On weak solutions to parabolic problem involving the fractional p-Laplacian via Young measures
Abstract
In this paper, we study the local existence of weak solutions for parabolic problem involving the fractional p-Laplacian. Our technique is based on the Galerkin method combined with the theory of Young measures. In addition, an example is given to illustrate the main results.
Keywords
weak solution; fractional p-Laplacian; existence; Young measure; Galerkin method
References
B. Abdellaoui, A. Attar, R. Bentifour, and I. Peral, On fractional p-Laplacian parabolic problem with general data, Ann. Mat. Pura Appl. (4) 197 (2018), no. 2, 329–356.
M. Abdellaoui, On the behavior of entropy solutions for a fractional p-Laplacian problem as t tends to infinity, Rend. Mat. Appl. (7) 43 (2022), no. 2, 103–132.
R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Elsevier/Academic Press, Amsterdam, 2003.
I. Athanasopoulos and L.A. Caffarelli, Optimal regularity of lower dimensional obstacle problems, J. Math. Sci. (N.Y.) 132 (2006), no. 3, 274–284.
I. Athanasopoulos, L.A. Caffarelli, and S. Salsa, The structure of the free boundary for lower dimensional obstacle problems, Amer. J. Math. 130 (2008), no. 2, 485–498.
E. Azroul and F. Balaadich, Strongly quasilinear parabolic systems in divergence form with weak monotonicity, Khayyam J. Math. 6 (2020), no. 1, 57–72.
F. Balaadich, On p-Kirchhoff-type parabolic problems, Rend. Circ. Mat. Palermo (2) 72 (2023), no. 2, 1005–1016.
F. Balaadich and E. Azroul, Existence results for fractional p-Laplacian systems via Young measures, Math. Model. Anal. 27 (2022), no. 2, 232–241.
B. Barrios, E. Colorado, A. de Pablo, and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations 252 (2012), no. 11, 6133–6162.
G.M. Bisci, Fractional equations with bounded primitive, Appl. Math. Lett. 27 (2014), 53–58.
G.M. Bisci and D. Repovš, Higher nonlocal problems with bounded potential, J. Math. Anal. Appl. 420 (2014), no. 1, 167–176.
L. Brasco, E. Lindgren, and E. Parini, The fractional Cheeger problem, Interfaces Free Boundaries 16 (2014), no. 3, 419–458.
L. Caffarelli, Non-local diffusions, drifts and games, in: H. Holden and K.H. Karlsen (eds.), Nonlinear Partial Differential Equations, Abel Symp., 7, Springer, Heidelberg, 2012, pp. 37–52.
L.A. Caffarelli, S. Salsa, and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008), no. 2, 425–461.
Q.-H. Choi and T. Jung, On the fractional p-Laplacian problems, J. Inequal. Appl. (2021), Paper No. 41, 17 pp.
E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955.
A. de Pablo, F. Quirós, A. Rodríguez, and J.L. Vázquez, A fractional porous medium equation, Adv. Math. 226 (2011), no. 2, 1378–1409.
E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573.
G. Dolzmann, N. Hungerbühler, and S. Müller, Non-linear elliptic systems with measure-valued right hand side, Math. Z. 226 (1997), no. 4, 545–574.
A. Fiscella, R. Servadei, and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 235–253.
J. Giacomoni and S. Tiwari, Existence and global behavior of solutions to fractional p-laplacian parabolic problems, Electron. J. Differential Equations (2018), Paper No. 44, 20 pp.
R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
N. Hungerbühler, A refinement of Ball’s theorem on Young measures, New York J. Math. 3 (1997), 48–53.
A. Iannizzotto, S.J. Mosconi, and M. Squassina, Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam. 32 (2016), no. 4, 1353–1392.
A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl. 414 (2014), no. 1, 372–385.
A.A. Kilbas, H.M. Srivastava, and J.J. Trujilo, Theory and Application of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.
R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), no. 3–4, 217–237.
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires (Some methods of nonlinear boundary value problems), Dunod, Paris; Gauthier-Villars, Paris, 1969.
J.M. Mazón, J.D. Rossi, and J. Toledo, Fractional p-Laplacian evolution equations, J. Math. Pures Appl. (9) 105 (2016), no. 6, 810–844.
S. Mosconi and M. Squassina, Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Math. Anal. Semin., 7, Univ. Bologna, Alma Mater Stud., Bologna, 2016, 147–164.
H. Qiu and M. Xiang, Existence of solutions for fractional p-Laplacian problems via Leray-Schauder’s nonlinear alternative, Bound. Value Probl. (2016), Paper No. 83, 8 pp.
R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), no. 2, 887–898.
E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA (2009), no. 49, 33–44.
J.L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S 7 (2014), no. 4, 857–885.
J.L. Vázquez, The Dirichlet problem for the fractional p-Laplacian evolution equation, J. Differential Equations 260 (2016), no. 7, 6038–6056.
M. Xiang, B. Zhang, and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian, J. Math. Anal. Appl. 424 (2015), no. 2, 1021–1041.
Applied Mathematics and Scientific Computing Laboratory, Sultan Moulay Slimane University Morocco
https://orcid.org/0000-0002-7583-0768
Applied Mathematics and Scientific Computing Laboratory, Sultan Moulay Slimane University Morocco
Applied Mathematics and Scientific Computing Laboratory, Sultan Moulay Slimane University Morocco
Applied Mathematics and Scientific Computing Laboratory, Sultan Moulay Slimane University Morocco
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