# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020101

## Lifespan of solutions to a damped plate equation with logarithmic nonlinearity

 School of Mathematics, Jilin University, Changchun, 130012, China

* Corresponding author: Yuzhu Han

Received  May 2020 Revised  August 2020 Published  October 2020

Fund Project: Supported by NSFC (11401252) and by Scientific Research Project of The Education Department of Jilin Province (JJKH20190018KJ)

This paper is devoted to the lifespan of solutions to a damped plate equation with logarithmic nonlinearity
 $u_{tt}+\Delta^2u-\Delta u-\Delta u_t+u_t = |u|^{p-2}u\ln|u|.$
Finite time blow-up criteria for solutions at both lower and high initial energy levels are established and an upper bound for the blow-up time is given for each case. Moreover, by constructing a new auxiliary functional and making full use of the strong damping term, a lower bound for the blow-up time is also derived.
Citation: Yuzhu Han, Qi Li. Lifespan of solutions to a damped plate equation with logarithmic nonlinearity. Evolution Equations & Control Theory, doi: 10.3934/eect.2020101
##### 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] 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.  Google Scholar [3] 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.  Google Scholar [4] L. J. An and A. Peirce, A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math., 55 (1995), 136-155.  doi: 10.1137/S0036139993255327.  Google Scholar [5] Y. Cao and C. Liu, Initial boundary value problem for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity, Electronic J. Differ. Equations, 116 (2018), 1-19.   Google Scholar [6] 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.  Google Scholar [7] H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.  Google Scholar [8] H. Di, Y. Shang and Z. Song, Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonl. Anal. RWA., 51 (2020), 102968, 22pp. doi: 10.1016/j.nonrwa.2019.102968.  Google Scholar [9] F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations,, Ann I H Poincaŕe-AN., 23 (2006), 185–207. doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar [10] B. Guo and X. Li, Bounds for the lifespan of solutions to fourth-order hyperbolic equations with initial data at arbitrary energy level, Taiwanese J. Math., 23 (2019), 1461-1477.  doi: 10.11650/tjm/190103.  Google Scholar [11] Y. Han, Blow-up at infinity of solutions to a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 474 (2019), 513-517.  doi: 10.1016/j.jmaa.2019.01.059.  Google Scholar [12] Y. Han, C. Cao and P. Sun, A $p$-Laplace equation with logarithmic nonlinearity at high initial energy level, Acta Appl. Math., 164 (2019), 155-164.  doi: 10.1007/s10440-018-00230-4.  Google Scholar [13] Y. Han, W. Gao, Z. Sun and H. Li, Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy, Comput. Math. Appl., 76 (2018), 2477-2483.  doi: 10.1016/j.camwa.2018.08.043.  Google Scholar [14] S. Ji, J. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equations, 261 (2016), 5446-5464.  doi: 10.1016/j.jde.2016.08.017.  Google Scholar [15] 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 [16] 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.   Google Scholar [17] H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equation of the form $Pu_t=-Au+\mathcal{F}u$, Arch. Ration. Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.  Google Scholar [18] F. Li and F. Liu, Blow-up of solutions to a quasilinear wave equarion for high initial energy, Comptes Rendus Mecanique, 346 (2018), 402-407.   Google Scholar [19] W. Lian, M. S. Ahmed and R. Xu, Global existence and blow up of solutions for semilinear hyperbolic equation with logarithmic nonlinearity, Nonl. Anal., 184 (2019), 239-257.  doi: 10.1016/j.na.2019.02.015.  Google Scholar [20] Q. Lin, Y. H. Wu and S. Lai, On global solution of an initial boundary value problem for a class of damped nonlinear equations, Nonl. Anal., 69 (2008), 4340-4351.  doi: 10.1016/j.na.2007.10.057.  Google Scholar [21] Y. Liu and R. Xu, A Class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differ. Equations, 244 (2008), 200-228.  doi: 10.1016/j.jde.2007.10.015.  Google Scholar [22] L. Ma and Z. B. Fang, Energy decay estimates and infinite blow-up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source, Math. Methods Appl. Sci., 41 (2018), 2639-2653.  doi: 10.1002/mma.4766.  Google Scholar [23] 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 [24] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942.  Google Scholar [25] S.-T. Wu, Lower and upper bounds for the blow-Up time of a class of damped fourth-order nonlinear evolution equations, J. Dyn. Control Syst., 24 (2018), 287-295.  doi: 10.1007/s10883-017-9366-7.  Google Scholar

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##### 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] 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.  Google Scholar [3] 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.  Google Scholar [4] L. J. An and A. Peirce, A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math., 55 (1995), 136-155.  doi: 10.1137/S0036139993255327.  Google Scholar [5] Y. Cao and C. Liu, Initial boundary value problem for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity, Electronic J. Differ. Equations, 116 (2018), 1-19.   Google Scholar [6] 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.  Google Scholar [7] H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.  Google Scholar [8] H. Di, Y. Shang and Z. Song, Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonl. Anal. RWA., 51 (2020), 102968, 22pp. doi: 10.1016/j.nonrwa.2019.102968.  Google Scholar [9] F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations,, Ann I H Poincaŕe-AN., 23 (2006), 185–207. doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar [10] B. Guo and X. Li, Bounds for the lifespan of solutions to fourth-order hyperbolic equations with initial data at arbitrary energy level, Taiwanese J. Math., 23 (2019), 1461-1477.  doi: 10.11650/tjm/190103.  Google Scholar [11] Y. Han, Blow-up at infinity of solutions to a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 474 (2019), 513-517.  doi: 10.1016/j.jmaa.2019.01.059.  Google Scholar [12] Y. Han, C. Cao and P. Sun, A $p$-Laplace equation with logarithmic nonlinearity at high initial energy level, Acta Appl. Math., 164 (2019), 155-164.  doi: 10.1007/s10440-018-00230-4.  Google Scholar [13] Y. Han, W. Gao, Z. Sun and H. Li, Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy, Comput. Math. Appl., 76 (2018), 2477-2483.  doi: 10.1016/j.camwa.2018.08.043.  Google Scholar [14] S. Ji, J. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equations, 261 (2016), 5446-5464.  doi: 10.1016/j.jde.2016.08.017.  Google Scholar [15] 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 [16] 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.   Google Scholar [17] H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equation of the form $Pu_t=-Au+\mathcal{F}u$, Arch. Ration. Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.  Google Scholar [18] F. Li and F. Liu, Blow-up of solutions to a quasilinear wave equarion for high initial energy, Comptes Rendus Mecanique, 346 (2018), 402-407.   Google Scholar [19] W. Lian, M. S. Ahmed and R. Xu, Global existence and blow up of solutions for semilinear hyperbolic equation with logarithmic nonlinearity, Nonl. Anal., 184 (2019), 239-257.  doi: 10.1016/j.na.2019.02.015.  Google Scholar [20] Q. Lin, Y. H. Wu and S. Lai, On global solution of an initial boundary value problem for a class of damped nonlinear equations, Nonl. Anal., 69 (2008), 4340-4351.  doi: 10.1016/j.na.2007.10.057.  Google Scholar [21] Y. Liu and R. Xu, A Class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differ. Equations, 244 (2008), 200-228.  doi: 10.1016/j.jde.2007.10.015.  Google Scholar [22] L. Ma and Z. B. Fang, Energy decay estimates and infinite blow-up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source, Math. Methods Appl. Sci., 41 (2018), 2639-2653.  doi: 10.1002/mma.4766.  Google Scholar [23] 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 [24] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942.  Google Scholar [25] S.-T. Wu, Lower and upper bounds for the blow-Up time of a class of damped fourth-order nonlinear evolution equations, J. Dyn. Control Syst., 24 (2018), 287-295.  doi: 10.1007/s10883-017-9366-7.  Google Scholar
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