2019 Volume 9 Issue 6
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Fushan Li, Shuai Xi, Ke Xu, Xiaomin Xue. DYNAMIC PROPERTIES FOR NONLINEAR VISCOELASTIC KIRCHHOFF-TYPE EQUATION WITH ACOUSTIC CONTROL BOUNDARY CONDITIONS Ⅱ[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2318-2332. doi: 10.11948/20190085
Citation: Fushan Li, Shuai Xi, Ke Xu, Xiaomin Xue. DYNAMIC PROPERTIES FOR NONLINEAR VISCOELASTIC KIRCHHOFF-TYPE EQUATION WITH ACOUSTIC CONTROL BOUNDARY CONDITIONS Ⅱ[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2318-2332. doi: 10.11948/20190085

DYNAMIC PROPERTIES FOR NONLINEAR VISCOELASTIC KIRCHHOFF-TYPE EQUATION WITH ACOUSTIC CONTROL BOUNDARY CONDITIONS Ⅱ

  • Corresponding author: Email address:fushan99@163.com(F. Li) 
  • Fund Project: This work was supported by Natural Science Foundation of Shandong Province of China(ZR2019MA067)
  • In this paper, we consider the nonlinear viscoelastic Kirchhoff-type equation with initial conditions and acoustic boundary conditions. Under suitable conditions on the initial data, the relaxation function h(·) and M(·), we prove that the solution blows up in finite time and give the upper bound of the blow-up time T*.
    MSC: 35L05, 35L15, 35L70
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  • [1] M. Aassila and A. Benaissa, Existence globale et comportement asymptotique des solutions des equations de Kirchhoff moyennement degenerees avce un terme nonlinear dissipatif, Funkc. Ekvacioj, 2000, 43, 309-333.

    Google Scholar

    [2] R. A. Admas, Sobolev Space, New York: Academac press, 1975.

    Google Scholar

    [3] J. T. Beale and S. I. Rosencrans, Acoustic boundary conditions, Bull. Amer. Math. Soc., 1974, 80, 1276-1278. doi: 10.1090/S0002-9904-1974-13714-6

    CrossRef Google Scholar

    [4] J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 1976, 25, 895-917. doi: 10.1512/iumj.1976.25.25071

    CrossRef Google Scholar

    [5] J. T. Beale, Acoustic scattering from locally reacting surfaces, Indiana Univ. Math. J., 1977, 26, 199-222. doi: 10.1512/iumj.1977.26.26015

    CrossRef Google Scholar

    [6] S. Berrimi and S. A. Messaoudi, Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping, Electron. J. Differ. Eq., 2004, 88, 1-10.

    Google Scholar

    [7] F. A. Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 2008, 254, 1342-1372. doi: 10.1016/j.jfa.2007.09.012

    CrossRef Google Scholar

    [8] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differ. Eq., 2002, 44, 1-14.

    Google Scholar

    [9] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and asymptotic stability for viscoelastic probiems, Differ. Integral Equ., 2002, 15, 731-748.

    Google Scholar

    [10] M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 2003, 42, 1310-1324. doi: 10.1137/S0363012902408010

    CrossRef Google Scholar

    [11] R. M. Christensen, Theory of viscoelasticity, Academic Press, New York, 1971.

    Google Scholar

    [12] L. C. Evans, Partial Differential Equations(Second Edition), Rhode Island: American Mathematical Society Providence, 2010.

    Google Scholar

    [13] C. L. Frota and J. A. Goldstein, Some nonlinear wave equations with acoustic boundary conditions, J. Differ. Equations, 2000, 164, 92-109. doi: 10.1006/jdeq.1999.3743

    CrossRef Google Scholar

    [14] Q. Gao, F. Li and Y. Wang, Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation, Cent. Eur. J. Math., 2011, 9(3), 686-698. doi: 10.2478/s11533-010-0096-2

    CrossRef Google Scholar

    [15] G. C. Gorain, Exponential energy decay estimates for the solutions of n-dimensional Kirchhoff type wave equation, Appl. Math. Comp., 2006, 177, 235-242. doi: 10.1016/j.amc.2005.11.003

    CrossRef Google Scholar

    [16] P. J. Graber and B. Said-Houari, On the wave equation with semilinear porous acoustic boundary conditions, J. Differ. Equations, 2012, 252, 4898-4941.

    Google Scholar

    [17] J. Jeong, J. Park, and Y. H.Kang, Global nonexistence of solutions for a nonlinear wave equation with time delay and acoustic boundary conditions, Comput. Math. Appl., 2018, 76, 661-671. doi: 10.1016/j.camwa.2018.05.006

    CrossRef Google Scholar

    [18] F. Li, Global existence and uniqueness of weak solution to nonlinear viscoelastic full Marguerre-von Kármán shallow shell equations, Acta. Math. Sini., 2009, 25, 2133-2156. doi: 10.1007/s10114-009-7048-4

    CrossRef Google Scholar

    [19] F. Li, Limit behavior of the solution to nonlinear viscoelastic Marguerre-von Kármán shallow shell system, J. Differ. Equations, 2010, 249, 1241-1257. doi: 10.1016/j.jde.2010.05.005

    CrossRef Google Scholar

    [20] F. Li and Y. Bai, Uniform decay rates for nonlinear viscoelastic Marguerre-von Kármán equations, J. Math. Anal. Appl. 2009, 351, 522-535. doi: 10.1016/j.jmaa.2008.10.045

    CrossRef Google Scholar

    [21] F. Li and Y. Bao, Uniform Stability of the Solution for a Memory-Type Elasticity System with Nonhomogeneous Boundary Control Condition, J. Dyn. Control. Syst., 2017, 23, 301-315. doi: 10.1007/s10883-016-9320-0

    CrossRef Google Scholar

    [22] F. Li, and G. Du, General energy decay for a degenerate viscoelastic Petrovsky-type plate equation with boundary feedback, J. Appl. Anal. Comput. 2018, 8(1), 390-401.

    Google Scholar

    [23] F. Li and Q. Gao, Blow-up of solution for a nonlinear Petrovsky type equation with memory, Appl. Math. Comput., 2016, 274, 383-392.

    Google Scholar

    [24] F. Li and F. Hu, Weighted integral inequality and applications in general energy decay estimate for a variable density wave equation with memory, Bound. Value. Probl. 2018, 2018:164.

    Google Scholar

    [25] F. Li and Z. Jia, Global existence and stability of a class of nonlinear evolution equations with hereditary memory and variable density, Bound. Value. Probl. 2019, 2019:37.

    Google Scholar

    [26] F. Li and J. Li, Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions, J. Math. Anal. Appl., 2012, 385, 1005-1014. doi: 10.1016/j.jmaa.2011.07.018

    CrossRef Google Scholar

    [27] F. Li and J Li, Global existence and blow-up phenomena for p-Laplacian heat equation with inhomogeneous Neumann boundary conditions, Bound. Value Probl. 2014, 2014:219. doi: 10.1186/s13661-014-0219-y

    CrossRef Google Scholar

    [28] F. Li, Z. Zhao and Y. Chen, Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation, Nonlinear Anal.: Real World Applications, 2011, 12, 1770-1784.

    Google Scholar

    [29] F. Li and C. Zhao, Uniform energy decay rates for nonlinear viscoelastic wave equation with nonlocal boundary damping, Nonlinear Anal., 2011, 74, 3468-3477. doi: 10.1016/j.na.2011.02.033

    CrossRef Google Scholar

    [30] F. Li and S. Xi, Dynamic properties for nonlinear viscoelastic Kirchhoff-type equation with acoustic control boundary conditions I, Mathematical Notes, Accepted.

    Google Scholar

    [31] K. Nishihara and Y. Yamada, On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms, Funkc. Ekvacioj, 1990, 33, 151-159.

    Google Scholar

    [32] K. Ono, Global existence, decay and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differ. Equations, 1997, 137, 273-301. doi: 10.1006/jdeq.1997.3263

    CrossRef Google Scholar

    [33] K. Ono and K. Nishihara, On a nonlinear degenerate integro-differential equation of hyperbolic type with a strong dissipation, Adv. Math. Seciencesn and Applications, 1995, 5, 457-476.

    Google Scholar

    [34] J. Y. Park and S. H. Park, Decay rate estimates for wave equations of memory type with acoustic boundary conditions, Nonlinear Anal. 74 (2011) 993-998. doi: 10.1016/j.na.2010.09.057

    CrossRef Google Scholar

    [35] J. Y. Park and J. J. Bae and Pusan, On the existence of solutions for some nondegenerate nonlinear wave equations of kirchhoff type, Czechoslovak Mathematical Journal, 2002, 52(127), 781-795.

    Google Scholar

    [36] S. T. Wu and L. Y. Tsai, Blow-up of solutions for some non-linear wave equations of Kirchhoff-type with some dissipation, Nonlinear Anal., 2006, 65, 243-264. doi: 10.1016/j.na.2004.11.023

    CrossRef Google Scholar

    [37] S. T. Wu, Exponential energy decay of solutions for an integro-differential equation with strong damping, J. Math. Anal. Appl., 2010, 364, 609-617. doi: 10.1016/j.jmaa.2009.11.046

    CrossRef Google Scholar

    [38] S. Xi and S. Zhu, Blow-Up Criterion for the 3D Non-resistive Compressible Magnetohydrodynamic Equations, J. Dyn. Diff. Equat., 2019, 1-22.

    Google Scholar

    [39] B. Yamna and B. Benyattou, Polynomial decay and blow up of solutions for variable coefficients viscoelastic wave equation with acoustic boundary conditions, Acta. Math. Sini., 2016, 32(2), 153-174. doi: 10.1007/s10114-016-5093-3

    CrossRef Google Scholar

    [40] Y. J. Ye, On the exponential decay of solutions for some Kirchhoff-type modelling equations with strong dissipation, Appl. Math., 2010, 1, 529-533. doi: 10.4236/am.2010.16070

    CrossRef Google Scholar

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