Long-Time Stabilization of Solutions to a Nonautonomous Semilinear Viscoelastic Equation

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• 作者：Hassan Yassine ; Ali Abbas
• 关键词：Evolutionary integral equation ; Semilinear ; Stabilization ; Łojasiewicz–Simon inequality
• 刊名：Applied Mathematics and Optimization
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：73
• 期：2
• 页码：251-269
• 全文大小：490 KB
• 参考文献：1.Ben Hassen, I.: Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations. Asymptot. Anal. 69, 31–44 (2010)MathSciNet MATH
  2.Cavalcanti, M.M., Domingos Cavalcanti, V.N., Ferreira, J.: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 24, 1043–1053 (2001)MathSciNet CrossRef MATH
  3.Chepyzhov, V.V., Gatti, S., Grasselli, M., Miranville, A., Pata, V.: Trajectory and global attractors for evolution equations with memory. Appl. Math. Lett. 19, 87–96 (2006)MathSciNet CrossRef MATH
  4.Dafermos, C.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)MathSciNet CrossRef MATH
  5.Giorgi, C., Muñoz Rivera, J.E.: Semilinear hyperbolic equation viscoelasticity. J. Math. Anal. Appl. 260, 83–99 (2001)MathSciNet CrossRef MATH
  6.Grasselli, M., Muñoz Rivera, J.E., Pata, V.: On the energy decay of the linear thermoelastic plate with memory. J. Math. Anal. Appl. 309, 1–14 (2005)MathSciNet CrossRef MATH
  7.Haraux, A.: Systèmes dynamiques dissipatifs et applications. Masson, Paris (1990)MATH
  8.Haraux, A., Jendoubi, M.A.: Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity. Calc. Var. Partial Differ. Equ. 9, 95–124 (1999)MathSciNet CrossRef MATH
  9.Han, X., Wang, M.: Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Nonlinear Anal. 70, 3090–3098 (2009)MathSciNet CrossRef MATH
  10.Jendoubi, M.A.: Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity. J. Differ. Equ. 144, 302–312 (1998)MathSciNet CrossRef MATH
  11.Jendoubi, M.A.: A simple unified approach to some convergence theorem of L. Simon. J. Funct. Anal. 153, 187–202 (1998)MathSciNet CrossRef MATH
  12.Łojasiewicz, S., Une propriété topologique des sous-ensembles analytiques réels, Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles, Paris, : Editions du C.N.R.S. Paris 1963, 87–89 (1962)
  13.Łojasiewicz, S.: Sur la géometrie semi- et sous-analytique. Ann. Inst. Fourier (Grenoble) 43, 1575–1595 (1963)CrossRef
  14.Łojasiewicz, S.: Ensembles semi-analytiques, Preprint. I.H.E.S, Bures-sur-Yvette (1965)
  15.Liu, Wenjun: Uniform decay of solutions for a quasilinear system of viscoelastic equations. Nonlinear Anal. 71, 2257–2267 (2009)MathSciNet CrossRef MATH
  16.Liu, Wenjun: Global existence and uniform decay of solutions for a system of wave equations with dispersive and dissipative terms. Front. Math. China 5(3), 555–574 (2010)MathSciNet CrossRef MATH
  17.Messaoudi, S.A., Tatar, N.-E.: Global existence and asymptotic behavior for a nonlinear viscoelastic problem. Math. Methods Sci. Res. J. 7(4), 136–149 (2003)MathSciNet MATH
  18.Messaoudi, S.A., Tatar, N.-E.: Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math. Methods Appl. Sci. 30, 665–680 (2007)MathSciNet CrossRef MATH
  19.Muñoz, J.E., Rivera Naso, M.G.: Optimal energy decay rate for a class of weakly dissipative second-order systems with memory. Appl. Math. Lett. 23, 743–746 (2010)MathSciNet CrossRef MATH
  20.Muñoz, J.E., Rivera Naso, M.G.: Asymptotic stability of semigroups associated with linear weak dissipative systems with memory. J. Math. Anal. Appl. 326, 691–707 (2007)MathSciNet CrossRef MATH
  21.Park, J.Y., Kang, J.R.: Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Acta Appl. Math. 110, 1393–1406 (2010)MathSciNet CrossRef MATH
  22.Simon, L.: Asymptotics for a class of non-linear evolution equation with applications to geometric problem. Ann. Math. 118, 525–571 (1983)CrossRef MATH
  23.Shuntang, Wu: General decay of solutions for a viscoelastic equation with nonlinear damping and source terms. Acta Mathematica Scientia 31B(4), 1436–1448 (2011)MathSciNet CrossRef MATH
  
• 作者单位：Hassan Yassine (1)
  Ali Abbas (1)
  
  1. Department of Mathematics, Faculty of Sciences, Lebanese University, Baalbek-Zahle, Lebanon
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Calculus of Variations and Optimal Control
  Systems Theory and Control
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  
• 出版者：Springer New York
• ISSN：1432-0606

文摘

We study the long-time behavior as time goes to infinity of global bounded solutions to the following nonautonomous semilinear viscoelastic equation: $$\begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u_{t}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)ds+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned}$$in \({\mathbb {R}}^+\times \Omega \), with Dirichlet boundary conditions, where \(\Omega \) is a bounded domain in \({\mathbb {R}}^n\) and the nonlinearity f is analytic. Based on an appropriate (perturbed) new Lyapunov function and the Łojasiewicz–Simon inequality we prove that any global bounded solution converges to a steady state. We discuss also the rate of convergence which is polynomial or exponential, depending on the Łojasiewicz exponent and the decay of the term g. Keywords Evolutionary integral equation Semilinear Stabilization Łojasiewicz–Simon inequality Mathematics Subject Classification 28C15 46E05 28C99 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 (23) References1.Ben Hassen, I.: Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations. Asymptot. Anal. 69, 31–44 (2010)MathSciNetMATH2.Cavalcanti, M.M., Domingos Cavalcanti, V.N., Ferreira, J.: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 24, 1043–1053 (2001)MathSciNetCrossRefMATH3.Chepyzhov, V.V., Gatti, S., Grasselli, M., Miranville, A., Pata, V.: Trajectory and global attractors for evolution equations with memory. Appl. Math. Lett. 19, 87–96 (2006)MathSciNetCrossRefMATH4.Dafermos, C.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)MathSciNetCrossRefMATH5.Giorgi, C., Muñoz Rivera, J.E.: Semilinear hyperbolic equation viscoelasticity. J. Math. Anal. Appl. 260, 83–99 (2001)MathSciNetCrossRefMATH6.Grasselli, M., Muñoz Rivera, J.E., Pata, V.: On the energy decay of the linear thermoelastic plate with memory. J. Math. Anal. Appl. 309, 1–14 (2005)MathSciNetCrossRefMATH7.Haraux, A.: Systèmes dynamiques dissipatifs et applications. Masson, Paris (1990)MATH8.Haraux, A., Jendoubi, M.A.: Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity. Calc. Var. Partial Differ. Equ. 9, 95–124 (1999)MathSciNetCrossRefMATH9.Han, X., Wang, M.: Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Nonlinear Anal. 70, 3090–3098 (2009)MathSciNetCrossRefMATH10.Jendoubi, M.A.: Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity. J. Differ. Equ. 144, 302–312 (1998)MathSciNetCrossRefMATH11.Jendoubi, M.A.: A simple unified approach to some convergence theorem of L. Simon. J. Funct. Anal. 153, 187–202 (1998)MathSciNetCrossRefMATH12.Łojasiewicz, S., Une propriété topologique des sous-ensembles analytiques réels, Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles, Paris, : Editions du C.N.R.S. Paris 1963, 87–89 (1962)13.Łojasiewicz, S.: Sur la géometrie semi- et sous-analytique. Ann. Inst. Fourier (Grenoble) 43, 1575–1595 (1963)CrossRef14.Łojasiewicz, S.: Ensembles semi-analytiques, Preprint. I.H.E.S, Bures-sur-Yvette (1965)15.Liu, Wenjun: Uniform decay of solutions for a quasilinear system of viscoelastic equations. Nonlinear Anal. 71, 2257–2267 (2009)MathSciNetCrossRefMATH16.Liu, Wenjun: Global existence and uniform decay of solutions for a system of wave equations with dispersive and dissipative terms. Front. Math. China 5(3), 555–574 (2010)MathSciNetCrossRefMATH17.Messaoudi, S.A., Tatar, N.-E.: Global existence and asymptotic behavior for a nonlinear viscoelastic problem. Math. Methods Sci. Res. J. 7(4), 136–149 (2003)MathSciNetMATH18.Messaoudi, S.A., Tatar, N.-E.: Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math. Methods Appl. Sci. 30, 665–680 (2007)MathSciNetCrossRefMATH19.Muñoz, J.E., Rivera Naso, M.G.: Optimal energy decay rate for a class of weakly dissipative second-order systems with memory. Appl. Math. Lett. 23, 743–746 (2010)MathSciNetCrossRefMATH20.Muñoz, J.E., Rivera Naso, M.G.: Asymptotic stability of semigroups associated with linear weak dissipative systems with memory. J. Math. Anal. Appl. 326, 691–707 (2007)MathSciNetCrossRefMATH21.Park, J.Y., Kang, J.R.: Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Acta Appl. Math. 110, 1393–1406 (2010)MathSciNetCrossRefMATH22.Simon, L.: Asymptotics for a class of non-linear evolution equation with applications to geometric problem. Ann. Math. 118, 525–571 (1983)CrossRefMATH23.Shuntang, Wu: General decay of solutions for a viscoelastic equation with nonlinear damping and source terms. Acta Mathematica Scientia 31B(4), 1436–1448 (2011)MathSciNetCrossRefMATH About this Article Title Long-Time Stabilization of Solutions to a Nonautonomous Semilinear Viscoelastic Equation Journal Applied Mathematics & Optimization Volume 73, Issue 2 , pp 251-269 Cover Date2016-04 DOI 10.1007/s00245-015-9301-9 Print ISSN 0095-4616 Online ISSN 1432-0606 Publisher Springer US Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Calculus of Variations and Optimal Control; Optimization Systems Theory, Control Theoretical, Mathematical and Computational Physics Mathematical Methods in Physics Numerical and Computational Physics Keywords Evolutionary integral equation Semilinear Stabilization Łojasiewicz–Simon inequality 28C15 46E05 28C99 Authors Hassan Yassine (1) Ali Abbas (1) Author Affiliations 1. Department of Mathematics, Faculty of Sciences, Lebanese University, Baalbek-Zahle, Lebanon Continue reading... 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