| Title:
|
Continuity of solutions of a quasilinear hyperbolic equation with hysteresis (English) |
| Author:
|
Kordulová, Petra |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
57 |
| Issue:
|
2 |
| Year:
|
2012 |
| Pages:
|
167-187 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
hysteresis |
| Keyword:
|
quasilinear hyperbolic equations |
| Keyword:
|
generalized play operator |
| Keyword:
|
discontinuous solution |
| MSC:
|
34C55 |
| MSC:
|
35L04 |
| MSC:
|
35L50 |
| MSC:
|
35L60 |
| MSC:
|
35L65 |
| MSC:
|
35S30 |
| idZBL:
|
Zbl 1249.35214 |
| idMR:
|
MR2899730 |
| DOI:
|
10.1007/s10492-012-0011-1 |
| . |
| Date available:
|
2012-03-05T07:04:40Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142034 |
| . |
| Reference:
|
[1] Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces.Editure Academiei/Noordhoff International Publishing Bucuresti/Leyden (1976). Zbl 0328.47035, MR 0390843 |
| Reference:
|
[2] Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions.Springer New York (1996). Zbl 0951.74002, MR 1411908 |
| Reference:
|
[3] Crandall, M. G.: An introduction to evolution governed by accretive operators. Proc. Int. Symp. Providence (1974).Dyn. Syst. 1 (1976), 131-156. MR 0636953 |
| Reference:
|
[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. Zbl 1163.35458, MR 2409229 |
| Reference:
|
[5] Gu, T.: Mathematical Modelling and Scale-up of Liquid Chromatography.Springer Berlin-New York (1995). |
| Reference:
|
[6] Kopfová, J.: Entropy condition for a quasilinear hyperbolic equation with hysteresis.Differ. Integral Equ. 18 (2005), 451-467. Zbl 1212.35295, MR 2122709 |
| Reference:
|
[7] Kopfová, J.: Hysteresis in a first order hyperbolic equation.Dissipative phase transitions, Ser. Adv. Math. Appl. Sci. Vol. 71 P. Colli et al. World Scientific Hackensack (2006), 141-150. MR 2223377, 10.1142/9789812774293_0008 |
| Reference:
|
[8] Kordulová, P.: Asymptotic behaviour of a quasilinear hyperbolic equation with hysteresis.Nonlinear Anal., Real World Appl. 8 (2007), 1398-1409. Zbl 1132.35312, MR 2344989 |
| Reference:
|
[9] Krasnosel'skij, M. A., Pokrovskij, A. V.: Systems with Hysteresis.Springer Berlin (1989). Zbl 0715.73026, MR 0987431 |
| Reference:
|
[10] Krejčí, P.: Hysteresis, Convexity and Dissipation in Hyperbolic Equations. GAKUTO International Series. Mathematical Sciences and Application.Gakkotosho Tokyo (1996). MR 2466538 |
| Reference:
|
[11] Pavel, N. H.: Nonlinear Evolution Operators and Semigroups.Springer Berlin (1987). Zbl 0626.35003, MR 0900380 |
| Reference:
|
[12] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44.Springer New York (1983). MR 0710486 |
| Reference:
|
[13] Peszyńska, M., Showalter, R. E.: A transport model with adsorption hysteresis.Differ. Integral Equ. 11 (1998), 327-340. Zbl 1004.35033, MR 1741849 |
| Reference:
|
[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). MR 0993982 |
| Reference:
|
[15] Ruthven, D. M.: Principles of Adsorption and Adsorption Processes.Wiley New York (1984). |
| Reference:
|
[16] Smoller, J.: Shock Waves and Reaction-Diffusion Equations.Springer New York-Heidelberg-Berlin (1983). Zbl 0508.35002, MR 0688146 |
| Reference:
|
[17] Visintin, A.: Differential Models of Hysteresis.Springer Berlin (1994). Zbl 0820.35004, MR 1329094 |
| Reference:
|
[18] Visintin, A.: Quasilinear first-order PDEs with hysteresis.J. Math. Anal. Appl. 312 (2005), 401-419. Zbl 1090.35117, MR 2179086, 10.1016/j.jmaa.2005.03.048 |
| . |
