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March  2020, 28(1): 263-289. doi: 10.3934/era.2020016

The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term

College of Science, Henan University of Technology, Zhengzhou 450001, China

* Corresponding author: Gongwei Liu

Received  November 2019

Fund Project: The first author is supported by NSFC (No. 11801145), Key Scientific Research Foundation of the Higher Education Institutions of Henan Province, China (Grant No.19A110004 and the Fund of Young Backbone Teacher in Henan Province (NO. 2018GGJS068, 21420048)

In this paper, we consider a plate equation with nonlinear damping and logarithmic source term. By the contraction mapping principle, we establish the local existence. The global existence and decay estimate of the solution at subcritical initial energy are obtained. We also prove that the solution with negative initial energy blows up in finite time under suitable conditions. Moreover, we also give the blow-up in finite time of solution at the arbitrarily high initial energy for linear damping (i.e. $ m = 2 $).

Citation: Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016
References:
[1]

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

[2]

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

[3]

M. M. Al-GharabliA. Guesmia and S. A. Messaoudi, Existence and a general decay results for a viscoelastic plate equation logarithmic nonlinearity, Commun. Pure Appl. Anal., 18 (2019), 159-180.  doi: 10.3934/cpaa.2019009.  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. Białynicki-Birula and J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 23 (1975), 461-466.   Google Scholar

[7]

I. Białynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.  Google Scholar

[8]

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

[9]

H. ChenP. Luo and G. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

P. Górka, Logarithmic Klein-Gordon equation, Acta Phys. Polon. B, 40 (2009), 59-66.   Google Scholar

[16]

P. GórkaH. Prado and E. G. Reyes, Nonlinear equations with infinitely many derivatives, Complex Anal. Oper. Theory, 5 (2011), 313-323.  doi: 10.1007/s11785-009-0043-z.  Google Scholar

[17]

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

[18]

Y. HeH. Gao and H. Wang, Blow-up and decay for a class of pseudo-parabolic $p$-Laplacian equation with logarithmic nonlinearity, Comput. Math. Appl., 75 (2018), 459-469.  doi: 10.1016/j.camwa.2017.09.027.  Google Scholar

[19]

T. Hiramatsu, M. Kawasaki and F. Takahashi, Numerical study of Q-ball formation in gravity mediation, Journal of Cosmology and Astroparticle Physics, 2010 (2010), 008. Google Scholar

[20]

Q. HuH. Zhang and G. Liu, Asymptotic behavior for a class of logarithmic wave equations with linear damping, Appl. Math. Optim., 79 (2019), 131-144.  doi: 10.1007/s00245-017-9423-3.  Google Scholar

[21]

R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475-491.  doi: 10.32917/hmj/1206127254.  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 (2019), 530-547.  doi: 10.1080/00036811.2018.1504029.  Google Scholar

[23]

C. N. Le and X. T. Le, Global solution and blow-up for a class of $p$-Laplacian evolution equations with logarithmic nonlinearity, Acta Appl. Math., 151 (2017), 149-169.  doi: 10.1007/s10440-017-0106-5.  Google Scholar

[24]

C. N. Le and X. T. Le, Global solution and blow-up for a class of pseudo p-Laplacian evolution equations with logarithmic nonlinearity, Comput. Math. Appl., 73 (2017), 2076-2091.  doi: 10.1016/j.camwa.2017.02.030.  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. 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

[27]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[28]

W. Liu and H. Zhuang, Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms, NoDEA Nonlinear Differential Equations Appl., 24 2017, Art. 67, 35 pp. doi: 10.1007/s00030-017-0491-5.  Google Scholar

[29]

Y. 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

[30]

Y. Liu and R. Xu, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 171-189.  doi: 10.3934/dcdsb.2007.7.171.  Google Scholar

[31]

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

[32]

S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58-66.  doi: 10.1002/mana.200310104.  Google Scholar

[33]

M. Nakao, A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan, 30 (1978), 747-762.  doi: 10.2969/jmsj/03040747.  Google Scholar

[34]

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

[35]

A. Peyravi, General stability and exponential growth for a class of semi-linear wave equations with logarithmic source and memeory terms, Appl. Math. Optim., (2018). doi: 10.1007/s00245-018-9508-7.  Google Scholar

[36]

V. S. Vladimirov, The equation of the $p$-adic open string for the scalar tachyon field, Izv. Math., 69 (2005), 487-512.  doi: 10.1070/IM2005v069n03ABEH000536.  Google Scholar

[37]

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

[38]

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.  Google Scholar

[39]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[40]

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

show all references

References:
[1]

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

[2]

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

[3]

M. M. Al-GharabliA. Guesmia and S. A. Messaoudi, Existence and a general decay results for a viscoelastic plate equation logarithmic nonlinearity, Commun. Pure Appl. Anal., 18 (2019), 159-180.  doi: 10.3934/cpaa.2019009.  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. Białynicki-Birula and J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 23 (1975), 461-466.   Google Scholar

[7]

I. Białynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.  Google Scholar

[8]

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

[9]

H. ChenP. Luo and G. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

P. Górka, Logarithmic Klein-Gordon equation, Acta Phys. Polon. B, 40 (2009), 59-66.   Google Scholar

[16]

P. GórkaH. Prado and E. G. Reyes, Nonlinear equations with infinitely many derivatives, Complex Anal. Oper. Theory, 5 (2011), 313-323.  doi: 10.1007/s11785-009-0043-z.  Google Scholar

[17]

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

[18]

Y. HeH. Gao and H. Wang, Blow-up and decay for a class of pseudo-parabolic $p$-Laplacian equation with logarithmic nonlinearity, Comput. Math. Appl., 75 (2018), 459-469.  doi: 10.1016/j.camwa.2017.09.027.  Google Scholar

[19]

T. Hiramatsu, M. Kawasaki and F. Takahashi, Numerical study of Q-ball formation in gravity mediation, Journal of Cosmology and Astroparticle Physics, 2010 (2010), 008. Google Scholar

[20]

Q. HuH. Zhang and G. Liu, Asymptotic behavior for a class of logarithmic wave equations with linear damping, Appl. Math. Optim., 79 (2019), 131-144.  doi: 10.1007/s00245-017-9423-3.  Google Scholar

[21]

R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475-491.  doi: 10.32917/hmj/1206127254.  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 (2019), 530-547.  doi: 10.1080/00036811.2018.1504029.  Google Scholar

[23]

C. N. Le and X. T. Le, Global solution and blow-up for a class of $p$-Laplacian evolution equations with logarithmic nonlinearity, Acta Appl. Math., 151 (2017), 149-169.  doi: 10.1007/s10440-017-0106-5.  Google Scholar

[24]

C. N. Le and X. T. Le, Global solution and blow-up for a class of pseudo p-Laplacian evolution equations with logarithmic nonlinearity, Comput. Math. Appl., 73 (2017), 2076-2091.  doi: 10.1016/j.camwa.2017.02.030.  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. 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

[27]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[28]

W. Liu and H. Zhuang, Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms, NoDEA Nonlinear Differential Equations Appl., 24 2017, Art. 67, 35 pp. doi: 10.1007/s00030-017-0491-5.  Google Scholar

[29]

Y. 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

[30]

Y. Liu and R. Xu, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 171-189.  doi: 10.3934/dcdsb.2007.7.171.  Google Scholar

[31]

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

[32]

S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58-66.  doi: 10.1002/mana.200310104.  Google Scholar

[33]

M. Nakao, A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan, 30 (1978), 747-762.  doi: 10.2969/jmsj/03040747.  Google Scholar

[34]

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

[35]

A. Peyravi, General stability and exponential growth for a class of semi-linear wave equations with logarithmic source and memeory terms, Appl. Math. Optim., (2018). doi: 10.1007/s00245-018-9508-7.  Google Scholar

[36]

V. S. Vladimirov, The equation of the $p$-adic open string for the scalar tachyon field, Izv. Math., 69 (2005), 487-512.  doi: 10.1070/IM2005v069n03ABEH000536.  Google Scholar

[37]

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

[38]

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.  Google Scholar

[39]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[40]

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

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