Riv. Mat. Univ. Parma, Vol. 5, No. 2, 2014

Bárta, Tomáš[1]

Global existence for a nonlinear model of 1D chemically reacting viscoelastic body

Pages: 399-423
Received: 19 August 2013   
Accepted: 18 December 2013
Mathematics Subject Classification (2010): 45K05, 45G10, 74D10.

Keywords: Nonconvolution integral equation, quasilinear hyperbolic integral equations, chemically reacting viscoelastic materials.
Author address:
[1] : Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Prague, Sokolovska 83, 180 00 Prague 8, Czech Republic

Abstract: In this paper we show existence of a global classical solution to a quasilinear hyperbolic integrodifferential equation of non-convolutionary type for small data. We apply the result to show global existence for a one-dimensional model of a chemically reacting viscoelastic body.

This work is supported by GACR 201/09/0917. Author is a researcher in the University Centre for Mathematical Modeling, Applied Analysis and Computational Mathematics (Math MAC) and a member of the Necas Center for Mathematical Modeling.


[1] T. Bárta, Nonmonotone nonconvolution functions of positive type and applications, Comment. Math. Univ. Carolin. 53 (2012), no. 2, 211-220. [MR3017255]
[2] T. Bárta, Global existence for a system of nonlocal PDEs with applications to chemically reacting incompressible fluids, Cent. Eur. J. Math. 11 (2013), no. 6, 1112-1128. [MR3036022]
[3] M. Bulicek, J. Malek and K. R. Rajagopal, Mathematical results concerning unsteady flows of chemically reacting incompressible fluids, Partial differential equations and fluid mechanics, London Math. Soc. Lecture Note Ser., 364, Cambridge Univ. Press, Cambridge 2009, 26-53. [MR2605756 ]
[4] C. M. Dafermos and J. A. Nohel, A nonlinear hyperbolic Volterra equation in viscoelasticity, Contributions to analysis and geometry (Baltimore, Md., 1980), Johns Hopkins Univ. Press, Baltimore, Md. 1981, 87-116. [MR0648457]
[5] W. J. Hrusa and J. A. Nohel, The Cauchy problem in one-dimensional nonlinear viscoelasticity, J. Differential Equations 59 (1985), no. 3, 388-412. [MR0807854]
[6] K. R. Rajagopal and A. S. Wineman, Applications of viscoelastic clock models in biomechanics, Acta Mech. 213 (2010), no. 3-4, 255-266. [DOI]
[7] M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, 35, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York 1987. [MR0919738]

Home Riv.Mat.Univ.Parma