On the Existence of Stationary Solutions for Some Systems of Integro-Differential Equations with Anomalous Diffusion
DOI:
https://doi.org/10.5890/DNC.2016.09.007Abstract
The article is devoted to the proof of the existence of solutions of a system of integro-differential equations appearing in the case of anomalous diffusion when the negative Laplacian is raised to some fractional power. The argument relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains along with the Sobolev inequality for a fractional Laplace operator are being used.References
[1] Solomon, T., Weeks, E., and Swinney, H. (1993), Phys. Rev. Lett., 71 3975-3978.
[2] Carreras, B., Lynch, V., and Zaslavsky, G. (2001), Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model, Phys. Plasmas, 8, 5096-5103.
[3] Manandhar, P., Jang, J., Schatz, G.C., Ratner, M.A., and Hong, S. (2003), Anomalous surface diffusion in nanoscale direct deposition processes, Phys. Rev. Lett., 90, 4043-4052.
[4] Sancho, J., Lacasta, A., Lindenberg, K., Sokolov, I., and Romero, A. (2004), Diffusion on a solid surface: Anomalous is normal, Phys. Rev. Lett., 92, 250601.
[5] Scher, H. and Montroll, E. (1975), Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B, 12, 2455- 2477.
[6] Vougalter, V. and Volpert, V. (2015), Existence of stationary solutions for some nonlocal reaction-diffusion equations, Dyn. Partial Differ. Equ., 12(1), 43-51.
[7] Amrouche, C., Girault, V., and Giroire, J. (1997), Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator: an approach in weighted Sobolev spaces, J. Math. Pures Appl., 76(1), 55-81.
[8] Amrouche, C. and Bonzom, F. (2008),Mixed exterior Laplace's problem, J. Math. Anal. Appl., 338, 124-140.
[9] Bolley, P. and Pham, T.L. (1993), Propriété d'indice en théorie Holderienne pour des opérateurs différentiels elliptiques dans Rn, J. Math. Pures Appl., 72 (1), 105-119.
[10] Bolley, P. and Pham, T.L. (2001), Propriété d'indice en théorie Hölderienne pour le problème extérieur de Dirichlet, Comm. Partial Differential Equations, 26(1-2), 315-334.
[11] Benkirane, N. (1988), Propriété d'indice en théorie Holderienne pour des opérateurs elliptiques dans Rn, CRAS, 307, Série I, 577-580.
[12] Volpert, V. (2011), Elliptic partial differential equations. Volume 1. Fredholm theory of elliptic problems in unbounded domains, Birkhauser.
[13] Vougalter, V. and Volpert, V. (2011), Solvability conditions for some non-Fredholm operators, Proc. Edinb.Math. Soc. (2), 54(1), 249-271.
[14] Vougalter, V. and Volpert, V. (2010), Solvability relations for some non Fredholm operators, Int. Electron. J. Pure Appl. Math., 2(1), 75-83.
[15] Vougalter, V. and Volpert, V. (2012), On the solvability conditions for the diffusion equation with convection terms, Commun. Pure Appl. Anal., 11(1), 365-373.
[16] Vougalter, V. and Volpert, V. (2011), On the solvability conditions for a linearized Cahn-Hilliard equation, Rend. Istit. Mat. Univ. Trieste, 43, 1-9.
[17] Vougalter, V. and Volpert, V. (2012), Solvability conditions for a linearized Cahn-Hilliard equation of sixth order, Math. Model. Nat. Phenom., 7(2), 146-154.
[18] Vougalter, V. and Volpert, V. (2011), On the existence of stationary solutions for some non-Fredholm integrodifferential equations, Doc. Math., 16 561-580.
[19] Vougalter, V. and Volpert, V. (2012), Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems, Anal. Math. Phys., 2(4), 473-496.
[20] Ducrot, A., Marion, M., and Volpert, V.(2005), Systemes de réaction-diffusion sans propriété de Fredholm, CRAS, 340 , 659-664.
[21] Ducrot, A., Marion, M., and Volpert, V. (2008), Reaction-diffusion problems with non Fredholm operators, Advances Diff. Equations, 13(11-12), 1151-1192.
[22] Alfimov, G.L., Medvedeva, E.V., and Pelinovsky, D.E. (2014), Wave Systems with an Infinite Number of Localized TravelingWaves, Phys. Rev. Lett., 112, 054103, 5pp.
[23] Volpert, V.A., Nec, Y., and Nepomnyashchy, A.A. (2010), Exact solutions in front propagation problems with superdiffusion, Phys. D, 239 (3-4), 134-144.
[24] Volpert, V.A., Nec, Y., and Nepomnyashchy, A.A. (2013), Fronts in anomalous diffusion-reaction systems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371(1982), 20120179, 18pp.
[25] Hajaiej, H., Yu, X. and Zhai, Z. (2012), Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms, J. Math. Anal. Appl., 396(2), 569-577.
[26] Lieb, E. (1983), Sharp constants in the Hardy-Littlewood-Sobolev inequalities and related inequalities, Ann. of Math., 118, 349-374.
[27] Vougalter, V. (2010), On threshold eigenvalues and resonances for the linearized NLS equation, Math. Model. Nat. Phenom., 5(4), 448-469.
[28] Cuccagna, S., Pelinovsky, D. and Vougalter, V. (2005), Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl. Math., 58(1), 1-29.
[29] Vougalter, V. and Volpert, V. (2015), Existence of stationary solutions for some integro-differential equations with anomalous diffusion, J. Pseudo-Differ. Oper. Appl., 6(4), 487-501.
[30] Bessonov, N., Reinberg, N. and Volpert, V.(2014), Mathematics of Darwin's Diagram, Math. Model. Nat. Phenom., 9(3), 5-25.
Article Metrics
Usage tracking begins September 1, 2026.