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Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source
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 |
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 $).
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. |
[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. |
[3] |
M. M. Al-Gharabli, A. 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. |
[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. |
[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. |
[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.
|
[7] |
I. Białynicki-Birula and J. Mycielski,
Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93.
doi: 10.1016/0003-4916(76)90057-9. |
[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. |
[9] |
H. Chen, P. 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. |
[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. |
[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] |
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. |
[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. |
[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. |
[15] |
P. Górka,
Logarithmic Klein-Gordon equation, Acta Phys. Polon. B, 40 (2009), 59-66.
|
[16] |
P. Górka, H. 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. |
[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. |
[18] |
Y. He, H. 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. |
[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. Hu, H. 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. |
[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. |
[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. |
[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. |
[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. |
[25] |
W. Lian, M. 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. |
[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] |
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[39] |
R. Xu, M. Zhang, S. Chen, Y. 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. |
[40] |
H. Zhang, G. 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. |
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. |
[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. |
[3] |
M. M. Al-Gharabli, A. 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. |
[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. |
[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. |
[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.
|
[7] |
I. Białynicki-Birula and J. Mycielski,
Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93.
doi: 10.1016/0003-4916(76)90057-9. |
[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. |
[9] |
H. Chen, P. 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. |
[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. |
[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] |
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. |
[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. |
[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. |
[15] |
P. Górka,
Logarithmic Klein-Gordon equation, Acta Phys. Polon. B, 40 (2009), 59-66.
|
[16] |
P. Górka, H. 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. |
[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. |
[18] |
Y. He, H. 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. |
[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. Hu, H. 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. |
[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. |
[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. |
[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. |
[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. |
[25] |
W. Lian, M. 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. |
[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] |
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[39] |
R. Xu, M. Zhang, S. Chen, Y. 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. |
[40] |
H. Zhang, G. 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. |
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