On time-periodic solutions of some nonlinear parabolic equations with nonmonotone multivalued terms

Mitsuharu Otani, Vasile Staicu

DOI: http://dx.doi.org/10.12775/TMNA.2019.092

Abstract


In this paper we study the existence of time periodic solutions to a class of nonlinear parabolic equations with multivalued nonlinear terms subject to the homogeneous Dirichlet boundary condition. We give two types of existence results: one for large periodic solutions with any large data, and the other for small periodic solutions with small data. Both concern the case where the nonlinear terms contain either a upper semicontinuous multivalued term or a lower semicontinuous multivalued term. Some applications of our results are also given.

Keywords


time-periodic solutions; nonlinear parabolic equations; nonmonotone multivalued terms

Full Text:

PREVIEW FULL TEXT

References


S. Aizicovici, N. S. Papageorgiou and V. Staicu, Periodic solutions to second order differential inclusions with the scalar p−Laplacian, J. Math. Anal. Appl. 322 (2006), 913–929.

J.P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984.

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publ., Leyden, 1976.

G. Bin, X. Xiaoping and Z. Qingmei, Existence of periodic solutions for a differential inclusion system involving the p(t)-Laplacian, Acta Math. Sci. Ser. B Engl. Ed. 31 (2011), no. 5, 1786–1802.

H. Brézis, Opérateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.

T. Cardinali and N. S. Papageorgiou, Periodic problems and problems with discontinuities for nonlinear parabolic equations, Csechoslovak Math. Journal 50 (2000), 467–497.

K.C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102–129.

K.C. Chang, The obstacle problem and partial differential equations with discontinuous nonlinearities, Comm. Pure Appl. Math. 33 (1980), 117–146.

K.C. Chang, Free boundary problems and the set-valued maps, J. Differential Equations 49 (1983), 1–28.

J. Duel and P. Hess Nonlinear parabolic boundary value problems with upper and lower solutions, Israel J. Math. 29 (1978), 92–104.

A. Fryszkowski, Continuous selections for a class of nonconvex multivalued maps, Studia Math. 76 (1981), 163–174.

D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1998.

S. Heikkila and S. Hu, On fixed points of multifunctions in ordered spaces, Appl. Anal. 51 (1993), 115–127.

C.J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53–72.

N. Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces, Proc. Amer. Math. Soc. 120 (1994), 185–192.

S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I. Theory, Kluwer, Dordrecht, 1997.

S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. II. Applications, Kluwer, Dordrecht, 2000.

D.A. Kandilakis and N.S. Papageorgiou, Periodic solutions of nonlinear evolution equations, Arch. Math. 32 (1996), 195–209.

S. Kyritsi, N. Matzakos and N.S. Papageorgiou, Periodic problems for strongly nonlinear second order differential inclusions, J. Differential Equations 183 (2002), 279–302.

L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 13 (1959), 115–162.

M. Ôtani, Non-monotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations 46(1982), 268–299.

M. Ôtani, Non-monotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Periodic problems, J. Differential Equations 54 (1984), 248–273.

M. Ôtani and V. Staicu, Existence results for quasilinear elliptic equations with multivalued nonlinear terms, Set-Valued Var. Anal. 22 (2014), 859–877.

M. Ôtani and V. Staicu On some nonlinear parabolic equations with nonmonotone multivalued terms, preprint.

V.N. Pavlenko, Strong solutions of periodic parabolic problem with discontinuous nonlinearities, Differential Equations 52 (2016), 505–516.

V.N. Pavlenko and M.S. Fedyashev, Periodic solutions of a parabolic equation with homogeneous Dirichlet boundary condition and linearly increasing discontinuous nonlinearity, Ukrainian Math. J. 64 (2013), 1231–1240.

I. Vrabie, Compactness Methods for Nonlinear Evolutions, Longman Scientific and Technical, Harlow, Essex, 1987.

I. Vrabie, Periodic for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (1990), 653–661.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism