# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020065

## 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

* Corresponding author: Bin Guo

Received  November 2019 Revised  February 2020 Published  June 2020

Fund Project: The first author is supported by The Scientific and Technological Project of Jilin Provinces's Education Department in Thirteenth-five-Year grant JJKH20180111KJ and supported by NSFC grant 11301211

The main aim of this paper is to deal with the upper and lower bounds for blow-up time of solutions to the following equation:
 $u_{tt}-\Delta u-\Delta u_{t} = |u|^{p-2}u\log|u|,$
which has been studied in [5]. For high initial energy, it is well known that the classical potential well method is not effective. In order to overcome this difficulty, the authors apply the new energy estimate method to establish the lower bound of the
 $L^{2}(\Omega)$
norm of the solution. Furthermore, the authors construct a new control functional and combine energy inequalities with the concavity argument to prove that the solution blows up in finite time for high initial energy. Meanwhile, an estimate of the upper bound of blow-up time is also obtained. Finally, a lower bound for blow-up time is obtained by introducing a new control functional. These results fill the gap of [5].
Citation: Ge Zu, Bin Guo. Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy. Evolution Equations & Control Theory, doi: 10.3934/eect.2020065
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] P. Górka, Logarithmic Klein-Gordon equation, Acta Phys. Polon. B, 40 (2009), 59-66.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] P. Górka, Logarithmic Klein-Gordon equation, Acta Phys. Polon. B, 40 (2009), 59-66.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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