev. A, 67 (2003), 042712. doi: 10.1103/PhysRevA.67.042712.  Google Scholar

[7]

M. Bawin, S. A. Coon and B. R. Holstein, Anions and anomalies, Int. J. Mod. Phys. A, (2007), 19–28. doi: 10.1142/9789812770301_0003.  Google Scholar

[8]

S. R. Beane, P. F. Bedaque, L. Childress, A. Kryjevski, J. McGuire and U. van Kolck, Singular potentials and limit cycles, Phys. Rev. A, 64 (2001), 042103. doi: 10.1103/PhysRevA.64.042103.  Google Scholar

[9]

D. BirminghamK. S. Gupta and S. Sen, Near-horizon conformal structure of black holes, Phys. Lett. B, 55 (2001), 191-196.  doi: 10.1016/S0370-2693(01)00354-9.  Google Scholar

[10]

H. Chen and S. Y. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.  Google Scholar

[11]

E. CsoboF. GenoudM. Ohta and J. Royer, Stability of standing waves for a nonlinear Klein-Gordon equation with delta potentials, J. Differential Equations, 268 (2019), 353-388.  doi: 10.1016/j.jde.2019.08.015.  Google Scholar

[12]

J. DenschlagG. Umshaus and J. Schmiedmayer, Probing a singular potential with cold atoms: A neutral atom and a charged wire, Phys. Rev. Lett., 81 (1998), 737-741.  doi: 10.1103/PhysRevLett.81.737.  Google Scholar

[13]

V. D. Dinh, Global existence and blowup for a class of the focusing nonlinear Schrödinger equation with inverse-square potential, J. Math. Anal. Appl., 468 (2018), 270-303.  doi: 10.1016/j.jmaa.2018.08.006.  Google Scholar

[14]

E. H. Dowell, Aeroelasticity of Plates and Shells, Nordhoff, Leyden, 1973. Google Scholar

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E. H. Dowell, A Modern Course in Aeroelasticity, Solid Mechanics and its Applications, 217. Springer, Cham, 2015. doi: 10.1007/978-3-319-09453-3.  Google Scholar

[16]

V. Efimov, Weakly-bound states of 3 resonantly-interacting particles, Sov. J. Nucl. Phys., 12 (1971), 589-595.   Google Scholar

[17]

P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping, J. Differential Equations, 132 (1996), 338-352.  doi: 10.1006/jdeq.1996.0183.  Google Scholar

[18]

J. Fröhlich and E. Lenzmann., Mean-field limit of quantum Bose gases and nonlinear Hartree equation, Séminaire: Équations aux Dérivées Partielles, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2004), 26 pp.  Google Scholar

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T. R. GovindarajanV. Suneeta and S. Vaidya, Horizon states for AdS black holes, Nucl. Phys. B, 583 (2000), 291-303.  doi: 10.1016/S0550-3213(00)00336-9.  Google Scholar

[20]

Y. Q. Guo and M. A. Rammaha, Systems of nonlinear wave equations with damping and supercritical boundary and interior sources, Trans. Amer. Math. Soc., 366 (2014), 2265-2325.  doi: 10.1090/S0002-9947-2014-05772-3.  Google Scholar

[21]

A. Khanmamedov and S. Simsek, Existence of the global attractor for the plate equation with nonlocal nonlinearity in $\mathbb{R}^n$, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 151-172.  doi: 10.3934/dcdsb.2016.21.151.  Google Scholar

[22]

R. KillipC. MiaoM. VisanJ. Zhang and J. Zheng, Sobolev spaces adapted to the Schrödinger operator with inverse-square potential, Math. Z., 288 (2018), 1273-1298.  doi: 10.1007/s00209-017-1934-8.  Google Scholar

[23]

R. KillipC. X. MiaoM. VisanJ. Y. Zhang and J. Q. Zheng, The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst., 37 (2017), 3831-3866.  doi: 10.3934/dcds.2017162.  Google Scholar

[24]

R. KillipJ. MurphyM. Visan and J. Q. Zheng, The focusing cubic NLS with inverse-square potential in three space dimensions, Differential Integral Equations, 30 (2017), 161-206.   Google Scholar

[25]

M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.  doi: 10.1007/s00033-011-0145-0.  Google Scholar

[26]

I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Differential Equations, 79 (1989), 340-381.  doi: 10.1016/0022-0396(89)90107-1.  Google Scholar

[27]

I. Lasiecka, Finite-dimensionality of attractors associated with von Kármán plate equations and boundary damping, J. Differential Equations, 117 (1995), 357-389.  doi: 10.1006/jdeq.1995.1057.  Google Scholar

[28]

J. M. L. Leblond, Electron capture by polar molecules, Phys. Rev., 153 (1967), 1-4.   Google Scholar

[29]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_tt = -Au+{F}(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.  Google Scholar

[30]

H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015.  Google Scholar

[31]

E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[32]

V. Liskevich, A. Shishkov and Z. Sobol, Singular solutions to the heat equations with nonlinear absorption and Hardy potentials, Commun. Contemp. Math., 14 (2012), 1250013, 28 pp. doi: 10.1142/S0219199712500137.  Google Scholar

[33]

G. W. Liu and H. W. Zhang, Well-posedness for a class of wave equation with past history and a delay, Z. Angew. Math. Phys., 67 (2016), Art. 6, 14 pp. doi: 10.1007/s00033-015-0593-z.  Google Scholar

[34]

T. T. Liu and Q. Z. Ma, Time-dependent asymptotic behavior of the solution for plate equations with linear memory, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4595-4616.  doi: 10.3934/dcdsb.2018178.  Google Scholar

[35]

Y. C. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169.  doi: 10.1016/S0022-0396(02)00020-7.  Google Scholar

[36]

Y. C. Liu and R. Z. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations, 244 (2008), 200-228.  doi: 10.1016/j.jde.2007.10.015.  Google Scholar

[37]

Y. Q. Liu and Y. Ueda, Decay estimate and asymptotic profile for a plate equation with memory, J. Differential Equations, 268 (2020), 2435-2463.  doi: 10.1016/j.jde.2019.09.007.  Google Scholar

[38]

Z. Y. Liu and S. M. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, 398. Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar

[39]

C. X. MiaoJ. Murphy and J. Q. Zheng, The energy-critical nonlinear wave equation with an inverse-square potential, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 417-456.  doi: 10.1016/j.anihpc.2019.09.004.  Google Scholar

[40]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.  Google Scholar

[41]

K. Ono, Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations, 137 (1997), 273-301.  doi: 10.1006/jdeq.1997.3263.  Google Scholar

[42]

J. J. Pan and J. Zhang, On the minimal mass blow-up solutions for the nonlinear Schrödinger equation with Hardy potential, Nonlinear Anal., 197 (2020), 111829. doi: 10.1016/j.na.2020.111829.  Google Scholar

[43]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[44]

R. Racke and Y. Ueda, Nonlinear thermoelastic plate equations-global existence and decay rates for the Cauchy problem, J. Differential Equations, 263 (2017), 8138-8177.  doi: 10.1016/j.jde.2017.08.036.  Google Scholar

[45]

B. Ben SlimeneS. Tayachi and F. B. Weissler, Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations, Nonlinear Anal., 152 (2017), 116-148.  doi: 10.1016/j.na.2016.12.008.  Google Scholar

[46]

H. Spohn, On the Vlasov hierarchy, Math. Methods Appl. Sci., 3 (1981), 445-455.  doi: 10.1002/mma.1670030131.  Google Scholar

[47]

T. Suzuki, Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities, Funkcial. Ekvac., 59 (2016), 1-34.  doi: 10.1619/fesi.59.1.  Google Scholar

[48]

T. Suzuki, Semilinear Schrödinger equations with a potential of some critical inverse-square type, J. Differential Equations, 268 (2020), 7629-7668.  doi: 10.1016/j.jde.2019.11.087.  Google Scholar

[49]

J. Vancostenoble and E. Zuazua, Hardy inequalities, observability, and control for the wave and Schrödinger equations with singular potentials, SIAM J. Math. Anal., 41 (2009), 1508-1532.  doi: 10.1137/080731396.  Google Scholar

[50]

X. F. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.  doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar

[51]

R. Z. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[52]

J. ZhangS. J. Zheng and S. H. Zhu, Orbital stability of standing waves for fractional Hartree equation with unbounded potentials, Nonlinear Dispersive Waves and Fluids, Contemp. Math., Amer. Math. Soc., Providence, RI, 725 (2019), 265-275.  doi: 10.1090/conm/725/14561.  Google Scholar

[53]

J. Y. Zhang and J. Q. Zheng, Scattering theory for nonlinear Schrödinger equations with inverse-square potential, J. Funct. Anal., 267 (2014), 2907-2932.  doi: 10.1016/j.jfa.2014.08.012.  Google Scholar

[54]

S. M. Zheng, Nonlinear Evolution Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 133. Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203492222.  Google Scholar

[55]

J. Zhou, Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping, Appl. Math. Comput., 265 (2015), 807-818.  doi: 10.1016/j.amc.2015.05.098.  Google Scholar

show all references

References:
[1]

B. AbdellaouiI. Peral and A. Primo, Influence of the Hardy potential in a semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 897-926.  doi: 10.1017/S0308210508000152.  Google Scholar

[2]

B. AbdellaouiI. Peral and A. Primo, Strong regularizing effect of a gradient term in the heat equation with the Hardy potential, J. Funct. Anal., 258 (2010), 1247-1272.  doi: 10.1016/j.jfa.2009.11.008.  Google Scholar

[3]

A. AttarS. Merchán and I. Peral, A remark on the existence properties of a semilinear heat equation involving a Hardy-Leray potential, J. Evol. Equ., 15 (2015), 239-250.  doi: 10.1007/s00028-014-0259-x.  Google Scholar

[4]

M. Bawin, Electron-bound states in the field of dipolar molecules, Phys. Rev. A, 70 (2004), 022505. doi: 10.1103/PhysRevA.70.022505.  Google Scholar

[5]

M. Bawin and S. A. Coon, Neutral atom and a charged wire: From elastic scattering to absorption, Phys. Rev. A, 63 (2001), 034701. doi: 10.1103/PhysRevA.63.034701.  Google Scholar

[6]

M. Bawin and S. A. Coon, Singular inverse square potential, limit cycles, and self-adjoint extensions, Phys. Rev. A, 67 (2003), 042712. doi: 10.1103/PhysRevA.67.042712.  Google Scholar

[7]

M. Bawin, S. A. Coon and B. R. Holstein, Anions and anomalies, Int. J. Mod. Phys. A, (2007), 19–28. doi: 10.1142/9789812770301_0003.  Google Scholar

[8]

S. R. Beane, P. F. Bedaque, L. Childress, A. Kryjevski, J. McGuire and U. van Kolck, Singular potentials and limit cycles, Phys. Rev. A, 64 (2001), 042103. doi: 10.1103/PhysRevA.64.042103.  Google Scholar

[9]

D. BirminghamK. S. Gupta and S. Sen, Near-horizon conformal structure of black holes, Phys. Lett. B, 55 (2001), 191-196.  doi: 10.1016/S0370-2693(01)00354-9.  Google Scholar

[10]

H. Chen and S. Y. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.  Google Scholar

[11]

E. CsoboF. GenoudM. Ohta and J. Royer, Stability of standing waves for a nonlinear Klein-Gordon equation with delta potentials, J. Differential Equations, 268 (2019), 353-388.  doi: 10.1016/j.jde.2019.08.015.  Google Scholar

[12]

J. DenschlagG. Umshaus and J. Schmiedmayer, Probing a singular potential with cold atoms: A neutral atom and a charged wire, Phys. Rev. Lett., 81 (1998), 737-741.  doi: 10.1103/PhysRevLett.81.737.  Google Scholar

[13]

V. D. Dinh, Global existence and blowup for a class of the focusing nonlinear Schrödinger equation with inverse-square potential, J. Math. Anal. Appl., 468 (2018), 270-303.  doi: 10.1016/j.jmaa.2018.08.006.  Google Scholar

[14]

E. H. Dowell, Aeroelasticity of Plates and Shells, Nordhoff, Leyden, 1973. Google Scholar

[15]

E. H. Dowell, A Modern Course in Aeroelasticity, Solid Mechanics and its Applications, 217. Springer, Cham, 2015. doi: 10.1007/978-3-319-09453-3.  Google Scholar

[16]

V. Efimov, Weakly-bound states of 3 resonantly-interacting particles, Sov. J. Nucl. Phys., 12 (1971), 589-595.   Google Scholar

[17]

P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping, J. Differential Equations, 132 (1996), 338-352.  doi: 10.1006/jdeq.1996.0183.  Google Scholar

[18]

J. Fröhlich and E. Lenzmann., Mean-field limit of quantum Bose gases and nonlinear Hartree equation, Séminaire: Équations aux Dérivées Partielles, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2004), 26 pp.  Google Scholar

[19]

T. R. GovindarajanV. Suneeta and S. Vaidya, Horizon states for AdS black holes, Nucl. Phys. B, 583 (2000), 291-303.  doi: 10.1016/S0550-3213(00)00336-9.  Google Scholar

[20]

Y. Q. Guo and M. A. Rammaha, Systems of nonlinear wave equations with damping and supercritical boundary and interior sources, Trans. Amer. Math. Soc., 366 (2014), 2265-2325.  doi: 10.1090/S0002-9947-2014-05772-3.  Google Scholar

[21]

A. Khanmamedov and S. Simsek, Existence of the global attractor for the plate equation with nonlocal nonlinearity in $\mathbb{R}^n$, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 151-172.  doi: 10.3934/dcdsb.2016.21.151.  Google Scholar

[22]

R. KillipC. MiaoM. VisanJ. Zhang and J. Zheng, Sobolev spaces adapted to the Schrödinger operator with inverse-square potential, Math. Z., 288 (2018), 1273-1298.  doi: 10.1007/s00209-017-1934-8.  Google Scholar

[23]

R. KillipC. X. MiaoM. VisanJ. Y. Zhang and J. Q. Zheng, The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst., 37 (2017), 3831-3866.  doi: 10.3934/dcds.2017162.  Google Scholar

[24]

R. KillipJ. MurphyM. Visan and J. Q. Zheng, The focusing cubic NLS with inverse-square potential in three space dimensions, Differential Integral Equations, 30 (2017), 161-206.   Google Scholar

[25]

M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.  doi: 10.1007/s00033-011-0145-0.  Google Scholar

[26]

I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Differential Equations, 79 (1989), 340-381.  doi: 10.1016/0022-0396(89)90107-1.  Google Scholar

[27]

I. Lasiecka, Finite-dimensionality of attractors associated with von Kármán plate equations and boundary damping, J. Differential Equations, 117 (1995), 357-389.  doi: 10.1006/jdeq.1995.1057.  Google Scholar

[28]

J. M. L. Leblond, Electron capture by polar molecules, Phys. Rev., 153 (1967), 1-4.   Google Scholar

[29]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_tt = -Au+{F}(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.  Google Scholar

[30]

H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015.  Google Scholar

[31]

E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[32]

V. Liskevich, A. Shishkov and Z. Sobol, Singular solutions to the heat equations with nonlinear absorption and Hardy potentials, Commun. Contemp. Math., 14 (2012), 1250013, 28 pp. doi: 10.1142/S0219199712500137.  Google Scholar

[33]

G. W. Liu and H. W. Zhang, Well-posedness for a class of wave equation with past history and a delay, Z. Angew. Math. Phys., 67 (2016), Art. 6, 14 pp. doi: 10.1007/s00033-015-0593-z.  Google Scholar

[34]

T. T. Liu and Q. Z. Ma, Time-dependent asymptotic behavior of the solution for plate equations with linear memory, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4595-4616.  doi: 10.3934/dcdsb.2018178.  Google Scholar

[35]

Y. C. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169.  doi: 10.1016/S0022-0396(02)00020-7.  Google Scholar

[36]

Y. C. Liu and R. Z. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations, 244 (2008), 200-228.  doi: 10.1016/j.jde.2007.10.015.  Google Scholar

[37]

Y. Q. Liu and Y. Ueda, Decay estimate and asymptotic profile for a plate equation with memory, J. Differential Equations, 268 (2020), 2435-2463.  doi: 10.1016/j.jde.2019.09.007.  Google Scholar

[38]

Z. Y. Liu and S. M. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, 398. Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar

[39]

C. X. MiaoJ. Murphy and J. Q. Zheng, The energy-critical nonlinear wave equation with an inverse-square potential, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 417-456.  doi: 10.1016/j.anihpc.2019.09.004.  Google Scholar

[40]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.  Google Scholar

[41]

K. Ono, Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations, 137 (1997), 273-301.  doi: 10.1006/jdeq.1997.3263.  Google Scholar

[42]

J. J. Pan and J. Zhang, On the minimal mass blow-up solutions for the nonlinear Schrödinger equation with Hardy potential, Nonlinear Anal., 197 (2020), 111829. doi: 10.1016/j.na.2020.111829.  Google Scholar

[43]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[44]

R. Racke and Y. Ueda, Nonlinear thermoelastic plate equations-global existence and decay rates for the Cauchy problem, J. Differential Equations, 263 (2017), 8138-8177.  doi: 10.1016/j.jde.2017.08.036.  Google Scholar

[45]

B. Ben SlimeneS. Tayachi and F. B. Weissler, Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations, Nonlinear Anal., 152 (2017), 116-148.  doi: 10.1016/j.na.2016.12.008.  Google Scholar

[46]

H. Spohn, On the Vlasov hierarchy, Math. Methods Appl. Sci., 3 (1981), 445-455.  doi: 10.1002/mma.1670030131.  Google Scholar

[47]

T. Suzuki, Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities, Funkcial. Ekvac., 59 (2016), 1-34.  doi: 10.1619/fesi.59.1.  Google Scholar

[48]

T. Suzuki, Semilinear Schrödinger equations with a potential of some critical inverse-square type, J. Differential Equations, 268 (2020), 7629-7668.  doi: 10.1016/j.jde.2019.11.087.  Google Scholar

[49]

J. Vancostenoble and E. Zuazua, Hardy inequalities, observability, and control for the wave and Schrödinger equations with singular potentials, SIAM J. Math. Anal., 41 (2009), 1508-1532.  doi: 10.1137/080731396.  Google Scholar

[50]

X. F. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.  doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar

[51]

R. Z. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[52]

J. ZhangS. J. Zheng and S. H. Zhu, Orbital stability of standing waves for fractional Hartree equation with unbounded potentials, Nonlinear Dispersive Waves and Fluids, Contemp. Math., Amer. Math. Soc., Providence, RI, 725 (2019), 265-275.  doi: 10.1090/conm/725/14561.  Google Scholar

[53]

J. Y. Zhang and J. Q. Zheng, Scattering theory for nonlinear Schrödinger equations with inverse-square potential, J. Funct. Anal., 267 (2014), 2907-2932.  doi: 10.1016/j.jfa.2014.08.012.  Google Scholar

[54]

S. M. Zheng, Nonlinear Evolution Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 133. Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203492222.  Google Scholar

[55]

J. Zhou, Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping, Appl. Math. Comput., 265 (2015), 807-818.  doi: 10.1016/j.amc.2015.05.098.  Google Scholar

[1]

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