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.
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References
V. Barbu: Nonlinear Semigroups and Differential Equations in Banach Spaces. Editure Academiei/Noordhoff International Publishing, Bucuresti/Leyden, 1976.
M. Brokate, J. Sprekels: Hysteresis and Phase Transitions. Springer, New York, 1996.
M.G. Crandall: An introduction to evolution governed by accretive operators. Proc. Int. Symp. Providence (1974). Dyn. Syst. 1 (1976), 131–156.
M. Eleuteri: An existence result for a p.d.e. with hysteresis, convection and a nonlinear boundary condition. Discrete Contin. Dyn. Syst., Suppl. 2007, pp. 344–353.
T. Gu: Mathematical Modelling and Scale-up of Liquid Chromatography. Springer, Berlin-New York, 1995.
J. Kopfová: Entropy condition for a quasilinear hyperbolic equation with hysteresis. Differ. Integral Equ. 18 (2005), 451–467.
J. Kopfová: Hysteresis in a first order hyperbolic equation. Dissipative phase transitions, Ser. Adv. Math. Appl. Sci. Vol. 71 (P. Colli et al., eds.). World Scientific, Hackensack, 2006, pp. 141–150.
P. Kordulová: Asymptotic behaviour of a quasilinear hyperbolic equation with hysteresis. Nonlinear Anal., Real World Appl. 8 (2007), 1398–1409.
M.A. Krasnosel’skij, A.V. Pokrovskij: Systems with Hysteresis. Springer, Berlin, 1989.
P. Krejčí: Hysteresis, Convexity and Dissipation in Hyperbolic Equations. GAKUTO International Series. Mathematical Sciences and Application. Gakkotosho, Tokyo, 1996.
N.H. Pavel: Nonlinear Evolution Operators and Semigroups. Springer, Berlin, 1987.
A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44. Springer, New York, 1983.
M. Peszyńska, R.E. Showalter: A transport model with adsorption hysteresis. Differ. Integral Equ. 11 (1998), 327–340.
H.-K. Rhee, R. Aris, N.R. Amundson: First-Order Partial Differential Equations. Vol. I: Theory and Applications of Single Equations. Prentice Hall, Englewood Cliffs, 1986.
D.M. Ruthven: Principles of Adsorption and Adsorption Processes. Wiley, New York, 1984.
J. Smoller: Shock Waves and Reaction-Diffusion Equations. Springer, New York-Heidelberg-Berlin, 1983.
A. Visintin: Differential Models of Hysteresis. Springer, Berlin, 1994.
A. Visintin: Quasilinear first-order PDEs with hysteresis. J. Math. Anal. Appl. 312 (2005), 401–419.
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This work was supported by the Grant Agency of the Czech Republic under grants 201/03/H152 and 201/02/P040.
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Kordulová, P. Continuity of solutions of a quasilinear hyperbolic equation with hysteresis. Appl Math 57, 167–187 (2012). https://doi.org/10.1007/s10492-012-0011-1
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DOI: https://doi.org/10.1007/s10492-012-0011-1