Blow up and asymptotic behavior for a system of viscoelastic wave equations of Kirchhoff type with a delay term

Year 2018, Volume: 2 Issue: 3, 146 - 167, 30.09.2018

Mahdi Fatima Zohra Ferhat Mohamed , Hakem Ali

https://doi.org/10.31197/atnaa.389865

Cited By: 1

Abstract

The focus of the current paper is to investigate the initial boundary value problem for a system of viscoelastic wave equations of Kirchho type with a delay term in a bounded domain. At first,

the energy decay rate is proved by Nakao's technique and expressed polynomially and exponentially depending on the parameter m. However and in the unstable set, for certain initial data, the blow-upof solutions is obtained.

Keywords

Global existence; Energy decay; Blow up of solutions; Nonlinear damping

References

• [1] V. Georgiev, D. Todorova, Existence of solutions of the wave equations with nonlinear damping and source terms, J. Differential Equations. 109 (1994) 295-308.
• [2] H.A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974) 138-146.
• [3] H.A. Levine, S. Ro Park, Global existence and global nonexistence of solutions of the Cauchy problems for a nonlinearly damped wave equation, J. Math. Anal. Appl. 228 (1998) 181-205.
• [4] K. Ono, Global existence, decay and blow up of solutions for some mildly degenerate nonlinear Kirchhoff string, J. Differential Equations. 137 (1997) 273-301.
• [5] Xiaosen Han, Mingxin Wang, global existence and blow up of solutions for a system of nonlinear viscoelastic wave equations with damping and source, Nonlinear Analysis. 71 (2009) 5427-5450.
• [6] Shun-Tang Wu, On decay and blow up of solutions for a system of nonlinear wave equations, J. Math. Anal. Appl. 394 (2012) 360-377.
• [7] M. Nakao, Existence and Decay of Global Solutions of Some Nonlinear Degenerate Parabolic Equation, Journal of Mathematical Analisis and Applications. 109(1985) 118-129.
• [8] M. Nakao, Y. Ohara, Gradient Estimates of Periodic Solutions for Some Quasilinear Parabolic Equations, Journal of Mathematical Analisis and Applications. 204(1996) 868-883.
• [9] M. Gobbino, Quasilinear Degenerate Parabolic Equations of Kirchhof Type, Mathematical Methods in Applied Sciences. 22 (1999) 375-388.
• [10] S. Berrimi, S. A. Messaoudi, A decay result for a quasilinear parabolic system, Progress in Nonlinear Diferential Equations and their Applications. 53 (2005) 43-50.
• [11] R. Datko, J. Lagnese, M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAMJ.Control Optim. 24(1986) 152-156.
• [12] S. Nicaise, C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations. 21(2008) 935-958.
• [13] S. A. Messaoudi, Blow up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl. 320 (2006) 902-915.
• [14] M. Nakao, A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan. 30 (1978) 747-762.
• [15] K. Ammari, Serge Nicaise, Cristina Pignotti, Feedback boundary stabilization of wave equations with interior delay, Systems and Control Letters. 59(2010) 623-628.
• [16] A. Benaissa, K. Zennir, Existence and asymptotic behavior of solutions for a nonlinear wave equation with a delay term, Math. Methods Appl. Sci. 59(2011) 620-645.
• [17] M. Aassila, Stability and asymptotic behaviour of solutions of the heat equation, IMAJ. Appl. Math. 69(1)(2004) 93-109.
• [18] T. Matsuyama, R. Ikehata, On global solutions and energy decay for the wave equation of Kirchhoff type with nonlinear damping terms, J. Math. Anal. Appl. 204(3) (1996) 729-753.
• [19] M. Nakao, Global existence of classical solutions to the initial-boundary value problem of the semilinear wave equations with a degenerate dissipative term, Nonlinear Anal. 15(2) (1990) 115-140.
• [20] M. Nakao, Existence of global smooth solutions to the initial-boundary value problem for the quasi-linear wave equation with a degenerate dissipative term, J. Differential Equations. 98(2) (1992) 299-327.
• [21] K. Ono, Blowing up and global existence of solutions for some degenerate nonlinear wave equations with some dissipation, Nonlinear Anal. 30(7) (1997) 4449-4457.
• [22] K. Ono, Global existence, decay and blow-up of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations. 137(2) (1997) 273-301.
• [23] Shun-Tang Wu, L.-Y. Tsai, On global existence and blow-up of solutions for an integro-differential equation with strong damping, Taiwanese J. Math. 10 (4)(2006) 979-1014.
• [24] R. Zeng, C. L. Mu, S. M. Zhou, A blow-up result for Kirchhoff-type equations with high energy, Math. Methods Appl. Sci. 34 (4)(2011) 479-486.
• [25] Shun-Tang Wu, Exponential energy decay of solutions for an integro-differential equation with strong damping, J. Math. Anal. Appl. 364 (2)(2010) 609-617.
• [26] Shun-Tang Wu, Asymptotic behavior for a viscolastic wave equation with a delay term, J. Taiwanese J. Math. 364 (2)(2013) 765-784.
• [27] G. Kirchhof, Vorlesongen iiber Mechanik, Leipzig, Teubner. (1883).
• [28] E. V. Stepanova, Decay of the solutions of parabolic equations with double nonlinearity and the degenerate absorption potential, Ukranian Math. J. 364 (4)(2013) 99-121.
• [29] S.M. Egnorov, E. Ya, Khruslov, Global weak solutions of the Navier-Stokes-Fokker-Planck system, Ukranian Math. J.364 (4)(2013) 212-248.
• [30] J. Liu, Global weak solutions for the weakly dissipative µ-Hunter-Saxton equation, Ukranian Math. J. 364 (4)(2013) 1217-1230.
• [31] D. Ouchenane, Kh. Zennir, M. Buyoud, Global weak solutions for the weakly dissipative µ-Hunter-Saxton equation, Ukranian. Math. J.364 (4)(2013) 723-739.