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: |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London,, 1975. | |
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. | |
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. | |
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. | |
S. Antontsev and S. Shmarev , Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math., 2010 () , 2633-2645. doi: 10.1016/j.cam.2010.01.026. | |
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. | |
D. Applebaum , Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004) , 1336-1347. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
P. Fife , Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003) , 153-191. | |
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. | |
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. | |
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. | |
G. Franzina and G. Palatucci , Fractional $p$-eigenvalues, Riv. Math. Univ. Parma, 5 (2014) , 373-386. | |
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. | |
N. Laskin , Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000) , 298-305. doi: 10.1016/S0375-9601(00)00201-2. | |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108. | |
E. Lindgren and P. Lindqvist , Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014) , 795-826. doi: 10.1007/s00526-013-0600-1. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. |