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.
Citation: Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term[J]. Electronic Research Archive, 2020, 28(1): 263-289. doi: 10.3934/era.2020016
| [1] | Alice Fiaschi . Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks and Heterogeneous Media, 2010, 5(2): 257-298. doi: 10.3934/nhm.2010.5.257 |
| [2] | Antonio DeSimone, Martin Kružík . Domain patterns and hysteresis in phase-transforming solids: Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation. Networks and Heterogeneous Media, 2013, 8(2): 481-499. doi: 10.3934/nhm.2013.8.481 |
| [3] | G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini . Globally stable quasistatic evolution in plasticity with softening. Networks and Heterogeneous Media, 2008, 3(3): 567-614. doi: 10.3934/nhm.2008.3.567 |
| [4] | G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini . Time-dependent systems of generalized Young measures. Networks and Heterogeneous Media, 2007, 2(1): 1-36. doi: 10.3934/nhm.2007.2.1 |
| [5] | Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli . A rate-independent model for permanent inelastic effects in shape memory materials. Networks and Heterogeneous Media, 2011, 6(1): 145-165. doi: 10.3934/nhm.2011.6.145 |
| [6] | Flavia Smarrazzo, Alberto Tesei . Entropy solutions of forward-backward parabolic equations with Devonshire free energy. Networks and Heterogeneous Media, 2012, 7(4): 941-966. doi: 10.3934/nhm.2012.7.941 |
| [7] | Adrian Muntean, Toyohiko Aiki . Preface to ``The Mathematics of Concrete". Networks and Heterogeneous Media, 2014, 9(4): i-ii. doi: 10.3934/nhm.2014.9.4i |
| [8] | Gianni Dal Maso, Francesco Solombrino . Quasistatic evolution for Cam-Clay plasticity: The spatially homogeneous case. Networks and Heterogeneous Media, 2010, 5(1): 97-132. doi: 10.3934/nhm.2010.5.97 |
| [9] | Mauro Garavello . Boundary value problem for a phase transition model. Networks and Heterogeneous Media, 2016, 11(1): 89-105. doi: 10.3934/nhm.2016.11.89 |
| [10] | Martin Heida, Stefan Neukamm, Mario Varga . Stochastic two-scale convergence and Young measures. Networks and Heterogeneous Media, 2022, 17(2): 227-254. doi: 10.3934/nhm.2022004 |
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.
| [1] |
Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term. J. Evol. Equ. (2018) 18: 105-125. 
|
| [2] |
The existence and the asymptotic behavior of a plate equation with frictional damping and a logarithmic source term. J. Math. Anal. Appl. (2017) 454: 1114-1128.
|
| [3] |
Existence and a general decay results for a viscoelastic plate equation logarithmic nonlinearity. Commun. Pure Appl. Anal. (2019) 18: 159-180.
|
| [4] |
Inflationary models with logarithmic potentials. Phys. Rev. D (1995) 52: 5576-5587.
|
| [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] | Wave equations with logarithmic nonlinearities. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. (1975) 23: 461-466. |
| [7] |
Nonlinear wave mechanics. Ann. Physics (1976) 100: 62-93.
|
| [8] |
Équations d'évolution avec non-linéarité logarithmique. Ann. Fac. Sci. Toulouse Math. (1980) 2: 21-51.
|
| [9] |
Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity. J. Math. Anal. Appl. (2015) 422: 84-98.
|
| [10] |
Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J. Differential Equations (2015) 258: 4424-4442.
|
| [11] |
Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete Contin. Dyn. Syst. (2019) 39: 1185-1203.
|
| [12] |
Global nonexistence for a semilinear Petrovsky equation. Nonlinear Anal. (2009) 70: 3203-3208.
|
| [13] |
Q-balls and baryogenesis in the MSSM. Physics Letters B (1998) 425: 309-321.
|
| [14] |
Global solutions and finite time blow up for damped semilinear wave equations. Ann. Inst. H. Poincaré Anal. Non Linéaire (2006) 23: 185-207.
|
| [15] | Logarithmic Klein-Gordon equation. Acta Phys. Polon. B (2009) 40: 59-66. |
| [16] |
Nonlinear equations with infinitely many derivatives. Complex Anal. Oper. Theory (2011) 5: 313-323.
|
| [17] |
Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics. Bull. Korean Math. Soc. (2013) 50: 275-283.
|
| [18] |
Blow-up and decay for a class of pseudo-parabolic -Laplacian equation with logarithmic nonlinearity. Comput. Math. Appl. (2018) 75: 459-469.
|
| [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. |
| [20] |
Asymptotic behavior for a class of logarithmic wave equations with linear damping. Appl. Math. Optim. (2019) 79: 131-144.
|
| [21] |
Stable and unstable sets for evolution equations of parabolic and hyperbolic type. Hiroshima Math. J. (1996) 26: 475-491.
|
| [22] |
Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay. Appl. Anal. (2019) 99: 530-547.
|
| [23] |
Global solution and blow-up for a class of -Laplacian evolution equations with logarithmic nonlinearity. Acta Appl. Math. (2017) 151: 149-169.
|
| [24] |
Global solution and blow-up for a class of pseudo p-Laplacian evolution equations with logarithmic nonlinearity. Comput. Math. Appl. (2017) 73: 2076-2091.
|
| [25] |
Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity. Nonlinear Anal. (2019) 184: 239-257.
|
| [26] |
Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv. Nonlinear Anal. (2020) 9: 613-632.
|
| [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] |
On potential wells and vacuum isolating of solutions for semilinear wave equations. J. Differential Equations (2003) 192: 155-169.
|
| [30] |
Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign. Discrete Contin. Dyn. Syst. Ser. B (2007) 7: 171-189.
|
| [31] |
Global existence and nonexistence in a system of Petrovsky. J. Math. Anal. Appl. (2002) 265: 296-308.
|
| [32] |
Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr. (2003) 260: 58-66.
|
| [33] |
A difference inequality and its application to nonlinear evolution equations. J. Math. Soc. Japan (1978) 30: 747-762.
|
| [34] |
Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. (1975) 22: 273-303.
|
| [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] |
The equation of the -adic open string for the scalar tachyon field. Izv. Math. (2005) 69: 487-512.
|
| [37] |
On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system. Taiwanese J. Math. (2009) 13: 545-558.
|
| [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] |
The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete Contin. Dyn. Syst. (2017) 37: 5631-5649.
|
| [40] |
Exponential decay of energy for a logarithmic wave equation. J. Partial Differ. Equ. (2015) 28: 269-277.
|
| 1. | Alice Fiaschi, Dorothee Knees, Ulisse Stefanelli, Young-Measure Quasi-Static Damage Evolution, 2012, 203, 0003-9527, 415, 10.1007/s00205-011-0474-3 | |
| 2. | Alice Fiaschi, Young-measure quasi-static damage evolution: The nonconvex and the brittle cases, 2013, 6, 1937-1179, 17, 10.3934/dcdss.2013.6.17 | |
| 3. | Alexander Mielke, Tomàš Roubíček, 2015, Chapter 2, 978-1-4939-2705-0, 45, 10.1007/978-1-4939-2706-7_2 | |
| 4. | Alice Fiaschi, Dorothee Knees, Sina Reichelt, Global higher integrability of minimizers of variational problems with mixed boundary conditions, 2013, 401, 0022247X, 269, 10.1016/j.jmaa.2012.11.040 | |
| 5. | Alice Fiaschi, Quasistatic evolution for a phase-transition model: a Young measure approach, 2011, 34, 09367195, 124, 10.1002/gamm.201110020 |
Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term[J]. Electronic Research Archive, 2020, 28(1): 263-289. doi: 10.3934/era.2020016
DownLoad: