December  2021, 14(12): 4439-4463. doi: 10.3934/dcdss.2021115

Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms

College of Mathematical Sciences, Harbin Engineering University, Heilongjiang, Harbin 150001, China

* Corresponding author: Xingchang Wang

Received  August 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

Fund Project: The first author is supported by the Ph.D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (3072021CF2404), the second author is supported by NSFC grant 11871017

We consider the well-posedness of solution of the initial boundary value problem to the fourth order wave equation with the strong and weak damping terms, and the logarithmic strain term, which was introduced to describe many complex physical processes. The local solution is obtained with the help of the Galerkin method and the contraction mapping principle. The global solution and the blowup solution in infinite time under sub-critical initial energy are also established, and then these results are extended in parallel to the critical initial energy. Finally, the infinite time blowup of solution is proved at the arbitrary positive initial energy.

Citation: Yue Pang, Xingchang Wang, Furong Wu. Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4439-4463. doi: 10.3934/dcdss.2021115
References:
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C. AlvesA. Moussaoui and L. Tavares, An elliptic system with logarithmic nonlinearity, Adv. Nonlinear Anal., 8 (2019), 928-945.  doi: 10.1515/anona-2017-0200.

[3]

L. J. An, Loss of hyperbolicity in elastic-plastic material at finite strains, SIAM J. Appl. Math., 53 (1993), 621-654.  doi: 10.1137/0153032.

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L. J. An and A. Peirce, The effect of microstructure on elastic-plastic models, SIAM J. Appl. Math., 54 (1994), 708-730.  doi: 10.1137/S0036139992238498.

[5]

L. J. An and A. Peirce, A weakly nonlinear analysis of elastoplastic-microstructure models, SIAM J. Appl. Math., 55 (1995), 136-155.  doi: 10.1137/S0036139993255327.

[6]

G. Andrews, On the existence of solutions to the equation $u_tt = u_xxt+\sigma (u_{x})_{x}$, J. Differential Equations, 35 (1980), 200-231.  doi: 10.1016/0022-0396(80)90040-6.

[7]

F. P. Bretherton, Resonant interactions between waves. The case of discrete oscillations, J. Fluid Mech., 20 (1964), 457-479.  doi: 10.1017/S0022112064001355.

[8]

E. Brué and Q.-H. Nguyen, On the Sobolev space of functions with derivative of logarithmic order, Adv. Nonlinear Anal., 9 (2020), 836-849.  doi: 10.1515/anona-2020-0027.

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P. BugielS. A. Wedrychowicz and B. Rzepka, Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in $\Bbb mathbb{R}^{d}$, Adv. Nonlinear Anal., 10 (2021), 972-981.  doi: 10.1515/anona-2020-0163.

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H. Buljan, A. Šiber, M. Soljačić, T. Schwartz, M. Segev and D. N. Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E (3), 68 (2003), 036607, 6 pp. doi: 10.1103/PhysRevE.68.036607.

[11]

H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.

[12]

Y. Chen and R. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal., 192 (2020), 111664, 39 pp. doi: 10.1016/j.na.2019.111664.

[13]

I. Chueshov and I. Lasiecka, Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Contin. Dyn. Syst., 15 (2006), 777-809.  doi: 10.3934/dcds.2006.15.777.

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I. Chueshov and I. Lasiecka, On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Comm. Partial Differential Equations, 36 (2011), 67-99.  doi: 10.1080/03605302.2010.484472.

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L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493-516.  doi: 10.1016/S0294-1449(98)80032-2.

[16]

S. De MartinoM. FalangaC. Godano and G. Lauro, Logarithmic Schrödinger-like equation as a model for magma transport, EPL, 63 (2003), 472-475.  doi: 10.1209/epl/i2003-00547-6.

[17]

H. DiY. Shang and J. Yu, Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source, Electron. Res. Arch., 28 (2020), 221-261.  doi: 10.3934/era.2020015.

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Z. Ding, Traveling waves in a suspension bridge system, SIAM J. Math. Anal., 35 (2003), 160-171.  doi: 10.1137/S0036141002412690.

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L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688.

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W. HeD. Qin and Q. Wu, Existence, multiplicity and nonexistence results for Kirchhoff type equations, Adv. Nonlinear Anal., 10 (2021), 616-635.  doi: 10.1515/anona-2020-0154.

[21]

A. C. Lazer and P. J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 243-274.  doi: 10.1016/S0294-1449(16)30368-7.

[22]

G. LiY. Chen and Y. Huang, A hybridized weak Galerkin finite element scheme for general second-order elliptic problems, Electron. Res. Arch., 28 (2020), 821-836.  doi: 10.3934/era.2020042.

[23]

W. LianM. S. Ahmed and R. Xu, Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity, Opuscula Math., 40 (2020), 111-130.  doi: 10.7494/OpMath.2020.40.1.111.

[24]

W. LianV. D. RădulescuR. XuY. Yang and N. Zhao, Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Adv. Calc. Var., 14 (2021), 589-611.  doi: 10.1515/acv-2019-0039.

[25]

W. LianJ. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.

[26]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.

[27]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[28]

Q. LinX. TianR. Xu and M. Zhang, Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2095-2107.  doi: 10.3934/dcdss.2020160.

[29]

A. Linde, Strings, textures, inflation and spectrum bending, Phys. Lett. B, 284 (1992), 215-222.  doi: 10.1016/0370-2693(92)90423-2.

[30]

X. Liu and J. Zhou, Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity, Electron. Res. Arch., 28 (2020), 599-625.  doi: 10.3934/era.2020032.

[31]

Y. Liu and R. Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl., 331 (2007), 585-607.  doi: 10.1016/j.jmaa.2006.09.010.

[32]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal., 98 (1987), 167-177.  doi: 10.1007/BF00251232.

[33]

R. L. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability, Arch. Rational Mech. Anal., 97 (1987), 353-394.  doi: 10.1007/BF00280411.

[34]

J. Shen, Y. Yang, S. Chen and R. Xu, Finite time blow up of fourth-order wave equations with nonlinear strain and source terms at high energy level, Internat. J. Math., 24 (2013), 1350043, 8 pp. doi: 10.1142/S0129167X13500432.

[35]

M.-P. Tran and T.-N. Nguyen, Pointwise gradient bounds for a class of very singular quasilinear elliptic equations, Discrete Contin. Dyn. Syst., 41 (2021), 4461-4476.  doi: 10.3934/dcds.2021043.

[36]

V. V. Varlamov, On the initial-boundary value problem for the damped Boussinesq equation, Discrete Contin. Dynam. Systems, 4 (1998), 431-444.  doi: 10.3934/dcds.1998.4.431.

[37]

X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288.  doi: 10.1515/anona-2020-0141.

[38]

Y. Wang and Y. Wang, On the initial-boundary problem for fourth order wave equations with damping, strain and source terms, J. Math. Anal. Appl., 405 (2013), 116-127.  doi: 10.1016/j.jmaa.2013.03.060.

[39]

R. XuW. LianX. Kong and Y. Yang, Fourth order wave equation with nonlinear strain and logarithmic nonlinearity, Appl. Numer. Math., 141 (2019), 185-205.  doi: 10.1016/j.apnum.2018.06.004.

[40]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.

[41]

R. 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.

[42]

R. Xu, X. Wang, Y. Yang and S. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728.

[43]

Y. YangM. Salik AhmedL. Qin and R. Xu, Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations, Opuscula Math., 39 (2019), 297-313.  doi: 10.7494/OpMath.2019.39.2.297.

[44]

Y. Zeng and K. Zhao, On the logarithmic Keller-Segel-Fisher/KPP system, Discrete Contin. Dyn. Syst., 39 (2019), 5365-5402.  doi: 10.3934/dcds.2019220.

[45]

M. ZhangQ. ZhaoY. Liu and W. Li, Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition, Electron. Res. Arch., 28 (2020), 369-381.  doi: 10.3934/era.2020021.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.

[2]

C. AlvesA. Moussaoui and L. Tavares, An elliptic system with logarithmic nonlinearity, Adv. Nonlinear Anal., 8 (2019), 928-945.  doi: 10.1515/anona-2017-0200.

[3]

L. J. An, Loss of hyperbolicity in elastic-plastic material at finite strains, SIAM J. Appl. Math., 53 (1993), 621-654.  doi: 10.1137/0153032.

[4]

L. J. An and A. Peirce, The effect of microstructure on elastic-plastic models, SIAM J. Appl. Math., 54 (1994), 708-730.  doi: 10.1137/S0036139992238498.

[5]

L. J. An and A. Peirce, A weakly nonlinear analysis of elastoplastic-microstructure models, SIAM J. Appl. Math., 55 (1995), 136-155.  doi: 10.1137/S0036139993255327.

[6]

G. Andrews, On the existence of solutions to the equation $u_tt = u_xxt+\sigma (u_{x})_{x}$, J. Differential Equations, 35 (1980), 200-231.  doi: 10.1016/0022-0396(80)90040-6.

[7]

F. P. Bretherton, Resonant interactions between waves. The case of discrete oscillations, J. Fluid Mech., 20 (1964), 457-479.  doi: 10.1017/S0022112064001355.

[8]

E. Brué and Q.-H. Nguyen, On the Sobolev space of functions with derivative of logarithmic order, Adv. Nonlinear Anal., 9 (2020), 836-849.  doi: 10.1515/anona-2020-0027.

[9]

P. BugielS. A. Wedrychowicz and B. Rzepka, Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in $\Bbb mathbb{R}^{d}$, Adv. Nonlinear Anal., 10 (2021), 972-981.  doi: 10.1515/anona-2020-0163.

[10]

H. Buljan, A. Šiber, M. Soljačić, T. Schwartz, M. Segev and D. N. Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E (3), 68 (2003), 036607, 6 pp. doi: 10.1103/PhysRevE.68.036607.

[11]

H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.

[12]

Y. Chen and R. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal., 192 (2020), 111664, 39 pp. doi: 10.1016/j.na.2019.111664.

[13]

I. Chueshov and I. Lasiecka, Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Contin. Dyn. Syst., 15 (2006), 777-809.  doi: 10.3934/dcds.2006.15.777.

[14]

I. Chueshov and I. Lasiecka, On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Comm. Partial Differential Equations, 36 (2011), 67-99.  doi: 10.1080/03605302.2010.484472.

[15]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493-516.  doi: 10.1016/S0294-1449(98)80032-2.

[16]

S. De MartinoM. FalangaC. Godano and G. Lauro, Logarithmic Schrödinger-like equation as a model for magma transport, EPL, 63 (2003), 472-475.  doi: 10.1209/epl/i2003-00547-6.

[17]

H. DiY. Shang and J. Yu, Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source, Electron. Res. Arch., 28 (2020), 221-261.  doi: 10.3934/era.2020015.

[18]

Z. Ding, Traveling waves in a suspension bridge system, SIAM J. Math. Anal., 35 (2003), 160-171.  doi: 10.1137/S0036141002412690.

[19]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688.

[20]

W. HeD. Qin and Q. Wu, Existence, multiplicity and nonexistence results for Kirchhoff type equations, Adv. Nonlinear Anal., 10 (2021), 616-635.  doi: 10.1515/anona-2020-0154.

[21]

A. C. Lazer and P. J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 243-274.  doi: 10.1016/S0294-1449(16)30368-7.

[22]

G. LiY. Chen and Y. Huang, A hybridized weak Galerkin finite element scheme for general second-order elliptic problems, Electron. Res. Arch., 28 (2020), 821-836.  doi: 10.3934/era.2020042.

[23]

W. LianM. S. Ahmed and R. Xu, Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity, Opuscula Math., 40 (2020), 111-130.  doi: 10.7494/OpMath.2020.40.1.111.

[24]

W. LianV. D. RădulescuR. XuY. Yang and N. Zhao, Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Adv. Calc. Var., 14 (2021), 589-611.  doi: 10.1515/acv-2019-0039.

[25]

W. LianJ. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.

[26]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.

[27]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[28]

Q. LinX. TianR. Xu and M. Zhang, Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2095-2107.  doi: 10.3934/dcdss.2020160.

[29]

A. Linde, Strings, textures, inflation and spectrum bending, Phys. Lett. B, 284 (1992), 215-222.  doi: 10.1016/0370-2693(92)90423-2.

[30]

X. Liu and J. Zhou, Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity, Electron. Res. Arch., 28 (2020), 599-625.  doi: 10.3934/era.2020032.

[31]

Y. Liu and R. Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl., 331 (2007), 585-607.  doi: 10.1016/j.jmaa.2006.09.010.

[32]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal., 98 (1987), 167-177.  doi: 10.1007/BF00251232.

[33]

R. L. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability, Arch. Rational Mech. Anal., 97 (1987), 353-394.  doi: 10.1007/BF00280411.

[34]

J. Shen, Y. Yang, S. Chen and R. Xu, Finite time blow up of fourth-order wave equations with nonlinear strain and source terms at high energy level, Internat. J. Math., 24 (2013), 1350043, 8 pp. doi: 10.1142/S0129167X13500432.

[35]

M.-P. Tran and T.-N. Nguyen, Pointwise gradient bounds for a class of very singular quasilinear elliptic equations, Discrete Contin. Dyn. Syst., 41 (2021), 4461-4476.  doi: 10.3934/dcds.2021043.

[36]

V. V. Varlamov, On the initial-boundary value problem for the damped Boussinesq equation, Discrete Contin. Dynam. Systems, 4 (1998), 431-444.  doi: 10.3934/dcds.1998.4.431.

[37]

X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288.  doi: 10.1515/anona-2020-0141.

[38]

Y. Wang and Y. Wang, On the initial-boundary problem for fourth order wave equations with damping, strain and source terms, J. Math. Anal. Appl., 405 (2013), 116-127.  doi: 10.1016/j.jmaa.2013.03.060.

[39]

R. XuW. LianX. Kong and Y. Yang, Fourth order wave equation with nonlinear strain and logarithmic nonlinearity, Appl. Numer. Math., 141 (2019), 185-205.  doi: 10.1016/j.apnum.2018.06.004.

[40]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.

[41]

R. 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.

[42]

R. Xu, X. Wang, Y. Yang and S. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728.

[43]

Y. YangM. Salik AhmedL. Qin and R. Xu, Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations, Opuscula Math., 39 (2019), 297-313.  doi: 10.7494/OpMath.2019.39.2.297.

[44]

Y. Zeng and K. Zhao, On the logarithmic Keller-Segel-Fisher/KPP system, Discrete Contin. Dyn. Syst., 39 (2019), 5365-5402.  doi: 10.3934/dcds.2019220.

[45]

M. ZhangQ. ZhaoY. Liu and W. Li, Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition, Electron. Res. Arch., 28 (2020), 369-381.  doi: 10.3934/era.2020021.

Table 1.  The values for $ m $ and $ m' $
The value Results
$ n $ $ \frac{2n}{n-2} $ $ m\in\left(n,\frac{2n}{n-2}\right) $ $ m'\in\left(n,\frac{2n}{n-2}\right) $ $ \frac{1}{m}+\frac{1}{m'}=\frac{1}{2} $
$ \le 2 $ $ \infty $ $ (n, \infty) $ $ (n,\infty) $ $ (m,m')=(3,6) $ valid
$ 3 $ $ 6 $ $ (3,6) $ $ (3,6) $ $ (m,m')=\left(5,\frac{10}{3}\right) $ valid
$ 4 $ $ 4 $ $ - $ $ - $ $ - $ invalid
$ 5 $ $ \frac{10}{3} $ $ - $ $ - $ $ - $ invalid
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ invalid
The value Results
$ n $ $ \frac{2n}{n-2} $ $ m\in\left(n,\frac{2n}{n-2}\right) $ $ m'\in\left(n,\frac{2n}{n-2}\right) $ $ \frac{1}{m}+\frac{1}{m'}=\frac{1}{2} $
$ \le 2 $ $ \infty $ $ (n, \infty) $ $ (n,\infty) $ $ (m,m')=(3,6) $ valid
$ 3 $ $ 6 $ $ (3,6) $ $ (3,6) $ $ (m,m')=\left(5,\frac{10}{3}\right) $ valid
$ 4 $ $ 4 $ $ - $ $ - $ $ - $ invalid
$ 5 $ $ \frac{10}{3} $ $ - $ $ - $ $ - $ invalid
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ invalid
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