# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022106
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## Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation

 1 Laboratory of Operator Theory and EDP: Foundations and Applications, Department of Mathematics, Faculty of Exact Sciences, University of El-Oued, P.O. Box 789, El Oued 39000, Algeria 2 Department of Mathematics, College of Sciences and Arts, ArRass, Qassim University, Saudi Arabia 3 Mascara University, Faculty of Economies Sciences, Mascara 29000, Algeria 4 Department of Mathematics, University of Swabi, Swabi 23430, KPK, Pakistan

* Corresponding author: Salah Boulaaras: s.boularas@qu.edu.sa

Received  March 2022 Revised  March 2022 Early access April 2022

Fund Project: This work is in memory of the scientist Sheikh Abd al-Aziz ibn Abdullah Ibn Baz: 21 November 1912 - 13 May 1999

In this paper, we consider a wave equation with logarithmic source term and fractional boundary dissipation. We study the global existence of the solution under some conditions and prove the general decay of the solution in this case by using the Lyapunov functional. Also, the blow-up of solution is established at three different levels of energy using the potential well method.

Citation: Nadjat Doudi, Salah Boulaaras, Nadia Mezouar, Rashid Jan. Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022106
##### References:
 [1] Z. Achouri, N. E. Amroun and A. Benaissa, The Euler-Bernoulli beam equation with boundary dissipation of fractional derivative type, Math. Methods Appl. Sci., 40 (2017), 3837-3854.  doi: 10.1002/mma.4267. [2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. [3] R. Aounallah, S. Boulaaras, A. Zarai and B. Cherif, General decay and blow up of solution for a nonlinear wave equation with a fractional boundary damping, Math. Methods Appl. Sci., 43 (2020), 7175-7193.  doi: 10.1002/mma.6455. [4] J. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D., 52 (1995), 5576-5587. [5] S. Boulaaras and N. Doudi, Global existence and exponential stability of coupled Lamé system with distributed delay and source term without memory term, Bound. Value Probl., (2020), Paper No. 173, 21 pp. doi: 10.1186/s13661-020-01471-9. [6] S. Boulaaras, R. Guefaifia and N. Mezouar, Global existence and decay for a system of two singular one-dimensional nonlinear viscoelastic equations with general source terms, Appl. Anal., 101 (2022), 824-848.  doi: 10.1080/00036811.2020.1760250. [7] S. Boulaaras and N. Mezouar, Global existence and decay of solutions of a singular nonlocal viscoelastic system with a nonlinear source term nonlocal boundary condition, and localized damping term, Math. Meth. Appl. Sci., 43 (2020), 1-25.  doi: 10.1002/mma.6361. [8] H. Dai and H. Zhang, Exponential growth for wave equation with fractional boundary dissipation and boundary source term, Bound. Value Probl., 2014 (2014), 8 pp. doi: 10.1186/s13661-014-0138-y. [9] N. Doudi and S. Boulaaras, Global existence combined with general decay of solutions for coupled Kirchhoff system with a distributed delay term, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020), Paper No. 204, 31 pp. doi: 10.1007/s13398-020-00938-9. [10] S. Gala, Q. Liu and M. A. Ragusa, A new regularity criterion for the nematic liquid crystal fows, Appl. Anal., 91 (2012), 1741-1747.  doi: 10.1080/00036811.2011.581233. [11] S. Gala and M. A. Ragusa, Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices, Appl. Anal., 95 (2016), 1271-1279.  doi: 10.1080/00036811.2015.1061122. [12] M.-R. Li and L.-Y. Tsai, Existence and nonexistence of global solutions of some system of semilinear wave equations, Nonlinear Anal., 54 (2003), 1397-1415.  doi: 10.1016/S0362-546X(03)00192-5. [13] W. Lian, M. S. Ahmed and R. Xu, Global existence and blow-up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity, Opuscula Math., 40 (2020), 111-130.  doi: 10.7494/OpMath.2020.40.1.111. [14] G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016. [15] Y. C. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169.  doi: 10.1016/S0022-0396(02)00020-7. [16] L. Lu and S. Li, Blow up of positive initial energy solutions for a wave equation with fractional boundary dissipation, Appl. Math. Lett., 24 (2011), 1729-1734.  doi: 10.1016/j.aml.2011.04.030. [17] B. Mbodje, Wave energy decay under fractional derivative controls, IMA J. Math. Control Inform., 23 (2006), 237-257.  doi: 10.1093/imamci/dni056. [18] N. Mezouar and S. Boulaaras, Global existence and exponential decay of solutions for generalized coupled non-degenerate Kirchhoff system with a time varying delay term, Boundary Value Problems, (2020), Paper No. 90, 28 pp. doi: 10.1186/s13661-020-01390-9. [19] N. Mezouar and S. Boulaaras, Global existence and decay of solutions for a class of viscoelastic Kirchhoff equation, Bull. Malays. Math. Sci. Soc., 43 (2020), 725-755.  doi: 10.1007/s40840-018-00708-2. [20] N. Mezouar, S. Boulaaras and A. Allahem, Global existence of solutions for the Viscoelastic Kirchhoff Equation with logarithmic source terms, Complexity, (2020), 7105387, 25 pp. doi: 10.1155/2020/7105387. [21] S.-H. Park, Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source, Adv. Difference Equ., (2020), Paper No. 631, 17 pp. doi: 10.1186/s13662-020-03037-6. [22] E. Piskin and N. Irkilb, Mathematical behaviour of solutions of the Kirchhoff type equation with logarithmic nonlinearity, AIP Conference Proceedings, 2183 (2019), 090008.  doi: 10.1063/1.5136208. [23] S. Polidoro and M. A. Ragusa, Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term, Rev. Mat. Iberoam., 24 (2008), 1011-1046.  doi: 10.4171/RMI/565. [24] B. Said-Houari and F. A. Falcão Nascimento, Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source integration, Commun. Pure Appl. Anal., 12 (2013), 375-403.  doi: 10.3934/cpaa.2013.12.375. [25] D. Valério, J. Machado and V. Kiryakova, Some pioneers of the applications of fractional calculus, Frac. Calc. Appl. Anal., 17 (2014), 552-578.  doi: 10.2478/s13540-014-0185-1. [26] A. Zarai, A. Draifia and S. Boulaaras, Blow up of solutions for a system of nonlocal singular viscoelastic equations, Appl. Anal., 97 (2018), 2231-2245.  doi: 10.1080/00036811.2017.1359564. [27] H.-C. Zhou and B.-Z. Guo, Boundary feedback stabilization for an unstable time fractional reaction diffusion equation, SIAM J. Control Optim., 56 (2018), 75-101.  doi: 10.1137/15M1048999.

show all references

##### References:
 [1] Z. Achouri, N. E. Amroun and A. Benaissa, The Euler-Bernoulli beam equation with boundary dissipation of fractional derivative type, Math. Methods Appl. Sci., 40 (2017), 3837-3854.  doi: 10.1002/mma.4267. [2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. [3] R. Aounallah, S. Boulaaras, A. Zarai and B. Cherif, General decay and blow up of solution for a nonlinear wave equation with a fractional boundary damping, Math. Methods Appl. Sci., 43 (2020), 7175-7193.  doi: 10.1002/mma.6455. [4] J. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D., 52 (1995), 5576-5587. [5] S. Boulaaras and N. Doudi, Global existence and exponential stability of coupled Lamé system with distributed delay and source term without memory term, Bound. Value Probl., (2020), Paper No. 173, 21 pp. doi: 10.1186/s13661-020-01471-9. [6] S. Boulaaras, R. Guefaifia and N. Mezouar, Global existence and decay for a system of two singular one-dimensional nonlinear viscoelastic equations with general source terms, Appl. Anal., 101 (2022), 824-848.  doi: 10.1080/00036811.2020.1760250. [7] S. Boulaaras and N. Mezouar, Global existence and decay of solutions of a singular nonlocal viscoelastic system with a nonlinear source term nonlocal boundary condition, and localized damping term, Math. Meth. Appl. Sci., 43 (2020), 1-25.  doi: 10.1002/mma.6361. [8] H. Dai and H. Zhang, Exponential growth for wave equation with fractional boundary dissipation and boundary source term, Bound. Value Probl., 2014 (2014), 8 pp. doi: 10.1186/s13661-014-0138-y. [9] N. Doudi and S. Boulaaras, Global existence combined with general decay of solutions for coupled Kirchhoff system with a distributed delay term, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020), Paper No. 204, 31 pp. doi: 10.1007/s13398-020-00938-9. [10] S. Gala, Q. Liu and M. A. Ragusa, A new regularity criterion for the nematic liquid crystal fows, Appl. Anal., 91 (2012), 1741-1747.  doi: 10.1080/00036811.2011.581233. [11] S. Gala and M. A. Ragusa, Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices, Appl. Anal., 95 (2016), 1271-1279.  doi: 10.1080/00036811.2015.1061122. [12] M.-R. Li and L.-Y. Tsai, Existence and nonexistence of global solutions of some system of semilinear wave equations, Nonlinear Anal., 54 (2003), 1397-1415.  doi: 10.1016/S0362-546X(03)00192-5. [13] W. Lian, M. S. Ahmed and R. Xu, Global existence and blow-up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity, Opuscula Math., 40 (2020), 111-130.  doi: 10.7494/OpMath.2020.40.1.111. [14] G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016. [15] Y. C. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169.  doi: 10.1016/S0022-0396(02)00020-7. [16] L. Lu and S. Li, Blow up of positive initial energy solutions for a wave equation with fractional boundary dissipation, Appl. Math. Lett., 24 (2011), 1729-1734.  doi: 10.1016/j.aml.2011.04.030. [17] B. Mbodje, Wave energy decay under fractional derivative controls, IMA J. Math. Control Inform., 23 (2006), 237-257.  doi: 10.1093/imamci/dni056. [18] N. Mezouar and S. Boulaaras, Global existence and exponential decay of solutions for generalized coupled non-degenerate Kirchhoff system with a time varying delay term, Boundary Value Problems, (2020), Paper No. 90, 28 pp. doi: 10.1186/s13661-020-01390-9. [19] N. Mezouar and S. Boulaaras, Global existence and decay of solutions for a class of viscoelastic Kirchhoff equation, Bull. Malays. Math. Sci. Soc., 43 (2020), 725-755.  doi: 10.1007/s40840-018-00708-2. [20] N. Mezouar, S. Boulaaras and A. Allahem, Global existence of solutions for the Viscoelastic Kirchhoff Equation with logarithmic source terms, Complexity, (2020), 7105387, 25 pp. doi: 10.1155/2020/7105387. [21] S.-H. Park, Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source, Adv. Difference Equ., (2020), Paper No. 631, 17 pp. doi: 10.1186/s13662-020-03037-6. [22] E. Piskin and N. Irkilb, Mathematical behaviour of solutions of the Kirchhoff type equation with logarithmic nonlinearity, AIP Conference Proceedings, 2183 (2019), 090008.  doi: 10.1063/1.5136208. [23] S. Polidoro and M. A. Ragusa, Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term, Rev. Mat. Iberoam., 24 (2008), 1011-1046.  doi: 10.4171/RMI/565. [24] B. Said-Houari and F. A. Falcão Nascimento, Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source integration, Commun. Pure Appl. Anal., 12 (2013), 375-403.  doi: 10.3934/cpaa.2013.12.375. [25] D. Valério, J. Machado and V. Kiryakova, Some pioneers of the applications of fractional calculus, Frac. Calc. Appl. Anal., 17 (2014), 552-578.  doi: 10.2478/s13540-014-0185-1. [26] A. Zarai, A. Draifia and S. Boulaaras, Blow up of solutions for a system of nonlocal singular viscoelastic equations, Appl. Anal., 97 (2018), 2231-2245.  doi: 10.1080/00036811.2017.1359564. [27] H.-C. Zhou and B.-Z. Guo, Boundary feedback stabilization for an unstable time fractional reaction diffusion equation, SIAM J. Control Optim., 56 (2018), 75-101.  doi: 10.1137/15M1048999.
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