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Lifespan of solutions to a damped plate equation with logarithmic nonlinearity
School of Mathematics, Jilin University, Changchun, 130012, China |
$ u_{tt}+\Delta^2u-\Delta u-\Delta u_t+u_t = |u|^{p-2}u\ln|u|. $ |
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] |
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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
D. H. Sattinger,
On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172.
doi: 10.1007/BF00250942. |
[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. |
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] |
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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
D. H. Sattinger,
On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172.
doi: 10.1007/BF00250942. |
[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. |
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