Journal of Integral Equations and Applications

Global existence and blow-Ups for certain ordinary integro-differential equations

Martin Saal

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We will study ordinary integro-differential equations of second order with nonlinearity given as a convolution, but differently from the widely investigated cases. In addition, the kernel depends on the solution. Such equations play a key role in the theory of glass-forming liquids, and we will establish results on global existence and investigate the long-term behavior. In contrast, we give examples where blow-ups occur.

Article information

Source
J. Integral Equations Applications, Volume 27, Number 4 (2015), 573-602.

Dates
First available in Project Euclid: 8 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1454939254

Digital Object Identifier
doi:10.1216/JIE-2015-27-4-573

Mathematical Reviews number (MathSciNet)
MR3457683

Zentralblatt MATH identifier
1334.45014

Subjects
Primary: 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 45G15: Systems of nonlinear integral equations 45J05: Integro-ordinary differential equations [See also 34K05, 34K30, 47G20]

Citation

Saal, Martin. Global existence and blow-Ups for certain ordinary integro-differential equations. J. Integral Equations Applications 27 (2015), no. 4, 573--602. doi:10.1216/JIE-2015-27-4-573. https://projecteuclid.org/euclid.jiea/1454939254


Export citation

References

  • J.M. Brader, T. Voigtmann, M. Fuchs, R.G. Larson and M.E. Cates, Glass rheology: From mode-coupling theory to a dynamical yield criterion, Proc. Nat. Acad. Sci. 106 (2009), 15186–15197.
  • T.A. Burton, Volterra integral and differential equations, Academic Press, Inc., New York, 1983.
  • S.P. Das, Mode-coupling theory and the glass transition in supercooled liquids, Rev. Mod. Phys. 76 (2004), 785–851.
  • M. Fuchs and M.E. Cates, Schematic models for dynamic yielding of sheared colloidal glasses, Faraday Discuss. 123 (2003), 267–286.
  • M.V. Gnann, I. Gazuz, A.M. Puertas, M. Fuchs and T. Voigtmann, Schematic models for active nonlinear microrheology, Soft Matter 7 (2011), 1390–1396.
  • W. Götze, The essentials of the mode-coupling theory for glassy dynamics, Cond. Matter Phys. 1 (1998), 873–904.
  • W. Götze, Complex dynamics of glass-forming liquids, Oxford University Press, Oxford, 2009.
  • W. Götze and L. Sjögren, General properties of certain non-linear integro-differential equations, J. Math. Anal. Appl. 195 (1995), 230–250.
  • G. Gripenberg, S.-O. Londen and O. Staffans, Volterra integral and functional equations, Cambridge University Press, Cambridge, 1990.
  • R. Haussmann, Some properties of mode coupling equations, Z. Phys.–Condensed Matter 79 (1990), 143–148.
  • W. Kob, The mode-coupling theory of the glass transition, Supercooled Liquids 4 (1997), 28–44
  • P. Kurth, On a new class of integro-differential equations, Konst. Schrift. Math. 326, 2014.
  • J. Prüß, Evolutionary integral equations and applications, Birkhäuser, New York, 2012.
  • M. Saal, Well-posedness and asymptotics of some nonlinear integro-differential equations, J. Integral Equations Appl. 25 (2013), 103–141.
  • ––––, Nichtlineare Integro-Differentialgleichungen zweiter Ordnung, Ph.D. thesis, University of Konstanz, Germany, 2014.