Continuity of solutions of a quasilinear hyperbolic equation with hysteresis

Petra Kordulová

Applications of Mathematics (2012)

  • Volume: 57, Issue: 2, page 167-187
  • ISSN: 0862-7940

Abstract

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This paper is devoted to the investigation of quasilinear hyperbolic equations of first order with convex and nonconvex hysteresis operator. It is shown that in the nonconvex case the equation, whose nonlinearity is caused by the hysteresis term, has properties analogous to the quasilinear hyperbolic equation of first order. Hysteresis is represented by a functional describing adsorption and desorption on the particles of the substance. An existence result is achieved by using an approximation of implicit time discretization scheme, a priori estimates and passage to the limit; in the convex case it implies the existence of a continuous solution.

How to cite

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Kordulová, Petra. "Continuity of solutions of a quasilinear hyperbolic equation with hysteresis." Applications of Mathematics 57.2 (2012): 167-187. <http://eudml.org/doc/246835>.

@article{Kordulová2012,
abstract = {This paper is devoted to the investigation of quasilinear hyperbolic equations of first order with convex and nonconvex hysteresis operator. It is shown that in the nonconvex case the equation, whose nonlinearity is caused by the hysteresis term, has properties analogous to the quasilinear hyperbolic equation of first order. Hysteresis is represented by a functional describing adsorption and desorption on the particles of the substance. An existence result is achieved by using an approximation of implicit time discretization scheme, a priori estimates and passage to the limit; in the convex case it implies the existence of a continuous solution.},
author = {Kordulová, Petra},
journal = {Applications of Mathematics},
keywords = {hysteresis; quasilinear hyperbolic equations; generalized play operator; discontinuous solution; hysteresis; quasilinear hyperbolic equation; generalized play operator; discontinuous solution},
language = {eng},
number = {2},
pages = {167-187},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Continuity of solutions of a quasilinear hyperbolic equation with hysteresis},
url = {http://eudml.org/doc/246835},
volume = {57},
year = {2012},
}

TY - JOUR
AU - Kordulová, Petra
TI - Continuity of solutions of a quasilinear hyperbolic equation with hysteresis
JO - Applications of Mathematics
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 2
SP - 167
EP - 187
AB - This paper is devoted to the investigation of quasilinear hyperbolic equations of first order with convex and nonconvex hysteresis operator. It is shown that in the nonconvex case the equation, whose nonlinearity is caused by the hysteresis term, has properties analogous to the quasilinear hyperbolic equation of first order. Hysteresis is represented by a functional describing adsorption and desorption on the particles of the substance. An existence result is achieved by using an approximation of implicit time discretization scheme, a priori estimates and passage to the limit; in the convex case it implies the existence of a continuous solution.
LA - eng
KW - hysteresis; quasilinear hyperbolic equations; generalized play operator; discontinuous solution; hysteresis; quasilinear hyperbolic equation; generalized play operator; discontinuous solution
UR - http://eudml.org/doc/246835
ER -

References

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  1. Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces, Editure Academiei/Noordhoff International Publishing Bucuresti/Leyden (1976). (1976) Zbl0328.47035MR0390843
  2. Brokate, M., Sprekels, J., Hysteresis and Phase Transitions, Springer New York (1996). (1996) Zbl0951.74002MR1411908
  3. Crandall, M. G., An introduction to evolution governed by accretive operators. Proc. Int. Symp. Providence (1974), Dyn. Syst. 1 (1976), 131-156. (1976) MR0636953
  4. Eleuteri, M., An existence result for a p.d.e. with hysteresis, convection and a nonlinear boundary condition, Discrete Contin. Dyn. Syst., Suppl (2007), 344-353. (2007) Zbl1163.35458MR2409229
  5. Gu, T., Mathematical Modelling and Scale-up of Liquid Chromatography, Springer Berlin-New York (1995). (1995) 
  6. Kopfová, J., Entropy condition for a quasilinear hyperbolic equation with hysteresis, Differ. Integral Equ. 18 (2005), 451-467. (2005) Zbl1212.35295MR2122709
  7. Kopfová, J., 10.1142/9789812774293_0008, Dissipative phase transitions, Ser. Adv. Math. Appl. Sci. Vol. 71 P. Colli et al. World Scientific Hackensack (2006), 141-150. (2006) MR2223377DOI10.1142/9789812774293_0008
  8. Kordulová, P., Asymptotic behaviour of a quasilinear hyperbolic equation with hysteresis, Nonlinear Anal., Real World Appl. 8 (2007), 1398-1409. (2007) Zbl1132.35312MR2344989
  9. Krasnosel'skij, M. A., Pokrovskij, A. V., Systems with Hysteresis, Springer Berlin (1989). (1989) Zbl0715.73026MR0987431
  10. Krejčí, P., Hysteresis, Convexity and Dissipation in Hyperbolic Equations. GAKUTO International Series. Mathematical Sciences and Application, Gakkotosho Tokyo (1996). (1996) MR2466538
  11. Pavel, N. H., Nonlinear Evolution Operators and Semigroups, Springer Berlin (1987). (1987) Zbl0626.35003MR0900380
  12. Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44, Springer New York (1983). (1983) MR0710486
  13. Peszyńska, M., Showalter, R. E., A transport model with adsorption hysteresis, Differ. Integral Equ. 11 (1998), 327-340. (1998) Zbl1004.35033MR1741849
  14. Rhee, H.-K., Aris, R., Amundson, N. R., First-Order Partial Differential Equations. Vol. I: Theory and Applications of Single Equations, Prentice Hall Englewood Cliffs (1986). (1986) MR0993982
  15. Ruthven, D. M., Principles of Adsorption and Adsorption Processes, Wiley New York (1984). (1984) 
  16. Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer New York-Heidelberg-Berlin (1983). (1983) Zbl0508.35002MR0688146
  17. Visintin, A., Differential Models of Hysteresis, Springer Berlin (1994). (1994) Zbl0820.35004MR1329094
  18. Visintin, A., 10.1016/j.jmaa.2005.03.048, J. Math. Anal. Appl. 312 (2005), 401-419. (2005) Zbl1090.35117MR2179086DOI10.1016/j.jmaa.2005.03.048

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