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March  2020, 28(1): 205-220. doi: 10.3934/era.2020014

## Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights

 1 Department of Mathematics, Federal University of Bahia, Salvador, 40170-115, Bahia, Brazil 2 Department of Mathematics, Federal University of São João del-Rei, São João del-Rei, 36307-352, Minas Gerais, Brazil

* Corresponding author: raposo@ufsj.edu.br

Received  January 2020 Revised  February 2020

Fund Project: The first author was partially supported by FCT project PTDC/MAT-PUR/28177/2017, with national funds, and by CMUP (UID/MAT/00144/2019), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. The second author was partially supported by CAPES (Brazil)

We consider the wave equation with a weak internal damping with non-constant delay and nonlinear weights given by
 $\begin{eqnarray*} \label{NLS} u_{tt}(x,t) - u_{xx}(x,t)+\mu_1(t)u_t(x,t) +\mu_2(t)u_t(x,t-\tau(t)) = 0 \end{eqnarray*}$
in a bounded domain. Under proper conditions on nonlinear weights $\mu_1(t), \mu_2(t)$ and non-constant delay $\tau(t)$, we prove global existence and estimative the decay rate for the energy.
Citation: Vanessa Barros, Carlos Nonato, Carlos Raposo. Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights. Electronic Research Archive, 2020, 28 (1) : 205-220. doi: 10.3934/era.2020014
##### References:
 [1] F. A. Mehmeti, Nonlinear Waves in Networks, Vol. 80, Mathematical Research, Akademie-Verlag, Berlin, 1994.  Google Scholar [2] A. Benaissa, A. Benguessoum and S. A. Messaoudi, Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback, Electron. J. Qual. Theory Differ. Equ., 11 (2014), 13 pp. doi: 10.14232/ejqtde.2014.1.11.  Google Scholar [3] F. Z. Benyoub, M. Ferhat and A. Hakem, Global existence and asymptotic stability for a coupled viscoelastic wave equation with a time-varying delay term, Electron. J. Math. Anal. Appl., 6 (2018), 119-156.   Google Scholar [4] H. Brézis, Opérators Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York.  Google Scholar [5] G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control Optim., 17 (1979), 66-81.  doi: 10.1137/0317007.  Google Scholar [6] G. Chen, Control and stabilization for the wave equation in a bounded domain. Ⅱ, SIAM J. Control Optim., 19 (1981), 114-122.  doi: 10.1137/0319009.  Google Scholar [7] R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.  Google Scholar [8] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040.  Google Scholar [9] B. Feng and X.-G. Yang, Long-time dynamics for a nonlinear Timoshenko system with delay, Appl. Anal., 96 (2017), 606-625.  doi: 10.1080/00036811.2016.1148139.  Google Scholar [10] B. Feng, Well-posedness and exponential decay for laminated Timoshenko beams with time delays and boundary feedbacks, Math. Methods Appl. Sci., 41 (2018), 1162-1174.  doi: 10.1002/mma.4655.  Google Scholar [11] M. Ferhat, Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 491-506.   Google Scholar [12] A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinity memory and time delay, IMA J. Math. Control Inform., 30 (2013), 507-526.  doi: 10.1093/imamci/dns039.  Google Scholar [13] A. Haraux, Two remarks on hyperbolic dissipative problems, Res. Notes in Math., 122 (1985), 161-179.   Google Scholar [14] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508-520.  doi: 10.2969/jmsj/01940508.  Google Scholar [15] T. Kato, Abstract Differential Equations and Nonlinear Mixed Problems, Lezioni Fermiane, [Fermi Lectures], Scuola Normale Superiore, Pisa; Accademia NAzionale dei Lincei, Rome, 1985.  Google Scholar [16] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar [17] I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with $L_2(0, \infty; L_2(\Gamma))$-feedback control in the Dirichlet boundary conditions, J. Differential Equations, 66 (1987), 340-390.  doi: 10.1016/0022-0396(87)90025-8.  Google Scholar [18] G. Liu, Well-posedness and exponential decay of solutions for a transmission problem with distributed delay, Electron. J. Differential Equations, 174 (2017), 13 pp.  Google Scholar [19] W. Liu, General decay rate estimate for the energy of a weak viscoelastic equation with an internal time-varying delay term, Taiwanese J. Math., 17 (2013), 2101-2115.  doi: 10.11650/tjm.17.2013.2968.  Google Scholar [20] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar [21] M. Nakao, Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl., 60 (1977), 542-549.  doi: 10.1016/0022-247X(77)90040-3.  Google Scholar [22] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.  Google Scholar [23] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations, 41 (2011), 20 pp.  Google Scholar [24] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958.   Google Scholar [25] S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.  Google Scholar [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [27] C. A. Raposo, H. Nguyen, J. O. Ribeiro and V. Barros, Well-posedness and exponential stability for a wave equation with nonlocal time-delay condition, Electron. J. Differential Equations, 279 (2017), 11 pp.  Google Scholar [28] M. Remil and A. Hakem, Global existence and asymptotic behavior of solutions to the viscoelastic wave equation with a constant delay term, Facta Univ. Ser. Math. Inform, 32 (2017), 485-502.  doi: 10.1007/s11766-017-3280-3.  Google Scholar [29] F. Tahamtani and A. Peyravi, Asymptotic behavior and blow-up of solutions for a nonlinear viscoelastic wave equation with boundary dissipation, Taiwanese J. Math., 17 (2013), 1921-1943.  doi: 10.11650/tjm.17.2013.3034.  Google Scholar [30] A. A. Than and J. Wang, Stabilization of the cascaded ODE-Schrödinger equations subject to observation with time delay, IEEE/CAA J. Autom. Sin., 6 (2019), 1027-1035.  doi: 10.1109/JAS.2019.1911588.  Google Scholar [31] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.  Google Scholar [32] K.-Y. Yang and J.-M. Wang, Pointwise feedback stabilization of an Euler-Bernoulli beam in observations with time delay, ESAIM Control Optim. Calc. Var., 25 (2019), 23 pp. doi: 10.1051/cocv/2017080.  Google Scholar

show all references

##### References:
 [1] F. A. Mehmeti, Nonlinear Waves in Networks, Vol. 80, Mathematical Research, Akademie-Verlag, Berlin, 1994.  Google Scholar [2] A. Benaissa, A. Benguessoum and S. A. Messaoudi, Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback, Electron. J. Qual. Theory Differ. Equ., 11 (2014), 13 pp. doi: 10.14232/ejqtde.2014.1.11.  Google Scholar [3] F. Z. Benyoub, M. Ferhat and A. Hakem, Global existence and asymptotic stability for a coupled viscoelastic wave equation with a time-varying delay term, Electron. J. Math. Anal. Appl., 6 (2018), 119-156.   Google Scholar [4] H. Brézis, Opérators Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York.  Google Scholar [5] G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control Optim., 17 (1979), 66-81.  doi: 10.1137/0317007.  Google Scholar [6] G. Chen, Control and stabilization for the wave equation in a bounded domain. Ⅱ, SIAM J. Control Optim., 19 (1981), 114-122.  doi: 10.1137/0319009.  Google Scholar [7] R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.  Google Scholar [8] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040.  Google Scholar [9] B. Feng and X.-G. Yang, Long-time dynamics for a nonlinear Timoshenko system with delay, Appl. Anal., 96 (2017), 606-625.  doi: 10.1080/00036811.2016.1148139.  Google Scholar [10] B. Feng, Well-posedness and exponential decay for laminated Timoshenko beams with time delays and boundary feedbacks, Math. Methods Appl. Sci., 41 (2018), 1162-1174.  doi: 10.1002/mma.4655.  Google Scholar [11] M. Ferhat, Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 491-506.   Google Scholar [12] A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinity memory and time delay, IMA J. Math. Control Inform., 30 (2013), 507-526.  doi: 10.1093/imamci/dns039.  Google Scholar [13] A. Haraux, Two remarks on hyperbolic dissipative problems, Res. Notes in Math., 122 (1985), 161-179.   Google Scholar [14] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508-520.  doi: 10.2969/jmsj/01940508.  Google Scholar [15] T. Kato, Abstract Differential Equations and Nonlinear Mixed Problems, Lezioni Fermiane, [Fermi Lectures], Scuola Normale Superiore, Pisa; Accademia NAzionale dei Lincei, Rome, 1985.  Google Scholar [16] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar [17] I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with $L_2(0, \infty; L_2(\Gamma))$-feedback control in the Dirichlet boundary conditions, J. Differential Equations, 66 (1987), 340-390.  doi: 10.1016/0022-0396(87)90025-8.  Google Scholar [18] G. Liu, Well-posedness and exponential decay of solutions for a transmission problem with distributed delay, Electron. J. Differential Equations, 174 (2017), 13 pp.  Google Scholar [19] W. Liu, General decay rate estimate for the energy of a weak viscoelastic equation with an internal time-varying delay term, Taiwanese J. Math., 17 (2013), 2101-2115.  doi: 10.11650/tjm.17.2013.2968.  Google Scholar [20] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar [21] M. Nakao, Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl., 60 (1977), 542-549.  doi: 10.1016/0022-247X(77)90040-3.  Google Scholar [22] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.  Google Scholar [23] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations, 41 (2011), 20 pp.  Google Scholar [24] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958.   Google Scholar [25] S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.  Google Scholar [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [27] C. A. Raposo, H. Nguyen, J. O. Ribeiro and V. Barros, Well-posedness and exponential stability for a wave equation with nonlocal time-delay condition, Electron. J. Differential Equations, 279 (2017), 11 pp.  Google Scholar [28] M. Remil and A. Hakem, Global existence and asymptotic behavior of solutions to the viscoelastic wave equation with a constant delay term, Facta Univ. Ser. Math. Inform, 32 (2017), 485-502.  doi: 10.1007/s11766-017-3280-3.  Google Scholar [29] F. Tahamtani and A. Peyravi, Asymptotic behavior and blow-up of solutions for a nonlinear viscoelastic wave equation with boundary dissipation, Taiwanese J. Math., 17 (2013), 1921-1943.  doi: 10.11650/tjm.17.2013.3034.  Google Scholar [30] A. A. Than and J. Wang, Stabilization of the cascaded ODE-Schrödinger equations subject to observation with time delay, IEEE/CAA J. Autom. Sin., 6 (2019), 1027-1035.  doi: 10.1109/JAS.2019.1911588.  Google Scholar [31] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.  Google Scholar [32] K.-Y. Yang and J.-M. Wang, Pointwise feedback stabilization of an Euler-Bernoulli beam in observations with time delay, ESAIM Control Optim. Calc. Var., 25 (2019), 23 pp. doi: 10.1051/cocv/2017080.  Google Scholar
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