Weak solution for nonlinear fractional p(.)-Laplacian problem with variable order via Rothe's time-discretization method
Abstract
In this paper, we prove the existence and uniqueness results of weak solutions to a class of nonlinear fractional parabolic p(.)-Laplacian problem with variable order. The main tool used here is the Rothe’s method combined with the theory of variable-order fractional Sobolev spaces with variable exponent.
Keyword : fractional p(.)-Laplacian, fractional Sobolev space, semi-discretization, Rothe’s method, variable exponent, variable order, weak solution
How to Cite
Sabri, A. (2022). Weak solution for nonlinear fractional p(.)-Laplacian problem with variable order via Rothe’s time-discretization method. Mathematical Modelling and Analysis, 27(4), 533–546. https://doi.org/10.3846/mma.2022.15740
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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L.M. Del Pezzo and J.D. Rossi. Traces for fractional Sobolev spaces with variable exponents. Advances in Operator Theory, 2(4):435–446, 2017. https://doi.org/10.22034/aot.1704-1152
G. Devillanova and G.C. Marano. A free fractional viscous oscillator as a forced standard damped vibration. Fractional Calculus and Applied Analysis, 19(2):319–356, 2016. https://doi.org/10.1515/fca-2016-0018
E. Di Nezza, G. Palatucci and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces. Bulletin des Sciences Math´ematiques, 136(5):521–573, 2012. https://doi.org/10.1016/j.bulsci.2011.12.004
A. El Hachimi and A. Jamea. Nonlinear parabolic problems with Neumann-type boundary conditions and L1-data. Electronic Journal of Qualitative Theory of Differential Equations, 27:1–22, 2007. https://doi.org/10.14232/ejqtde.2007.1.27
X. Fan and D. Zhao. On the spaces Lp(x)(Ω) and W m,p(x)(Ω). Journal of Mathematical Analysis and Applications, 263(2):424–446, 2001. https://doi.org/10.1006/jmaa.2000.7617
J. Giacomoni and S. Tiwari. Existence and global behavior of solutions to fractional p-Laplacian parabolic problems. Electronic Journal of Differential Equations, 2018(44):1–20, 2018.
U. Kaufmann, J.D. Rossi and R. Vidal. Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians. Electronic Journal of Qualitative Theory of Differential Equations, 2017(76):1–10, 2017. https://doi.org/10.14232/ejqtde.2017.1.76
O. Kovácık and J. Rákosnık. On spaces lp(x) and wk,p(x). Czechoslovak Mathematical Journal, 41(4):592–618, 1991. https://doi.org/10.21136/CMJ.1991.102493
J.M. Mazón, J.D. Rossi and J. Toledo. Fractional p-Laplacian evolution equations. Journal de Mathématiques Pures et Appliquées, 105(6):810–844, 2016. https://doi.org/10.1016/j.matpur.2016.02.004
G. Molica Bisci, V.D. Radulescu and R. Servadei. Variational methods for nonlocal fractional problems, volume 162 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781316282397
V.D. Rǎdulescu and D.D. Repovš. Partial differential equations with variable exponents. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015. https://doi.org/10.1201/b18601
A. Sabri, H.T. Alaoui and A. Jamea. Existence of weak solutions for fractional p-Laplacian problem with Dirichlet-type boundary condition. Discussiones Mathematicae: Differential Inclusions, Control and Optimization, 39(1):69–80, 2019. https://doi.org/10.7151/dmdico.1211
A. Sabri and A. Jamea. Rothe time-discretization method for a nonlinear parabolic p(u)-Laplacian problem with Fourier-type boundary condition and L1-data. Ricerche di Matematica, pp. 1–24, 2020. https://doi.org/10.1007/s11587-020-00544-2
A. Sabri, A. Jamea and H.T. Alaoui. Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and L1-data. Communications in Mathematics, 28(1):67–88, 2020. https://doi.org/10.2478/cm-2020-0006
A. Sabri, A. Jamea and H.T. Alaoui. Weak solution for fractional p(x)-Laplacian problem with Dirichlet-type boundary condition. Methods of Functional Analysis and Topology, 26(3):272–282, 2020. https://doi.org/10.31392/MFAT-npu26_3.2020.08
A. Sabri, A. Jamea and H.T. Alaoui. Weak solution for fractional (p1, . . . , pm)-Laplacian system with Dirichlet-type boundary conditions. Rendiconti del Circolo Matematico di Palermo. II Series, 70(3):1541–1560, 2021. https://doi.org/10.1007/s12215-020-00573-8
R. Servadei and E. Valdinoci. Variational methods for non-local operators of elliptic type. Discrete and Continuous Dynamical Systems, 33(5):2105–2137, 2013. https://doi.org/10.3934/dcds.2013.33.2105
J. Simon. Regularite de la solution d’une equation non lineaire dans RN . In Journées d’Analyse Non Linéaire (Proc. Conf., Besan¸con, 1977), volume 665 of Lecture Notes in Math., pp. 205–227, Berlin, Heidelberg, 1978. Springer, Berlin. https://doi.org/10.1007/BFb0061807
C. Zhang and X. Zhang. Renormalized solutions for the fractional p(x)-Laplacian equation with L1 data. Nonlinear Analysis, 190:111610, 2020. https://doi.org/10.1016/j.na.2019.111610
J. Zuo, T. An and A. Fiscella. A critical Kirchhoff-type problem driven by a p(·)-fractional Laplace operator with variable s(·)-order. Mathematical Methods in the Applied Sciences, 44(1):1071–1085, 2021. https://doi.org/10.1002/mma.6813
J. Zuo, A. Fiscella and A. Bahrouni. Existence and multiplicity results for p(·) & q(·) fractional Choquard problems with variable order. Complex Variables and Elliptic Equations, 67(2):500–516, 2022. https://doi.org/10.1080/17476933.2020.1835878
B. Abdellaoui, A. Attar, R. Bentifour and I. Peral. On fractional p-Laplacian parabolic problem with general data. Annali di Matematica, 197(2):329–356, 2018. https://doi.org/10.1007/s10231-017-0682-z
D. Applebaum. Lévy processes and stochastic calculus, volume 116 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2009. https://doi.org/10.1017/CBO9780511809781
A. Bahrouni. Comparison and sub-supersolution principles for the fractional p(x)-Laplacian. Journal of Mathematical Analysis and Applications, 458(2):1363–1372, 2018. https://doi.org/10.1016/j.jmaa.2017.10.025
A. Bahrouni and V.D. Rǎdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete and Continuous Dynamical Systems - S, 11(3):379–389, 2018. https://doi.org/10.3934/dcdss.2018021
R. Biswas and S. Tiwari. Variable order nonlocal Choquard problem with variable exponents. Complex Variables and Elliptic Equations, 66(5):853–875, 2021. https://doi.org/10.1080/17476933.2020.1751136
C. Bucur and E. Valdinoci. Nonlocal diffusion and applications, volume 20 of Lecture Notes of the Unione Matematica Italiana. Springer, Unione Matematica Italiana, Bologna, 2016. https://doi.org/10.1007/978-3-319-28739-3
L. Caffarelli. Non-local diffusions, drifts and games. In Nonlinear partial differential equations, volume 7 of Abel Symp., pp. 37–52. Springer, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-25361-4_3
L.M. Del Pezzo and J.D. Rossi. Traces for fractional Sobolev spaces with variable exponents. Advances in Operator Theory, 2(4):435–446, 2017. https://doi.org/10.22034/aot.1704-1152
G. Devillanova and G.C. Marano. A free fractional viscous oscillator as a forced standard damped vibration. Fractional Calculus and Applied Analysis, 19(2):319–356, 2016. https://doi.org/10.1515/fca-2016-0018
E. Di Nezza, G. Palatucci and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces. Bulletin des Sciences Math´ematiques, 136(5):521–573, 2012. https://doi.org/10.1016/j.bulsci.2011.12.004
A. El Hachimi and A. Jamea. Nonlinear parabolic problems with Neumann-type boundary conditions and L1-data. Electronic Journal of Qualitative Theory of Differential Equations, 27:1–22, 2007. https://doi.org/10.14232/ejqtde.2007.1.27
X. Fan and D. Zhao. On the spaces Lp(x)(Ω) and W m,p(x)(Ω). Journal of Mathematical Analysis and Applications, 263(2):424–446, 2001. https://doi.org/10.1006/jmaa.2000.7617
J. Giacomoni and S. Tiwari. Existence and global behavior of solutions to fractional p-Laplacian parabolic problems. Electronic Journal of Differential Equations, 2018(44):1–20, 2018.
U. Kaufmann, J.D. Rossi and R. Vidal. Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians. Electronic Journal of Qualitative Theory of Differential Equations, 2017(76):1–10, 2017. https://doi.org/10.14232/ejqtde.2017.1.76
O. Kovácık and J. Rákosnık. On spaces lp(x) and wk,p(x). Czechoslovak Mathematical Journal, 41(4):592–618, 1991. https://doi.org/10.21136/CMJ.1991.102493
J.M. Mazón, J.D. Rossi and J. Toledo. Fractional p-Laplacian evolution equations. Journal de Mathématiques Pures et Appliquées, 105(6):810–844, 2016. https://doi.org/10.1016/j.matpur.2016.02.004
G. Molica Bisci, V.D. Radulescu and R. Servadei. Variational methods for nonlocal fractional problems, volume 162 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781316282397
V.D. Rǎdulescu and D.D. Repovš. Partial differential equations with variable exponents. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015. https://doi.org/10.1201/b18601
A. Sabri, H.T. Alaoui and A. Jamea. Existence of weak solutions for fractional p-Laplacian problem with Dirichlet-type boundary condition. Discussiones Mathematicae: Differential Inclusions, Control and Optimization, 39(1):69–80, 2019. https://doi.org/10.7151/dmdico.1211
A. Sabri and A. Jamea. Rothe time-discretization method for a nonlinear parabolic p(u)-Laplacian problem with Fourier-type boundary condition and L1-data. Ricerche di Matematica, pp. 1–24, 2020. https://doi.org/10.1007/s11587-020-00544-2
A. Sabri, A. Jamea and H.T. Alaoui. Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and L1-data. Communications in Mathematics, 28(1):67–88, 2020. https://doi.org/10.2478/cm-2020-0006
A. Sabri, A. Jamea and H.T. Alaoui. Weak solution for fractional p(x)-Laplacian problem with Dirichlet-type boundary condition. Methods of Functional Analysis and Topology, 26(3):272–282, 2020. https://doi.org/10.31392/MFAT-npu26_3.2020.08
A. Sabri, A. Jamea and H.T. Alaoui. Weak solution for fractional (p1, . . . , pm)-Laplacian system with Dirichlet-type boundary conditions. Rendiconti del Circolo Matematico di Palermo. II Series, 70(3):1541–1560, 2021. https://doi.org/10.1007/s12215-020-00573-8
R. Servadei and E. Valdinoci. Variational methods for non-local operators of elliptic type. Discrete and Continuous Dynamical Systems, 33(5):2105–2137, 2013. https://doi.org/10.3934/dcds.2013.33.2105
J. Simon. Regularite de la solution d’une equation non lineaire dans RN . In Journées d’Analyse Non Linéaire (Proc. Conf., Besan¸con, 1977), volume 665 of Lecture Notes in Math., pp. 205–227, Berlin, Heidelberg, 1978. Springer, Berlin. https://doi.org/10.1007/BFb0061807
C. Zhang and X. Zhang. Renormalized solutions for the fractional p(x)-Laplacian equation with L1 data. Nonlinear Analysis, 190:111610, 2020. https://doi.org/10.1016/j.na.2019.111610
J. Zuo, T. An and A. Fiscella. A critical Kirchhoff-type problem driven by a p(·)-fractional Laplace operator with variable s(·)-order. Mathematical Methods in the Applied Sciences, 44(1):1071–1085, 2021. https://doi.org/10.1002/mma.6813
J. Zuo, A. Fiscella and A. Bahrouni. Existence and multiplicity results for p(·) & q(·) fractional Choquard problems with variable order. Complex Variables and Elliptic Equations, 67(2):500–516, 2022. https://doi.org/10.1080/17476933.2020.1835878