doi: 10.3934/era.2021066
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A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms

School of Science, Changchun University of Science and Technology, Changchun 130022, China

* Corresponding author: Ying Chu

Received  June 2021 Revised  July 2021 Early access September 2021

Fund Project: The second author is supported by the fund of the "Thirteen Five" Scientific and Technological Research Planning Project of the Department of Education of Jilin Province in China [number JJKH20190547KJ and JJKH20200727KJ]

In this paper, we study a class of hyperbolic equations of the fourth order with strong damping and logarithmic source terms. Firstly, we prove the local existence of the weak solution by using the contraction mapping principle. Secondly, in the potential well framework, the global existence of weak solutions and the energy decay estimate are obtained. Finally, we give the blow up result of the solution at a finite time under the subcritical initial energy.

Citation: Yi Cheng, Ying Chu. A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms. Electronic Research Archive, doi: 10.3934/era.2021066
References:
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M. M. Al-Gharabli and S. A. Messaoudi, The existence and the asymptotic behavior of a plate equation with frictional damping and a logarithmic source term, J. Math. Anal. Appl., 454 (2017), 1114-1128.  doi: 10.1016/j.jmaa.2017.05.030.  Google Scholar

[2]

M. M. Al-Gharabli and S. A. Messaoudi, Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term, J. Evol. Equ., 18 (2018), 105-125.  doi: 10.1007/s00028-017-0392-4.  Google Scholar

[3]

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.  Google Scholar

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J. D. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), 5576-5587.  doi: 10.1103/PhysRevD.52.5576.  Google Scholar

[5]

K. Bartkowski and P. Górka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A, 41 (2008), 355201, 11 pp. doi: 10.1088/1751-8113/41/35/355201.  Google Scholar

[6]

I. Bialynicki-Birula and J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 23 (1975), 461-466.   Google Scholar

[7]

G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343 (2008), 1166-1176.  doi: 10.1016/j.jmaa.2008.01.049.  Google Scholar

[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.  Google Scholar

[9]

T. Cazenave and A. Haraux, Équations d'évolution avec non-linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51.  doi: 10.5802/afst.543.  Google Scholar

[10]

J. Chabrowski and J. Marcos do Ó, On some fourth-order semilinear elliptic problems in $R^N$, Nonlinear Anal., 49 (2002), 861-884.  doi: 10.1016/S0362-546X(01)00144-4.  Google Scholar

[11]

H. Chen and S. 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

[12]

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.  Google Scholar

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[14]

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.  Google Scholar

[15]

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[18]

P. Górka, Logarithmic Klein-Gordon equation, Acth Physica Polonica B, 40 (2009), 59-66.   Google Scholar

[19]

X. Han, Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics, Bull. Korean Math. Soc., 50 (2013), 275-283.  doi: 10.4134/BKMS.2013.50.1.275.  Google Scholar

[20]

Y. Han, Finite time blow up for a semilinear pseudo-parabolic equation with general nonlinearity, Appl. Math. Lett., 99 (2020), 105986, 7 pp. doi: 10.1016/j.aml.2019.07.017.  Google Scholar

[21]

T. HiramatsuM. Kawasaki and F. Takahashi, Numerical study of Q-ball formation in gravity mediation, J. Cosmol. Astropart. P., 6 (2010), 008-008.   Google Scholar

[22]

M. Kafini and S. Messaoudi, Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay, Appl. Anal., 99 (2020), 530-547.  doi: 10.1080/00036811.2018.1504029.  Google Scholar

[23]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au+ F(u)$, Arch. Ration. Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.  Google Scholar

[24]

P. Li and C. Liu, A class of fourth-order parabolic equation with logarithmic nonlinearity, J. Inequal. Appl., (2018), Paper No. 328, 21 pp. doi: 10.1186/s13660-018-1920-7.  Google Scholar

[25]

W. LianM. S. Ahmed and R. Xu, Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal., 184 (2019), 239-257.  doi: 10.1016/j.na.2019.02.015.  Google Scholar

[26]

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.  Google Scholar

[27]

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

[28]

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.  Google Scholar

[29]

M. LiaoB. Guo and Q. Li, Global existence and energy decay estimates for weak solutions to the pseudo-parabolic equation with variable exponents, Math. Method. Appl. Sci., 43 (2020), 2516-2527.  doi: 10.1002/mma.6060.  Google Scholar

[30]

J.-L. Lions, Quelque Méthodes de Résolution des Problemes aux Limites non Linéaires, Dunod, Paris, 1969.  Google Scholar

[31]

G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.  Google Scholar

[32]

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.  Google Scholar

[33]

S. A. Messaoudi, Global existence and nonexistence in a system of petrovsky, J. Math. Anal. Appl., 265 (2002), 296-308.  doi: 10.1006/jmaa.2001.7697.  Google Scholar

[34]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.  doi: 10.1088/0951-7715/19/7/001.  Google Scholar

[35]

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

[36]

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.  Google Scholar

[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.  Google Scholar

[38]

S.-T. Wu and L.-Y. Tsai, On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Taiwan. J. Math., 13 (2009), 545-558.  doi: 10.11650/twjm/1500405355.  Google Scholar

[39]

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.  Google Scholar

[40]

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.  Google Scholar

[41]

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.  Google Scholar

[42]

H. ZhangG. Liu and Q. Hu, Exponential decay of energy for a logarithmic wave equation, J. Partial Differ. Equ., 28 (2015), 269-277.  doi: 10.4208/jpde.v28.n3.5.  Google Scholar

[43]

X. Zhu, B. Guo and M. Liao, Global existence and blow-up of weak solutions for a pseudo-parabolic equation with high initial energy, Appl. Math. Lett., 104 (2020), 106270, 7 pp. doi: 10.1016/j.aml.2020.106270.  Google Scholar

show all references

References:
[1]

M. M. Al-Gharabli and S. A. Messaoudi, The existence and the asymptotic behavior of a plate equation with frictional damping and a logarithmic source term, J. Math. Anal. Appl., 454 (2017), 1114-1128.  doi: 10.1016/j.jmaa.2017.05.030.  Google Scholar

[2]

M. M. Al-Gharabli and S. A. Messaoudi, Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term, J. Evol. Equ., 18 (2018), 105-125.  doi: 10.1007/s00028-017-0392-4.  Google Scholar

[3]

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.  Google Scholar

[4]

J. D. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), 5576-5587.  doi: 10.1103/PhysRevD.52.5576.  Google Scholar

[5]

K. Bartkowski and P. Górka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A, 41 (2008), 355201, 11 pp. doi: 10.1088/1751-8113/41/35/355201.  Google Scholar

[6]

I. Bialynicki-Birula and J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 23 (1975), 461-466.   Google Scholar

[7]

G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343 (2008), 1166-1176.  doi: 10.1016/j.jmaa.2008.01.049.  Google Scholar

[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.  Google Scholar

[9]

T. Cazenave and A. Haraux, Équations d'évolution avec non-linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51.  doi: 10.5802/afst.543.  Google Scholar

[10]

J. Chabrowski and J. Marcos do Ó, On some fourth-order semilinear elliptic problems in $R^N$, Nonlinear Anal., 49 (2002), 861-884.  doi: 10.1016/S0362-546X(01)00144-4.  Google Scholar

[11]

H. Chen and S. 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

[12]

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.  Google Scholar

[13]

W. Chen and Y. Zhou, Global nonexistence for a semilinear Petrovsky equation, Nonlinear Anal., 70 (2009), 3203-3208.  doi: 10.1016/j.na.2008.04.024.  Google Scholar

[14]

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.  Google Scholar

[15]

K. Enqvist and J. McDonald, Q-balls and baryogenesis in the MSSM, Phys. Lett. B, 425 (1998), 309-321.  doi: 10.1016/S0370-2693(98)00271-8.  Google Scholar

[16]

L. C. Evans, Partial Differential Equations, Second ed., in: Graduate Studies in Mathematics, vol. 19, 2010. doi: 10.1090/gsm/019.  Google Scholar

[17]

F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar

[18]

P. Górka, Logarithmic Klein-Gordon equation, Acth Physica Polonica B, 40 (2009), 59-66.   Google Scholar

[19]

X. Han, Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics, Bull. Korean Math. Soc., 50 (2013), 275-283.  doi: 10.4134/BKMS.2013.50.1.275.  Google Scholar

[20]

Y. Han, Finite time blow up for a semilinear pseudo-parabolic equation with general nonlinearity, Appl. Math. Lett., 99 (2020), 105986, 7 pp. doi: 10.1016/j.aml.2019.07.017.  Google Scholar

[21]

T. HiramatsuM. Kawasaki and F. Takahashi, Numerical study of Q-ball formation in gravity mediation, J. Cosmol. Astropart. P., 6 (2010), 008-008.   Google Scholar

[22]

M. Kafini and S. Messaoudi, Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay, Appl. Anal., 99 (2020), 530-547.  doi: 10.1080/00036811.2018.1504029.  Google Scholar

[23]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au+ F(u)$, Arch. Ration. Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.  Google Scholar

[24]

P. Li and C. Liu, A class of fourth-order parabolic equation with logarithmic nonlinearity, J. Inequal. Appl., (2018), Paper No. 328, 21 pp. doi: 10.1186/s13660-018-1920-7.  Google Scholar

[25]

W. LianM. S. Ahmed and R. Xu, Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal., 184 (2019), 239-257.  doi: 10.1016/j.na.2019.02.015.  Google Scholar

[26]

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.  Google Scholar

[27]

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

[28]

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.  Google Scholar

[29]

M. LiaoB. Guo and Q. Li, Global existence and energy decay estimates for weak solutions to the pseudo-parabolic equation with variable exponents, Math. Method. Appl. Sci., 43 (2020), 2516-2527.  doi: 10.1002/mma.6060.  Google Scholar

[30]

J.-L. Lions, Quelque Méthodes de Résolution des Problemes aux Limites non Linéaires, Dunod, Paris, 1969.  Google Scholar

[31]

G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.  Google Scholar

[32]

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.  Google Scholar

[33]

S. A. Messaoudi, Global existence and nonexistence in a system of petrovsky, J. Math. Anal. Appl., 265 (2002), 296-308.  doi: 10.1006/jmaa.2001.7697.  Google Scholar

[34]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.  doi: 10.1088/0951-7715/19/7/001.  Google Scholar

[35]

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

[36]

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.  Google Scholar

[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.  Google Scholar

[38]

S.-T. Wu and L.-Y. Tsai, On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Taiwan. J. Math., 13 (2009), 545-558.  doi: 10.11650/twjm/1500405355.  Google Scholar

[39]

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.  Google Scholar

[40]

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.  Google Scholar

[41]

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.  Google Scholar

[42]

H. ZhangG. Liu and Q. Hu, Exponential decay of energy for a logarithmic wave equation, J. Partial Differ. Equ., 28 (2015), 269-277.  doi: 10.4208/jpde.v28.n3.5.  Google Scholar

[43]

X. Zhu, B. Guo and M. Liao, Global existence and blow-up of weak solutions for a pseudo-parabolic equation with high initial energy, Appl. Math. Lett., 104 (2020), 106270, 7 pp. doi: 10.1016/j.aml.2020.106270.  Google Scholar

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