Strongly nonlinear periodic parabolic equation in Orlicz spaces
DOI:
https://doi.org/10.24193/subbmath.2025.1.04Keywords:
The periodic solution, nonlinear parabolic equation, Galerkin method, Orlicz spaces, weak solutionsAbstract
In this paper, we prove the existence of a weak solution to the following nonlinear periodic parabolic equations in Orlicz-spaces: $$\frac{\partial u}{\partial t}-div(a(x,t,\nabla u))=f(x,t)$$ where $-div(a (x,t,\nabla u))$ is a Leray-Lions operator defined on a subset of $W^{1,x}_{0}L_{M}(Q)$. The $\Delta_{2}$-condition is not assumed and the data $f$ belongs to $W^{-1,x}E_{\overline{M}}(Q)$.\\ The Galerkin method and the fixed point argument are employed in the proof.
Mathematics Subject Classification (2010): 35B10, 35A01, 35D30.
Received 26 August 2023; Accepted 05 February 2024.
References
[1] Agarwal, R.P., Alghamdi, A.M., Gala, S., Ragusa, M.A., On the regularity criterionon one velocity component for the micropolar fluid equations, Math. Model. Anal., 28(2)(2023), 271-284.
[2] Azroul, E., Redwane, H., Rhoudaf, M., Existence of a renormalized solution for a class of nonlinear parabolic equations in Orlicz spaces, Port. Math., 66(1)(2009), 29-63.
[3] Benkirane, A., Elmahi, A., Almost everywhere convergence of the gradients of solutions to elliptic equations in Orlicz spaces and application, Nonlinear Anal, Theory Methods Appl., 28(1997), 1769-1784.
[4] Boldrini, J.L., Crema, J., On forced periodic solutions of superlinear quasiparabolic problems, Electron. J. Differential Equations, 14(1998), 1-18.
[5] Brézis, H., Browder, F.E., Strongly nonlinear parabolic initial boundary value problems, Proc. Nat. Acad. Sci. U.S.A., 76(1979), 38-40.
[6] Donaldson, T., Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial boundary value problems, J. Differential Equations, 16(1974), 201-256.
[7] El Hachimi, A., Lamrani Alaoui, A., Existence of stable periodic solutions for quasilinear parabolic problems in the presence of well-ordered lower and upper-solutions, Electron.
J. Differential Equations, 9(2002), 117-126.
[8] El Hachimi, A., Lamrani Alaoui, A., Time periodic solutions to a nonhomogeneous Dirichlet periodic problem, Appl. Math. E-Notes, 8)(2008), 1-8.
[9] El Hachimi, A., Lamrani Alaoui, A., Periodic solutions of nonlinear parabolic equations with measure data and polynomial growth in |∇u|, Recent Developments in Nonlinear Analysis, (2010).
[10] El-Houari, H., Chadli, L.S., Moussa, H., On a class of Schrodinger system Problem in Orlicz-Sobolev spaces, J. Funct. Spaces, 2022(2022).
[11] Elmahi, A., Strongly nonlinear parabolic initial-boundary value problems in Orlicz spaces, Electron. J. Differ. Equ. Conf., 9(2002), 203-220.
[12] Elmahi, A., Meskine, D., Parabolic initial-boundary value problems in Orlicz spaces, Ann. Polon. Math., 85(2005), 99-119.
[13] Gossez, J.-P., Some approximation properties in Orlicz-Sobolev spaces, Studia Math.,
74(1982), 17-24.
[14] Gwiazda, Swierczewska-Gwiazda, P., Wóblewska-Kamińska, A., Generalized Stokes system in Orlicz space, Discrete Contin. Dyn. Syst., 32(6)(2012), 2125-2146.
[15] Gwiazda, Wittbold, P., Wóblewska-Kamińska, P., Zimmermann, A., Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces, J. Differential Equations, 253(2012), 635-666.
[16] Gwiazda, Wittbold, P., Wóblewska-Kamińska, P., Zimmermann, A., Renormalized solutions to nonlinear parabolic problems in generalized Musielak Orlicz spaces, Nonlinear Anal., 129(2015), 1-36.
[17] Krasnosel’skii, M., Rutickii, Y., Convex Functions and Orlicz Spaces, P. Noordhoff Groningen, 1969.
[18] Landes, R., On Galerkin’s method in the existence theory of quasilinear elliptic equations, J. Funct., 39(1980), 123-148.
[19] Landes, R., Mustonen, V., A strongly nonlinear parabolic initial boundary value problem, Ark. Mat., 25(1987) 29-40.
[20] Robert, J., Équations d’évolution; paraboliques fortement non linéaires, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 1.3-4(1974), 247-259.
[21] Yee, T.L., Cheung, K.L., Ho, K.P., Integral operators on local Orlicz-Morrey spaces, Filomat, 36(4)(2022), 1231-1243.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
