# American Institute of Mathematical Sciences

July  2017, 37(7): 4035-4051. doi: 10.3934/dcds.2017171

## A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian

 1 Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, 06123 Perugia, Italy 2 College of Science, Civil Aviation University of China, Tianjin 300300, China 3 Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China

* Corresponding author

Received  March 2016 Revised  February 2017 Published  April 2017

In this paper, we study an anomalous diffusion model of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we are concerned with the following initial-boundary value problem involving the fractional $p$-Laplacian $\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+M([u]_{s, p}^{p}\text{)}(-\Delta)_{p}^{s}u=f(x, t) & \text{in }\Omega \times {{\mathbb{R}}^{+}}, {{\partial }_{t}}u=\partial u/\partial t, \\ u(x, 0)={{u}_{0}}(x) & \text{in }\Omega, \\ u=0\ & \text{in }{{\mathbb{R}}^{N}}\backslash \Omega, \\\end{array}\text{ }\ \ \right.$ where $[u]_{s, p}$ is the Gagliardo $p$-seminorm of $u$, $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partialΩ$, $1 < p < N/s$, with $0 < s < 1$, the main Kirchhoff function $M:\mathbb{R}^{ + }_{0} \to \mathbb{R}^{ + }$ is a continuous and nondecreasing function, $(-Δ)_p^s$ is the fractional $p$-Laplacian, $u_0$ is in $L^2(Ω)$ and $f∈ L^2_{\rm loc}(\mathbb{R}^{ + }_0;L^2(Ω))$. Under some appropriate conditions, the well-posedness of solutions for the problem above is studied by employing the sub-differential approach. Finally, the large-time behavior and extinction of solutions are also investigated.

Citation: Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
##### References:
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Math.(234), 2010 (), 2633-2645. doi: 10.1016/j.cam.2010.01.026. Google Scholar [6] S. Antontsev, S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, J. Math. Anal. Appl., 361 (2010), 371-391. doi: 10.1016/j.jmaa.2009.07.019. Google Scholar [7] D. Applebaum, Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. Google Scholar [8] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714. doi: 10.1016/j.na.2015.06.014. Google Scholar [9] G. Autuori, P. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516. doi: 10.1007/s00205-009-0241-x. Google Scholar [10] H. 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Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191. Google Scholar [21] M. Fila, Boundedness of global solutions of nonlinear diffusion equations, J. Differential Equations, 98 (1992), 226-240. doi: 10.1016/0022-0396(92)90091-Z. Google Scholar [22] A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253. doi: 10.5186/aasfm.2015.4009. Google Scholar [23] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170. doi: 10.1016/j.na.2013.08.011. Google Scholar [24] G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma, 5 (2014), 373-386. Google Scholar [25] M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type, Math. Meth. Appl. Sci., 22 (1999), 375-388. doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7. Google Scholar [26] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar [27] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108. Google Scholar [28] E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826. doi: 10.1007/s00526-013-0600-1. Google Scholar [29] T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977. doi: 10.1016/j.na.2005.03.021. Google Scholar [30] X. Mingqi, G. Molica Bisci, G. H. Tian and B. L. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 357-374. doi: 10.1088/0951-7715/29/2/357. Google Scholar [31] M. Pérez-Llanosa and J. D. Rossi, Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal., 70 (2009), 1629-1640. doi: 10.1016/j.na.2008.02.076. Google Scholar [32] P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22. doi: 10.4171/RMI/879. Google Scholar [33] P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203-214. doi: 10.1006/jdeq.1998.3477. Google Scholar [34] P. Pucci, M. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogenous Schrodinger-Kirchhoff type equations involving the fractional $p-$Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806. doi: 10.1007/s00526-015-0883-5. Google Scholar [35] P. Pucci, M. Q. Xiang and B. L. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55. doi: 10.1515/anona-2015-0102. Google Scholar [36] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs Vol. 49, American Mathematical Society, Providence, RI, 1997, xiv + 278 pp. Google Scholar [37] J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Abel Symp., Springer, Heidelberg, 7 (2012), 271-298. doi: 10.1007/978-3-642-25361-4_15. Google Scholar [38] M. Q. Xiang, B. L. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional $p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041. doi: 10.1016/j.jmaa.2014.11.055. Google Scholar [39] M. Q. Xiang, B. L. Zhang and M. Ferrara, Multiplicity results for the nonhomogeneous fractional $p$-Kirchhoff equations with concave-convex nonlinearities, Proc. Roy. Soc. A, 471 (2015), 20150034, 14 pp. doi: 10.1098/rspa.2015.0034. Google Scholar [40] M. Q. Xiang, B. L. Zhang and V. Rădulescu, Existence of solutions for perturbed fractional $p$-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413. doi: 10.1016/j.jde.2015.09.028. Google Scholar

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##### References:
 [1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London,, 1975. Google Scholar [2] G. Akagi and K. Matsuura, Well-posedness and large-time behaviors of solutions for a parabolic equations involving $p(x)$-Laplacian, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications, 8th AIMS Conference.Suppl., 1 (2011), 22-31. Google Scholar [3] G. Akagi, K. Matsuura, Nonlinear diffusion equations driven by the $p(·)$-Laplacian, Nonlinear Differential Equations Appl. NoDEA, 20 (2013), 37-64. doi: 10.1007/s00030-012-0153-6. Google Scholar [4] F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo, A nonlocal $p$-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions, SIAM J. Math. Anal., 40 (2009), 1815-1851. doi: 10.1137/080720991. Google Scholar [5] S. Antontsev and S. Shmarev, Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math.(234), 2010 (), 2633-2645. doi: 10.1016/j.cam.2010.01.026. Google Scholar [6] S. Antontsev, S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, J. Math. Anal. Appl., 361 (2010), 371-391. doi: 10.1016/j.jmaa.2009.07.019. Google Scholar [7] D. Applebaum, Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. Google Scholar [8] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714. doi: 10.1016/j.na.2015.06.014. Google Scholar [9] G. Autuori, P. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516. doi: 10.1007/s00205-009-0241-x. Google Scholar [10] H. Brézis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Math Studies, Vol.5 North-Holland, Amsterdam, New York, 1973. Google Scholar [11] L. Caffarelli, Some nonlinear problems involving non-local diffusions, ICIAM 07-6th International Congress on Industrial and Applied Mathematics, Eur. Math. Soc., Zürich, (2009), 43-56. doi: 10.4171/056-1/3. Google Scholar [12] L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 37-52. doi: 10.1007/978-3-642-25361-4_3. Google Scholar [13] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behaviour for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. Google Scholar [14] F. Colasuonno and P. Pucci, Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974. doi: 10.1016/j.na.2011.05.073. Google Scholar [15] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390. doi: 10.1016/j.jde.2006.12.002. Google Scholar [16] A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836. doi: 10.1016/j.jfa.2014.05.023. Google Scholar [17] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299. doi: 10.1016/j.anihpc.2015.04.003. Google Scholar [18] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [19] J. M. do'O, O. H. Miyagaki and M. Squassina, Nonautonomous fractional problems with exponential growth, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1395-1410. doi: 10.1007/s00030-015-0327-0. Google Scholar [20] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191. Google Scholar [21] M. Fila, Boundedness of global solutions of nonlinear diffusion equations, J. Differential Equations, 98 (1992), 226-240. doi: 10.1016/0022-0396(92)90091-Z. Google Scholar [22] A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253. doi: 10.5186/aasfm.2015.4009. Google Scholar [23] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170. doi: 10.1016/j.na.2013.08.011. Google Scholar [24] G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma, 5 (2014), 373-386. Google Scholar [25] M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type, Math. Meth. Appl. Sci., 22 (1999), 375-388. doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7. Google Scholar [26] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar [27] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108. Google Scholar [28] E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826. doi: 10.1007/s00526-013-0600-1. Google Scholar [29] T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977. doi: 10.1016/j.na.2005.03.021. Google Scholar [30] X. Mingqi, G. Molica Bisci, G. H. Tian and B. L. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 357-374. doi: 10.1088/0951-7715/29/2/357. Google Scholar [31] M. Pérez-Llanosa and J. D. Rossi, Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal., 70 (2009), 1629-1640. doi: 10.1016/j.na.2008.02.076. Google Scholar [32] P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22. doi: 10.4171/RMI/879. Google Scholar [33] P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203-214. doi: 10.1006/jdeq.1998.3477. Google Scholar [34] P. Pucci, M. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogenous Schrodinger-Kirchhoff type equations involving the fractional $p-$Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806. doi: 10.1007/s00526-015-0883-5. Google Scholar [35] P. Pucci, M. Q. Xiang and B. L. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55. doi: 10.1515/anona-2015-0102. Google Scholar [36] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs Vol. 49, American Mathematical Society, Providence, RI, 1997, xiv + 278 pp. Google Scholar [37] J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Abel Symp., Springer, Heidelberg, 7 (2012), 271-298. doi: 10.1007/978-3-642-25361-4_15. Google Scholar [38] M. Q. Xiang, B. L. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional $p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041. doi: 10.1016/j.jmaa.2014.11.055. Google Scholar [39] M. Q. Xiang, B. L. Zhang and M. Ferrara, Multiplicity results for the nonhomogeneous fractional $p$-Kirchhoff equations with concave-convex nonlinearities, Proc. Roy. Soc. A, 471 (2015), 20150034, 14 pp. doi: 10.1098/rspa.2015.0034. Google Scholar [40] M. Q. Xiang, B. L. Zhang and V. Rădulescu, Existence of solutions for perturbed fractional $p$-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413. doi: 10.1016/j.jde.2015.09.028. Google Scholar
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