Existence of solutions for nonlinear parabolic problems
Halidias, Nikolaos ; Papageorgiou, Nikolaos S.
Archivum Mathematicum, Tome 035 (1999), p. 255-274 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

We consider nonlinear parabolic boundary value problems. First we assume that the right hand side term is discontinuous and nonmonotone and in order to have an existence theory we pass to a multivalued version by filling in the gaps at the discontinuity points. Assuming the existence of an upper solution $\phi $ and of a lower solution $\psi $ such that $\psi \le \phi $, and using the theory of nonlinear operators of monotone type, we show that there exists a solution $x \in [\psi ,\phi ]$ and that the set of all such solutions is compact in $W_{pq}(T)$. For the problem with a Caratheodory right hand side we show the existence of extremal solutions in $[\psi ,\phi ]$.

Publié le : 1999-01-01
Classification:  35K55,  35K60 
@article{107700,
     author = {Nikolaos Halidias and Nikolaos S. Papageorgiou},
     title = {Existence of solutions for nonlinear parabolic problems},
     journal = {Archivum Mathematicum},
     volume = {035},
     year = {1999},
     pages = {255-274},
     zbl = {1046.35054},
     mrnumber = {1725842},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107700}
}

Halidias, Nikolaos; Papageorgiou, Nikolaos S. Existence of solutions for nonlinear parabolic problems. Archivum Mathematicum, Tome 035 (1999) pp. 255-274. http://gdmltest.u-ga.fr/item/107700/

Aizicovici S.; Papageorgiou N. S. Infinite dimensional parametric optimal control problems, Math. Nachr. 162 (1993), 17–38. (1993) | MR 1239573 | Zbl 0807.49001

Boccardo L.; Murat F.; Puel J.-P. Existence results for some quasilinear parabolic problems, Nonlin. Anal-TMA 13 (1989), 373–392. (1989) | MR 0987375

Carl S. On the existence of extremal weak solutions for a class of quasilinear parabolic problems, Diff. Integ. Eqns 6 (1993), 1493–1505. (1993) | MR 1235207 | Zbl 0805.35057

Carl S. Enclosure of solution for quasilinear dynamic hemivariational inequalities, Nonlin. World 3 (1996), 281–298. (1996) | MR 1411356

Chang K.-C. Variational methods for nondifferentiable functions and their applications to partial differential equations, J. Math. Anal. 80 (1981), 102–129. (1981) | MR 0614246

Chipot M.; Rodrigues; J-E. Comparison and stability of solutions to a class of quasilinear parabolic problems, Proc. Royal Soc. Edinburgh 110 A (1988), 275–285. (1988) | MR 0974743 | Zbl 0669.35052

Costa D. G.; Goncalves J. V. A. Critical point theory for nondifferentiable functionals and applications, J. Math. Anal. 153 ( 1990), 470–485. (1990) | MR 1080660 | Zbl 0717.49007

Dancer E. N.; Sweers G. On the existence of maximal weak solution for a semilinear elliptic equation, Diff. Integr. Eqns 2 (1989), 533–540. (1989) | MR 0996759

Deuel J.; Hess P. Nonlinear parabolic boundary value problems with upper and lower solutions, Israel J. Math. 29 (1978), 92–104. (1978) | MR 0492636 | Zbl 0372.35045

Dunford N.; Schwartz J. Linear Operators I, Wiley, New York (1958). (1958)

Hu S.; Papageorgiou N. S. Handbook of Multivalued Analysis, Volume I: Theory, Kluwer, Dordrecht, The Netherlands (1997). (1997) | MR 1485775 | Zbl 0887.47001

Evans L.; Gariepy R. Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton (1992). (1992) | MR 1158660 | Zbl 0804.28001

Halidias N.; Papageorgiou N. S. Second order multivalued boundary value problems, Archivum Math. (Brno) 34 (1998), 267–284. (1998) | MR 1645320 | Zbl 0915.34021

Kandilakis D.; Papageorgiou N. S. Nonlinear periodic parabolic problems with nonmonotone discontinuities, Proc. Edinburgh Math. Soc. 41 (1998), 117–132. (1998) | MR 1604345 | Zbl 0909.35074

Kesavan S. Topics in Functional Analysis and Applications, Wiley, New York (1989). (1989) | MR 0990018 | Zbl 0666.46001

Lions J.-L. Quelques Methodes de Resolution des Problems aux Limits Non-Lineaires, Dunod, Paris (1969). (1969) | MR 0259693

Miettinen M. Approximation of hemivariational inequalities and optimal control problem, Univ. of Jyvaskyla, Math. Department, Finland, Report 59 (1993). (1993) | MR 1248824

Miettinen M. A parabolic hemivariational inequality, Nonl. Anal-TMA 26 (1996), 725–734. (1996) | MR 1362746 | Zbl 0858.35072

Mokrane A. Existence of bounded solutions for some nonlinear parabolic equations, Proc. Royal Soc. Edinburgh 107 (1987), 313–326. (1987) | MR 0924524

Panagiotopoulos P. D. Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer Verlag, New York, Berlin (1994). (1994) | MR 1385670

Rauch J. Discontinuous semilinear differential equations and multiple-valued maps, Proc. AMS 64 (1977), 272–282. (1977) | MR 0442453 | Zbl 0413.35031

Stuart C. Maximal and minimal solutions of elliptic equations with discontinuous nonlinearities, Math. Zeitschrift 163 (1978), 239–249. (1978) | MR 0513729

Zeidler E. Nonlinear Functional Analysis and its Applications, Springer Verlag, New York (1990). (1990) | Zbl 0684.47029