Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T02:47:16.827Z Has data issue: false hasContentIssue false
  • English
  • Français

Parabolic equations with discontinuous nonlinearities

Published online by Cambridge University Press:  17 April 2009

Giuseppina Barletta
Giuseppina Barletta
Affiliation:
Dipartimento di Matematica, Università di Messina, contrade Papardo, salita Sperone n.31, 98166 Sant' Agata (ME), Italy e-mail: [email protected] Dipartimento di Informatica, Matematica, Elettronica e Trasporti, Università di Reggio Calabria, Via Graziella (Feo di Vito), 89100 Reggio Calabria, Italy e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we deal with the homogeneous Cauchy-Dirichlet problem for a class of parabolic equations with either Carathéodory or discontinuous nonlinear terms. We then present an application and explicitly point out an existence result for a differential inclusion, which can be applied to the classical Stefan problem.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 63 , Issue 2 , April 2001 , pp. 219 - 228
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Adams, R.A., Sobolev space, Pure and Applied Mathematics (Academic Press, New York, London, 1975).Google Scholar
[2]Arino, O., Gautier, S. and Penot, J.P., ‘A fixed point theorem for sequentially continuous mappings with applications to ordinary differential equations’, Funkcial. Ekvac. 27 (1984), 273–279.Google Scholar
[3]Averna, D. and Marano, S.A., ‘Existence theorems for inclusions of the type Ψ(u)(t)∈F(t, Φ (u)(t))’, Appl. Anal. 72 (1999), 449–458.CrossRefGoogle Scholar
[4]Carl, S., ‘Enclosure of solutions for quasilinear dynamic hemivariational inequalities’, Nonlinear World 3 (1996), 281–298.Google Scholar
[5]Carl, S. and Heikkilä, S. ‘Extremal solutions of quasilinear parabolic boundary value problems with discontinuous nonlinearities’, Dynamic Systems Appl. 3 (1994), 251–258.Google Scholar
[6]Chang, K.C., ‘Free boundary problems and the set-valued mappings’, J. Differential Equations 49 (1983), 1–28.CrossRefGoogle Scholar
[7]De Giorgi, E., Buttazzo, G. and Dal Maso, G., ‘On the lower semicontinuity of certain integral functionals’, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (8) Mat. Appl. 74 (1983), 274–282.Google Scholar
[8]Deimling, K., Multivalued differential equations, de Gruyter Series Nonlin. Anal. Appl. 1 (Walter de Gruyter, Berlin, 1990).Google Scholar
[9]Frigon, M. and Saccon, C., ‘Heat equations with discontinuous nonlinearities on convex and non convex constraints’, Nonlinear Anal. 17 (1991), 923–946.CrossRefGoogle Scholar
[10]Gilbarg, D. and Trudinger, N.S., Elliptic partial differential equations of second order, Second edition (Springer-Verlag, Berlin, Heidelberg, New York, 1983).Google Scholar
[11]Heikkilä, S. and Lakshmikantham, V., Monotone iterative techniques for discontinuous nonlinear differential equations (Marcel Dekker, New York, 1994).Google Scholar
[12]Klein, E. and Thompson, A.C., Theory of correspondences (Wiley, New York, 1984).Google Scholar
[13]Ladyženskaja, O.A., Solonnikov, V.A. and Ural'ceva, N.N., Linear and quasilinear equations of parabolic type, Transl. Math. Monographs 23 (Amer. Math. Soc., Providence, R.I., 1968).CrossRefGoogle Scholar
[14]Marano, S.A., ‘Elliptic boundary-value problems with discontinuous nonlinearities’, Set-Valued Anal. 3 (1995), 167–180.CrossRefGoogle Scholar
[15]Marano, S.A., ‘Existence results for the equation − Δu = f(x, u) in Rn’, Z. Anal. Anwendungen 19 (2000) (to appear).CrossRefGoogle Scholar
[16]Papageorgiou, N.S., ‘On the existence of solutions for nonlinear parabolic problems with nonmonotone discontinuities’, J. Math. Anal. Appl. 205 (1997), 434–453.CrossRefGoogle Scholar