Weak Periodic Solutions of Some Quasilinear Parabolic Equations with Data Measures

Volume 3, Issue 3, 2002

Article 46

WEAK PERIODIC SOLUTIONS OF SOME QUASILINEAR PARABOLIC EQUATIONS WITH DATA MEASURES

FACULTÉ DES SCIENCES ET TÉCHNIQUES GUELIZ,
DÉPARTEMENT DE MATHÉMATIQUES ET INFORMATIQUE.
B.P.618 MARRAKECH-MAROC.
E-Mail: alaa@fstg-marrakech.ac.ma

FACULTÉ DES SCIENCES SEMLALIA,
DÉPARTEMENT DE MATHÉMATIQUES.
B.P.2390 MARRAKECH-MAROC.
E-Mail: iguernane@ucam.ac.ma

Received 03, August, 2001; Accepted 17 April, 2002.
Communicated by: D. Bainov

ABSTRACT. The goal of this paper is to study the existence of weak periodic solutions for some quasilinear parabolic equations with data measures and critical growth nonlinearity with respect to the gradient. The classical techniques based on C $^{\alpha }$ -estimates for the solution or its gradient cannot be applied because of the lack of regularity and a new approach must be considered. Various necessary conditions are obtained on the data for existence. The existence of at least one weak periodic solution is proved under the assumption that a weak periodic super solution is known.The results are applied to reaction-diffusion systems arising from chemical kinetics.

Key words:
Quasilinear equations, Periodic, Parabolic, Convex nonlinearities, Data measures, Nonlinear capacities.

2000 Mathematics Subject Classification:
35K55, 35K57, 35B10, 35D05, 31C15.

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