On a new class of integro-differential equations. (English) Zbl 1307.45008
Summary: We consider various initial-value problems for ordinary integro-differential equations of first order that are characterized by convolution-terms, where all factors depend on the solutions of the equations. Applications of such problems are descriptions of certain glass-transition phenomena based on mode-coupling theory, for instance. We will prove results concerning well-posedness of such problems and the asymptotic behaviour of their solutions.
MSC:
| 45J05 | Integro-ordinary differential equations |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 45M05 | Asymptotics of solutions to integral equations |
| 45D05 | Volterra integral equations |
| 45G10 | Other nonlinear integral equations |
| 82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |
Keywords:
well-posedness; integro-differential equations; asymptotic behaviour; glass-transition; nonlinear Volterra equation; phase transition; initial value problem; convolutionReferences:
| [1] | J.A.D. Appleby, I. Györi and D.W. Reynolds, Subexponential solutions of linear scalar integrodifferential equations with a delay , Funct. Diff. Eq. 11 (2004), 11–18. · Zbl 1063.45005 |
| [2] | J.A.D. Appleby and D.W. Reynolds, On the non-exponential convergence of asymptotically stable solutions of linear scalar volterra integrodifferential equations , J. Int. Eq. Appl. 14 (2002), 109–118. · Zbl 1041.45009 · doi:10.1216/jiea/1031328362 |
| [3] | —-, On necessary and sufficient conditions for exponential stability in linear Volterra integro-differential equations , J. Int. Eq. Appl. 16 (2004), 221–240. · Zbl 1088.45005 · doi:10.1216/jiea/1181075283 |
| [4] | —-, On the non-exponential decay to equilibrium of solutions of nonlinear scalar Volterra integro-differential equations , Electr. J. Qual. Theor. Diff. Eq., Proc. 7th Coll. QTDE (2004), 1–14. |
| [5] | J.M. Brader, M. Siebenbürger, M. Ballauff, K. Reinheimer, M. Wilhelm, S.J. Frey, F. Weysser and M. Fuchs, Nonlinear response of dense colloidal suspensions under oscillatory shear : Mode-coupling theory and ft-rheology experiments , Phys. Rev. E 82 (061401), 2010. |
| [6] | 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. Natl. Acad. Sci. 106 (2009), 15186-15191. |
| [7] | H. Brunner and Z.W. Yang, Blow-up behavior of Hammerstein-type Volterra integral equations , J. Int. Eq. Appl. 24 (2012), 487. · Zbl 1261.45003 · doi:10.1216/JIE-2012-24-4-487 |
| [8] | P.J. Bushell and W. Okrasinski, On the maximal interval of existence for solutions to some non-linear Volterra integral equations with convolution kernel , Bull. Lond. Math. Soc. 28 (1996), 59–65. · Zbl 0841.45008 · doi:10.1112/blms/28.1.59 |
| [9] | M. Fuchs and M.E. Cates, Schematic models for dynamic yielding of sheared colloidal glasses , Faraday Disc. 123 (2003), 267. |
| [10] | 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. |
| [11] | W. Götze, Complex dynamics of glass-forming liquids , Oxford University Press, Oxford, 2009. |
| [12] | W. Götze and L. Sjögren, General properties of certain non-linear integrodifferential equations , J. Math. Anal. Appl. 195 (1995), 230–250. · Zbl 0853.45011 · doi:10.1006/jmaa.1995.1352 |
| [13] | G. Gripenberg, S.-O. Londen and O. Staffans, Volterra integral and functional equations , Cambridge University Press, Cambridge, 2009. · Zbl 1159.45001 |
| [14] | R.A. Handelsman and W.E. Olmstead, Asymptotic analysis of a class of nonlinear integral equations , SIAM J. Appl. Math. 22 (1972), 373–384. · Zbl 0237.45019 · doi:10.1137/0122035 |
| [15] | C.B. Holmes, M.E. Cates, M. Fuchs and P. Sollich, Glass transitions and shear thickening suspension rheology , J. Rheol. 49 (2005), 237–269. |
| [16] | P. Kurth, Nichtlineare Integro-Differentialgleichungen aus der mathematischen Physik , Ph.D. thesis, University of Konstanz, 2013. |
| [17] | P. Kurth, On a new class of partial integro-differential equations , Konstanzer Schrift. Math. 327 (2014), 1-24. |
| [18] | W.A.J. Luxemburg, On an asymptotic problem concerning the Laplace transform , Appl. Anal. 8 (1978), 67–70. · Zbl 0394.44001 · doi:10.1080/00036817808839212 |
| [19] | J. Prüss, Evolutionary integral equations and applications , Birkhäuser, Basel, 2012. |
| [20] | M. Saal, Well-posedness and asymptotics of some nonlinear integrodifferential equations , J. Int. Eq. Appl. 25 (2013), 103–141. · Zbl 1277.45014 · doi:10.1216/JIE-2013-25-1-103 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
