In this paper, we consider the following viscoelastic wave equation with delay
|ut|ρ utt-△u-△utt + ƒ0t g(t - s)△u(s)ds + μ1ut(x, t) + μ2ut(x, t - τ)=b|u|p-2 u
in a bounded domain. Under appropriate conditions on μ1, μ2, the kernel function g, the nonlinear source and the initial data, there are solutions that blow up in finite time.
Shun-Tang WU
. BLOW-UP OF SOLUTION FOR A VISCOELASTIC WAVE EQUATION WITH DELAY[J]. Acta mathematica scientia, Series B, 2019
, 39(1)
: 329
-338
.
DOI: 10.1007/s10473-019-0124-7
[1] Cavalcanti M M, Domingos Cavalcanti V N, Ferreira J. Existence and uniform decay of nonlinear viscoelastic equation with strong damping. Math Methods Appl Sci, 2001, 24:1043-1053
[2] Datko R. Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J Control Optim, 1988, 26:697-713
[3] 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, 1986, 24(1):152-156
[4] Datko R. Two examples of ill-posedness with respect to time delays revisited. IEEE Trans Automatic Control, 1997, 42:511-515
[5] Kafini M, Messaoudi S A. A blow-up result in a nonlinear wave equation with delay. Mediterr J Math, 2016, 13:237-247
[6] Kafini M, Messaoudi S A, Nicaise S. A blow-up result in a nonlinear abstract evolution system with delay. NoDEA Nonlinear Differential Equations Appl, 2016, 23:13
[7] Kirane M, Belkacem S H. Existence and asymptotic stability of a viscoelastic wave equation with a delay. Z Angew Math Phys, DOI 10.1007/s00033-011-0145-0
[8] Li J, Chai S G. Energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback. Nonlinear Analysis:Theory, Methods & Applications, 2015, 112:105-117
[9] Liu W J. General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Analysis:Theory, Methods & Applications, 2010, 73:1890-1904
[10] Liu W J, Li G, Hong L H. General decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping. Journal of Function Spaces, 2014, 2014:Article ID 284809
[11] Liu W J, Yu J. On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms. Nonlinear Analysis:Theory, Methods & Applications, 2011, 74(6):2175-2190
[12] Liu W J, Chen K, Yu J. Asymptotic stability for a non-autonomous full von Karman beam with thermoviscoelastic damping. Applicable Analysis, 2018, 97(3):400-414
[13] Liu W J, Zhu B Q, Li G, Wang D H. General decay for a viscoelastic Kirchhoff equation with BalakrishnanTaylor damping, dynamic boundary conditions and a time-varying delay term. Evolution Equations and Control Theory, 2017, 6(2):239-260
[14] Liu W J, Sun Y, Li G. On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term. Topological Methods in Nonlinear Analysis, 2017, 49(1):299-323
[15] Messaoudi S A, Tatar N-e. Global existence and asymptotic behavior for a nonlinear viscoelastic problem. Mathematical Science Research Journal, 2003, 7(4):136-149
[16] Mohamed Ferhat, Ali Hakem. Asymptotic behavior for a weak viscoelastic wave equations with a dynamic boundary and time varying delay term. J Appl Math Comput, 2016, 51(1/2):509-526
[17] 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, 2006, 45(5):1561-1585
[18] Nicaise S, Pignotti C. Stabilization of the wave equation with boundary or internal distributed delay. Diff Int Equs, 2008, 21(9/10):935-958
[19] Nicaise S, Valein J. Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Netw Heterog Media, 2007, 2(3):425-479
[20] Nicaise S, Valein J, Fridman E. Stabilization of the heat and the wave equations with boundary time-varying delays. Discrete and Continuous Dynamical Systems-Series S, 2009, 2(3):559-581
[21] Nicaise S, Pignotti C, Valein J. Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems-Series S, 2011, 4(3):693-722
[22] Pignotti C. A note on stabilization of locally damped wave equations with time delay. Syst Control Lett, 2012, 61(1):92-97
[23] Song H T. Global nonexistence of positive initial energy solutions for a viscoelastic wave equation. Nonlinear Analysis:Theory, Methods & Applications, 2015, 125:260-269
[24] Suh I H, Bien Z. Use of time delay action in the controller design. IEEE Trans Automat Control, 1980, 25:600-603
[26] Xu G Q, Yung S P, Li L K. Stabilization of the wave systems with input delay in the boundary control. ESAIM:Control Optim Calc Var, 2006, 12:770-785
[26] Wu S T. Asymptotic behavior for a viscoelastic wave equation with a delay term. Taiwanese J Math, 2013, 17:765-784
