Well-posedness for a class of wave equation with past history and a delay

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• 作者：Gongwei Liu ; Hongwei Zhang
• 关键词：Viscoelasticity ; Energy decay ; Delay ; Memory
• 刊名：Zeitschrift f¨¹r angewandte Mathematik und Physik
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：67
• 期：1
• 全文大小：552 KB
• 参考文献：1.Alabau-Boussouira F., Cannarsa P., Sforza D.: Decay estimates for second order evolution equations with memory. J. Funct. Anal. 254, 1342–1372 (2008)MathSciNet CrossRef MATH
  2.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)
  3.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)MathSciNet CrossRef MATH
  4.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)MathSciNet CrossRef MATH
  5.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)MathSciNet MATH
  6.Cavalcanti M.M., Oquendo H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310–1324 (2003)MathSciNet CrossRef MATH
  7.Dafermos C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)MathSciNet CrossRef MATH
  8.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)MathSciNet CrossRef MATH
  9.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)MathSciNet CrossRef MATH
  10.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)MathSciNet CrossRef MATH
  11.Georgiev V., Todorova G.: Existence of solutions of the wave equations with nonlinear damping and source terms. J. Differ. Equ. 109, 295–308 (1994)MathSciNet CrossRef MATH
  12.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)MathSciNet CrossRef MATH
  13.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)MathSciNet CrossRef MATH
  14.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)MathSciNet CrossRef MATH
  15.Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, (1969)
  16.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)MATH
  17.Levine H.A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal. 5, 138–146 (1974)MathSciNet CrossRef MATH
  18.Liu G., Zhang H.: Blow up at infinity of solutions for integro-differential equation. Appl. Math. Comput. 230, 303–314 (2014)MathSciNet CrossRef
  19.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)MathSciNet CrossRef MATH
  20.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)MathSciNet CrossRef MATH
  21.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)MathSciNet CrossRef MATH
  22.Nicaise S., Pignotti C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21(9–10), 935–958 (2008)MathSciNet MATH
  23.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)MathSciNet CrossRef MATH
  24.Nicaise S., Pignotti C.: Interior feedback stabilization of wave equations with time dependent delay. Electron. J. Differ. Equ. 41, 1–20 (2011)MathSciNet MATH
  25.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)MathSciNet CrossRef MATH
  26.Pata V., Zucchi A.: Attractors for a damped hyperbolic equation with linear memory. Adv. Math. Sci. Appl. 11, 505–529 (2001)MathSciNet MATH
  27.Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44 of Applied Mathematical Sciences. Springer, New York (1983)CrossRef MATH
  28.Pignotti C.: A note on stabilization of locally damped wave equations with time delay. Syst. Control Lett. 61, 92–97 (2012)MathSciNet CrossRef MATH
  29.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)MathSciNet CrossRef MATH
  30.Vitillaro E.: Global nonexistence theorems for a class of evolution equations with dissipation. Arch. Ration. Mech. Anal. 149, 155–182 (1999)MathSciNet CrossRef MATH
  31.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)MathSciNet CrossRef MATH
  32.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)MathSciNet CrossRef MATH
  33.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)MathSciNet CrossRef MATH
  
• 作者单位：Gongwei Liu (1)
  Hongwei Zhang (1)
  
  1. College of Science, Henan University of Technology, Zhengzhou, 450001, People’s Republic of China
  
• 刊物主题：Theoretical and Applied Mechanics; Mathematical Methods in Physics;
• 出版者：Springer Basel
• ISSN：1420-9039

文摘

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 $$u_{tt} - \alpha\triangle u + \int\limits_{-\infty}^{t} \mu(t - s)\triangle u(s){\rm d}s+\mu_1u_t + \mu_2u_t(x, t - \tau) + f(u) = h,$$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 \({h(x) = 0}\). Keywords Viscoelasticity Energy decay Delay Memory Mathematics Subject Classification 35B40 35L05 74Dxx 93D20 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (33) References1.Alabau-Boussouira F., Cannarsa P., Sforza D.: Decay estimates for second order evolution equations with memory. J. Funct. Anal. 254, 1342–1372 (2008)MathSciNetCrossRefMATH2.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)3.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)MathSciNetCrossRefMATH4.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)MathSciNetCrossRefMATH5.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)MathSciNetMATH6.Cavalcanti M.M., Oquendo H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310–1324 (2003)MathSciNetCrossRefMATH7.Dafermos C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)MathSciNetCrossRefMATH8.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)MathSciNetCrossRefMATH9.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)MathSciNetCrossRefMATH10.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)MathSciNetCrossRefMATH11.Georgiev V., Todorova G.: Existence of solutions of the wave equations with nonlinear damping and source terms. J. Differ. Equ. 109, 295–308 (1994)MathSciNetCrossRefMATH12.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)MathSciNetCrossRefMATH13.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)MathSciNetCrossRefMATH14.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)MathSciNetCrossRefMATH15.Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, (1969)16.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)MATH17.Levine H.A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal. 5, 138–146 (1974)MathSciNetCrossRefMATH18.Liu G., Zhang H.: Blow up at infinity of solutions for integro-differential equation. Appl. Math. Comput. 230, 303–314 (2014)MathSciNetCrossRef19.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)MathSciNetCrossRefMATH20.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)MathSciNetCrossRefMATH21.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)MathSciNetCrossRefMATH22.Nicaise S., Pignotti C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21(9–10), 935–958 (2008)MathSciNetMATH23.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)MathSciNetCrossRefMATH24.Nicaise S., Pignotti C.: Interior feedback stabilization of wave equations with time dependent delay. Electron. J. Differ. Equ. 41, 1–20 (2011)MathSciNetMATH25.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)MathSciNetCrossRefMATH26.Pata V., Zucchi A.: Attractors for a damped hyperbolic equation with linear memory. Adv. Math. Sci. Appl. 11, 505–529 (2001)MathSciNetMATH27.Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44 of Applied Mathematical Sciences. Springer, New York (1983)CrossRefMATH28.Pignotti C.: A note on stabilization of locally damped wave equations with time delay. Syst. Control Lett. 61, 92–97 (2012)MathSciNetCrossRefMATH29.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)MathSciNetCrossRefMATH30.Vitillaro E.: Global nonexistence theorems for a class of evolution equations with dissipation. Arch. Ration. Mech. Anal. 149, 155–182 (1999)MathSciNetCrossRefMATH31.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)MathSciNetCrossRefMATH32.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)MathSciNetCrossRefMATH33.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)MathSciNetCrossRefMATH About this Article Title Well-posedness for a class of wave equation with past history and a delay Journal Zeitschrift für angewandte Mathematik und Physik 67:6 Online DateMarch 2016 DOI 10.1007/s00033-015-0593-z Print ISSN 0044-2275 Online ISSN 1420-9039 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Theoretical and Applied Mechanics Mathematical Methods in Physics Keywords 35B40 35L05 74Dxx 93D20 Viscoelasticity Energy decay Delay Memory Industry Sectors Aerospace Engineering Oil, Gas & Geosciences Authors Gongwei Liu (1) Hongwei Zhang (1) Author Affiliations 1. College of Science, Henan University of Technology, Zhengzhou, 450001, People’s Republic of China Continue reading... To view the rest of this content please follow the download PDF link above.