Existence of periodic solutions to fractional \(p(z)\)-Laplacian parabolic problems

Ghizlane Zineddaine, Said Melliani, Abderrazak Kassidi

Abstract


We consider a class of nonlinear parabolic initial boundary value problems having the fractional $p(z)$-Laplacian operator. By combining variable exponent fractional Sobolev spaces with topological degree theory, we establish the existence of a time-periodic non-trivial weak solution.

Keywords


Periodic solutions; Fractional $p(z)$-Laplacian; Topological degree; Parabolic equations

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References


Abbassi, A., Allalou, C., Kassidi, A., Topological degree methods for a Neumann problem governed by nonlinear elliptic equation, Moroccan Journal of Pure and Applied Analysis, 6(2)(2020a), 231-242.

Abbassi, A., Allalou, C., Kassidi, A., Existence of weak solutions for nonlinear p-elliptic problem by topological degree, Nonlinear Dyn. Syst. Theory, 20(3)(2020b), 229-241.

Abbassi, A., Allalou, C., Kassidi, A., Existence results for some nonlinear elliptic equations via topological degree methods, J. Elliptic Parabol. Equ., 7(1)(2021), 121-136.

Adhikari, D.R., Asfaw, T.M., Stachura, E., A topological degree theory for perturbed AG (S+)-operators and applications to nonlinear problems, J. Math. Anal. Appl., 497(2)(2021), 124912.

Adhikari, D.R., Kartsatos, A.G., Strongly quasibounded maximal monotone perturbations for the Berkovits-Mustonen topological degree theory, J. Math. Anal. Appl., 348(1)(2008), 122-136.

Allalou, C., Abbassi, A., Kassidi, A., The discontinuous nonlinear Dirichlet boundary value problem with p-Laplacian, Azerb. J. Math., 11(2021), no. 2.

Applebaum, V., Levy processes-from probability to finance quantum groups, Notices Am. Math. Soc., 51(2004), 1336-1347.

Asfaw, T.M., A degree theory for compact perturbations of monotone type operators and application to nonlinear parabolic problem, Abstract and Appl. Anal, (2017).

Asfaw, T.M., Topological degree theories for continuous perturbations of resolvent compact maximal monotone operators, existence theorems and applications, J. Nonlinear Sci. Appl., (JNSA), 13(5)(2020).

Azroul, E., Benkirane, A., Shimi, M., Eigenvalue problems involving the fractional p(x)- Laplacian operator, Adv. Oper. Theory, 4(2019), no. 2, 539-555.

Bahrouni, A., R adulescu, V.D., On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. Ser. S, 11(2018), no. 3, 379-389.

Berkovits, J., Extension of the Leray-Schauder degree for abstract Hammerstein type mappings, J. Differential Equations, 234(2007), no. 1, 289-310.

Berkovits, J., Mustonen, V., Topological degree for perturbations of linear maximal

monotone mappings and applications to a class of parabolic problems, Rend. Mat. Appl.,

(1992), no. 3, 597-621.

Bhauryal, N., Koley, U, Vallet, G., A fractional degenerate parabolic-hyperbolic Cauchy problem with noise, Journal of Di erential Equations, 284(2021), 433-521.

Bisci, G.M., R adulescu, V.D., Servadei, R., Variational methods for nonlocal fractional problems, Vol. 162, Cambridge University Press, 2016.

Boudjeriou, T., Global existence and blow-up of solutions for a parabolic equation involving the fractional p(x)-Laplacian, Appl. Anal., 101(8)(2022), 2903-2921.

Brezis, H., Lieb, E.H., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88(1983), 486-490.

Caffarelli, L., Non-local diffusions, drifts and games, in nonlinear partial differential equations, Abel Symp., 7(2012) 37-52.

Chen, Y., Levine, S., Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66(2006), no. 4, 1383-1406.

Cho, Y.J., Chen, Y.Q., Topological Degree Theory and Applications, Chapman and Hall/CRC, 2006.

Fan, X., Zhao, D., On the spaces $L^{p(x)}(Omega)$ and $W^{m,p(x)}(Omega)$}+AG3 J. Math. Anal. Appl., {bf 263}(2001), no. 2, 424--446.

Fu, Y., Pucci, P., On solutions of space-fractional di usion equations by means of potential wells, Electron. J. Qual. Theory Di er. Equ., 70(2016), 1-17.

El Hammar, H., Allalou, C., Abbassi, A., Kassidi, A., The topological degree methods for the fractional p( )-Laplacian problems with discontinuous nonlinearities, Cubo (Temuco), 24(1)(2022), 63-82.

Hammou, M.A., Azroul, E., Existence of weak solutions for a nonlinear parabolic equations by topological degree, Advances in the Theory of Nonlinear Analysis and its Applications, 4(4) (2020), 292-298.

Hammou, M.A., Quasilinear parabolic problem with p(z)-Laplacian operator by topological degree, Kragujevac J. Math., 47(4)(2023), 523-529.

Kaufmann, U., Rossi, J.D., Vidal, R., Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians, Electron. J. Qual. Theory Differ. Equ., Paper no. 76(2017), 10.

Kovacik, O., Rakosnk, J., On spaces Lp(x) and Wk;p(x), Czechoslovak Math. J., 41(116)(1991), no. 4, 592{618.

Laskin, N., Fractional quantum mechanics and L evy path integrals, Physics Letters A, 268(2000), no. 4-6, 298-305.

Lions, J.L., Quelques M ethodes de Resolution des Probl emes aux Limites Non-Lineaires, Dunod, Paris, 1969.

Liu, J.J., Pucci, P., Zhang, Q.H., Wave breaking analysis for the periodic rotation-two-component Camassa-Holm system, Nonlinear Anal., 187(2019), 2146228.

Mazon, J.M., Rossi, J.D., Toledo, J., Fractional p-Laplacian evolution equation, J. Math. Pures Appl., 165(9)(2016), 810-844.

Metzler, R., Klafter, J., The random walk's guide to anomalous di usion: A fractional dynamics approach, Phys. Rep., 339(2000), no. 1, 77.

Pan, N., Pucci, P., Xu, R.Z., Zhang, B.L., Degenerate Kirchho -type wave problems involving the fractional Laplacian with nonlinear damping and source terms, J. Evol. Equ., 19(3)(2019), 615-643.

Perera, K., Agarwal, R.P., O'Regan, D., Morse theoretic aspects of p-Laplacian type operators, vol. 161 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010.

Pucci, P., Xiang, M., Zhang, B., A diffusion problem of Kirchho type involving the nonlocal fractional p-Laplacian, Discrete Contin. Dyn. Syst., 37(7)(2017), 4035.

Tao, M., Zhang, B., Positive solutions for perturbed fractional p-Laplacian problems, Fractal and Fractional, 6(10)(2022), 571.

Temghart, S.A, Kassidi, A., Allalou, C., Abbassi, A., On an elliptic equation of Kirchho type problem via topological degree, Nonlinear Stud., 28.4(2021), 1179-1193.

Tsutsumi, M., Existence and nonexistence of global solutions of nonlinear parabolic equations, Publ. Res. Inst. Math. Sci., 8(1972/1973), 211-229.

Yacini, S., Allalou, C., Hilal, K., Kassidi, A., Weak solutions Kirchho type problem via topological degree, Advanced Mathematical Models & Applications, Jomard Publishing, 6(3)(2021), 309-321.

Zeider, E., Nonlinear Functional Analysis and its Applications, IInB: Nonlinear Monotone Operators, Springer, New York, 1990.




DOI: http://dx.doi.org/10.24193/subbmath.2025.1.05

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