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Decay rate of global solutions to three dimensional generalized MHD system
Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy
School of Mathematics, Jilin University, Changchun 130012, China |
$ u_{tt}-\Delta u-\Delta u_{t} = |u|^{p-2}u\log|u|, $ |
$ L^{2}(\Omega) $ |
References:
[1] |
J. M. Ball,
Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser., 28 (1977), 473-486.
doi: 10.1093/qmath/28.4.473. |
[2] |
H. Chen and S. Y. 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. |
[3] |
Y. Cao and C. H. Liu, Initial boundary value problem for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity, Electron. J. Differential Equations, (2018), Paper No. 116, 1–19. |
[4] |
P. Dai, C. L. Mu and G. Y. Xu, Blow-up phenomena for a pseudo-parabolic equation with $p$-Laplacian and Logarithmic nonlinearity terms, J. Math. Anal. Appl., 481 (2020), no. 1, 123439, 27 pp.
doi: 10.1016/j.jmaa.2019.123439. |
[5] |
H. F. Di, Y. D. Shang and Z. F. Song, Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonlinear Anal. Real World Appl., 51 (2020), 102968, 22 pp.
doi: 10.1016/j.nonrwa.2019.102968. |
[6] |
P. Górka,
Logarithmic Klein-Gordon equation, Acta Phys. Polon. B, 40 (2009), 59-66.
|
[7] |
B. Guo and F. Liu,
A lower bound for the blow-up time to a viscoelastic hyperbolic equation with nonlinear sources, Appl. Math. Lett., 60 (2016), 115-119.
doi: 10.1016/j.aml.2016.03.017. |
[8] |
Y. J. He, H. 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. |
[9] |
M. A. Hamza and H. Zaag, The blow-up rate for a non-scaling invariant semilinear wave equations, J. Math. Anal. Appl., 483 (2020), 123652, 34 pp.
doi: 10.1016/j.jmaa.2019.123652. |
[10] |
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. |
[11] |
H. A. Levine, Remarks on the growth and nonexistence of solutions to nonlinear wave equations, A Seminar on PDEs - 1973, Rutgers Univ., New Brunswick, N. J., 1973, 59–70. |
[12] |
H. A. Levine,
Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.
doi: 10.1137/0505015. |
[13] |
H. A. Levine,
Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_tt=-Au+F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.
doi: 10.2307/1996814. |
[14] |
L. W. 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. |
[15] |
L. C. Nhan and L. X. Truong,
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. |
[16] |
V. Pata and S. Zelik,
Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.
doi: 10.1088/0951-7715/19/7/001. |
[17] |
L. L. Sun, B. Guo and W. J. Gao,
A lower bound for the blow-up time to a damped semilinear wave equation, Appl. Math. Lett., 37 (2014), 22-25.
doi: 10.1016/j.aml.2014.05.009. |
[18] |
J. Zhou,
Lower bounds for blow-up time of two nonlinear wave equations, Appl. Math. Lett., 45 (2015), 64-68.
doi: 10.1016/j.aml.2015.01.010. |
show all references
References:
[1] |
J. M. Ball,
Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser., 28 (1977), 473-486.
doi: 10.1093/qmath/28.4.473. |
[2] |
H. Chen and S. Y. 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. |
[3] |
Y. Cao and C. H. Liu, Initial boundary value problem for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity, Electron. J. Differential Equations, (2018), Paper No. 116, 1–19. |
[4] |
P. Dai, C. L. Mu and G. Y. Xu, Blow-up phenomena for a pseudo-parabolic equation with $p$-Laplacian and Logarithmic nonlinearity terms, J. Math. Anal. Appl., 481 (2020), no. 1, 123439, 27 pp.
doi: 10.1016/j.jmaa.2019.123439. |
[5] |
H. F. Di, Y. D. Shang and Z. F. Song, Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonlinear Anal. Real World Appl., 51 (2020), 102968, 22 pp.
doi: 10.1016/j.nonrwa.2019.102968. |
[6] |
P. Górka,
Logarithmic Klein-Gordon equation, Acta Phys. Polon. B, 40 (2009), 59-66.
|
[7] |
B. Guo and F. Liu,
A lower bound for the blow-up time to a viscoelastic hyperbolic equation with nonlinear sources, Appl. Math. Lett., 60 (2016), 115-119.
doi: 10.1016/j.aml.2016.03.017. |
[8] |
Y. J. He, H. 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. |
[9] |
M. A. Hamza and H. Zaag, The blow-up rate for a non-scaling invariant semilinear wave equations, J. Math. Anal. Appl., 483 (2020), 123652, 34 pp.
doi: 10.1016/j.jmaa.2019.123652. |
[10] |
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. |
[11] |
H. A. Levine, Remarks on the growth and nonexistence of solutions to nonlinear wave equations, A Seminar on PDEs - 1973, Rutgers Univ., New Brunswick, N. J., 1973, 59–70. |
[12] |
H. A. Levine,
Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.
doi: 10.1137/0505015. |
[13] |
H. A. Levine,
Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_tt=-Au+F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.
doi: 10.2307/1996814. |
[14] |
L. W. 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. |
[15] |
L. C. Nhan and L. X. Truong,
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. |
[16] |
V. Pata and S. Zelik,
Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.
doi: 10.1088/0951-7715/19/7/001. |
[17] |
L. L. Sun, B. Guo and W. J. Gao,
A lower bound for the blow-up time to a damped semilinear wave equation, Appl. Math. Lett., 37 (2014), 22-25.
doi: 10.1016/j.aml.2014.05.009. |
[18] |
J. Zhou,
Lower bounds for blow-up time of two nonlinear wave equations, Appl. Math. Lett., 45 (2015), 64-68.
doi: 10.1016/j.aml.2015.01.010. |
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