Publication Cover
Applicable Analysis
An International Journal
Volume 101, 2022 - Issue 13
114
Views
2
CrossRef citations to date
0
Altmetric
Articles

Blow-up result for an abstract evolution problem with infinite memory and time-varying delay

&
Pages 4574-4597 | Received 05 Dec 2018, Accepted 04 Dec 2020, Published online: 22 Dec 2020

References

  • Levine HA. Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J Math Anal. 1974;5:138–146.
  • Levine HA. Instability and nonexistence of global solutions of nonlinear wave equation of the form Putt=Au+F(u). Trans Am Math Soc. 1974;192:1–21.
  • Georgiev V, Todorova G. Existence of solutions of the wave equation with nonlinear damping and source terms. J Differ Equ. 1994;109:295–308.
  • Vitillaro E. Global nonexistence theorems for a class of evolution equations with dissipation. Arch Rational Mech Anal. 1999;149:155–182.
  • Boukhatem Y, Benabderrahmanne B. Blow up of solutions for a semilinear hyperbolic equation. Electron J Qual Theory Differ Equ. 2012;40:1–12.
  • Boukhatem Y, Benabderrahmanne B. Polynomial decay and blow up of solutions for variable coefficients viscoelastic wave equation with acoustic boundary conditions. Acta Math Sin. 2016;32(2):153–174.
  • Levine HA, Ro Park S, Serrin J. Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation. J Math Anal Appl. 1998;228:181–205.
  • Messaoudi SA, Said-Houari B. Blow up of solutions of a class of wave equations with nonlinear damping and source terms. Math Meth Appl Sci. 2004;27:1687–1696.
  • Wu ST. Blow-up of solutions for an integro-differential equation with a nonlinear source. Electron J Differ Equ. 2006;45:1–9.
  • Wu ST, Lin CY. Global nonexistence for an integro-differential equation. Math Meth Appl Sci. 2012;35(1):72–83.
  • Messaoudi SA. Blow up of solutions with positive initial energy in a nonlinear viscoelastic equation. J Math Anal Appl. 2006;320:902–915.
  • Song H. Global nonexistence of positive initial energy solutions for a viscoelastic wave equation. Nonlinear Anal. 2015;125:260–269.
  • Datko R, Lagnese J, Polis MP. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J Control Optim. 1986;24:152–156.
  • 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.
  • Boukhatem Y, Benabderrahmanne B. General decay for a viscoelastic equation of variable coefficients in the presence of past history with delay term in the boundary feedback and acoustic boundary conditions. Acta Appl Math. 2017;37(5):1453–1471.
  • Boukhatem Y, Benabderrahmanne B. Asymptotic behavior for a past history viscoelastic problem with acoustic boundary conditions. Appl Anal. 2020;99(2):249–269.
  • Nicaise S, Pignotti C. Stabilization of the wave equation with boundary or internal distributed delay. Differ Integral Equ. 2008;21:935–958.
  • Xu CQ, Yung SP, Li LK. Stabilization of the wave system with input delay in the boundary control. ESAIM Control Optim Calc Var. 2006;12:770–785.
  • Nicaise S, Valein J, Fridman E. Stability of the heat and of the wave equations with boundary time-varying delays. Disc Cont Dyn Syst Ser 5. 2009;2(3):559–581.
  • Nicaise S, Pignotti C. Interior feedback stabilization of wave equations with time dependence delay. Electron J Differ Equ. 2011;2011(41):1–20.
  • Boukhatem Y, Benabderrahmanne B. General decay for a viscoelastic equation of variable coefficients with a time-varying delay in the boundary feedback and acoustic boundary conditions. Acta Math Sci. 2017;37(5):1453–1471.
  • Fridman E. Stabilization of second order evolution equations with unbounded feedback with time-dependent delay. SIAM J Control Optim. 2010;48(8):5028–5052.
  • Wang P, Hao J. Viscoelastic versus frictional dissipation in a variable coefficients plate system with time-varying delay. Z Angew Math Phys. 2019;70(5):148.
  • Kafini M, Messaoudi SA, Nicaise S. A blow-up result in a nonlinear abstract evolution system with delay. Nonlinear Differ Equ Appl. 2016;23(2):13.
  • Kang JR. Global nonexistence of solutions for viscoelastic wave equation with delay. Math Meth Appl Sci. 2018;41(16):6834–6841.
  • Kafini M, Messaoudi SA. Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay. Appl Anal. 2020;99(3):530–547.
  • Dafermos CM. Asymptotic stability in viscoelasticity. Arch Rational Mech Anal. 1970;37(4):297–308.
  • Kato T. Linear and quasilinear equations of evolution of hyperbolic type. In: G da Prato, G Geymonat, editors. Hyperbolicity: lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.). Vol. 2. Cortona (Arezzo): Liguori Editore; 1977. p. 125–191.
  • Kato T. Abstract differential equations and nonlinear mixed problems. Pisa: Scuola Normale Superiore; 1985. Lezioni Fermiane [Fermi Lectures].
  • Pazy A. Semigroups of linear operators and applications to partial differential equations. New York (NY): Springer; 1983.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.