American Institute of Mathematical Sciences

Advanced
doi: 10.3934/eect.2020065

Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy

Ge Zu and  Bin Guo ,

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].
Keywords: Wave equations, Logarithmic nonlinearity, concavity argument, high initial energy, bounds for lifespan.
Mathematics Subject Classification: Primary: 35L20, 35B44; Secondary: 35D30.
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

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.  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

[1]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[2]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116
[3]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273
[4]

Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265
[5]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018
[6]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252
[7]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445
[8]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379
[9]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347
[10]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268
[11]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258
[12]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158
[13]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264
[14]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215
[15]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[16]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466
[17]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380
[18]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073
[19]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297
[20]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

2019 Impact Factor: 0.953

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences