Abstract
In this paper, we consider a class of wave equation with past history and a delay term in the internal feedback. Namely, we investigate the following equation
together with some suitable initial data and boundary conditions. The problem was considered by several authors, with \({\mu_2 = 0}\). The project of the present paper is to provide the well-posedness for this problem in a general setting which includes a delay term (\({\mu_2 \neq 0}\)). We also establish the exponential stability result when \({h(x) = 0}\).
Similar content being viewed by others
References
Alabau-Boussouira F., Cannarsa P., Sforza D.: Decay estimates for second order evolution equations with memory. J. Funct. Anal. 254, 1342–1372 (2008)
Alabau-Boussouira F., Nicaise S., Pignotti C.: Exponential stability of the wave equation with memory and time delay. Springer Indam Series, Vol. 10, pp. 1–22 (2014)
Araújo R.O., Ma T.F., Qin Y.: Long-time behavior of a quasilinear viscoelastic equation with past history. J. Differ. Equ. 254, 4066–4087 (2013)
Berrimi S., Messaoudi S.A.: Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. 64(10), 2314–2331 (2006)
Cavalcanti M.M., Domingos Cavalcanti V.N., Soriano J.A.: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 44, 1–14 (2002)
Cavalcanti M.M., Oquendo H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310–1324 (2003)
Dafermos C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)
Dai Q., Yang Z.: Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay. Z. Angew. Math. Phys. 65(5), 885–903 (2014)
Datko R., Lagnese J., Polis M.P.: An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24(1), 152–156 (1986)
Datko R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim. 26(3), 697–713 (1988)
Georgiev V., Todorova G.: Existence of solutions of the wave equations with nonlinear damping and source terms. J. Differ. Equ. 109, 295–308 (1994)
Giorgi C., Muñoz Rivera J.E., Pata V.: Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl. 260, 83–99 (2001)
Guesmia A.: Well-posedness and exponential stability of an abstract evolution equation with infinity memory and time delay. IMA J. Math. Control Inform. 30, 507–526 (2013)
Kirane M., Said-Houari B.: Existence and asymptotic stability of a viscoelastic wave equation with a delay. Z. Angew. Math. Phys. 62, 1065–1082 (2011)
Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, (1969)
Levine H.A.: Instability and nonexistence of global solutions of nonlinear wave equation of the form \({Pu_{tt} = -Au + \mathfrak{F}(u)}\). Trans. Am. Math. Soc. 192, 1–21 (1974)
Levine H.A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal. 5, 138–146 (1974)
Liu G., Zhang H.: Blow up at infinity of solutions for integro-differential equation. Appl. Math. Comput. 230, 303–314 (2014)
Liu W.J.: General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms. J. Math. Phys. 50(11), 113506 (2009)
Liu W.J.: General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback. J. Math. Phys. 54(4), 043504 (2013)
Nicaise S., Pignotti C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45(5), 1561–1585 (2006)
Nicaise S., Pignotti C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21(9–10), 935–958 (2008)
Nicaise S., Valein J., Fridman E.: Stabilization of the heat and the wave equations with boundary time-varying delays. Discrete Contin. Dyn. Syst. Ser. S 2(3), 559–581 (2009)
Nicaise S., Pignotti C.: Interior feedback stabilization of wave equations with time dependent delay. Electron. J. Differ. Equ. 41, 1–20 (2011)
Park J.Y., Park S.H.: Decay rate estimates for wave equations of memory type with acoustic boundary conditions. Nonlinear Anal. 74(3), 993–998 (2011)
Pata V., Zucchi A.: Attractors for a damped hyperbolic equation with linear memory. Adv. Math. Sci. Appl. 11, 505–529 (2001)
Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44 of Applied Mathematical Sciences. Springer, New York (1983)
Pignotti C.: A note on stabilization of locally damped wave equations with time delay. Syst. Control Lett. 61, 92–97 (2012)
Sun F.Q., Wang M.X.: Global and blow-up solutions for a system of nonlinear hyperbolic equations with dissipative terms. Nonlinear Anal. 64(4), 739–761 (2006)
Vitillaro E.: Global nonexistence theorems for a class of evolution equations with dissipation. Arch. Ration. Mech. Anal. 149, 155–182 (1999)
Wu S.T.: General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions. Z. Angew. Math. Phys. 63(1), 65–106 (2012)
Xu G.Q., Yung S.P., Li L.K.: Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var. 12(4), 770–785 (2006)
Yang Z.: Existence and energy decay of solutions for the Euler-Bernoulli viscoelastic equation with a delay. Z. Angew. Math. Phys. 66(3), 727–745 (2015)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, G., Zhang, H. Well-posedness for a class of wave equation with past history and a delay. Z. Angew. Math. Phys. 67, 6 (2016). https://doi.org/10.1007/s00033-015-0593-z
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-015-0593-z