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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


On the Existence of Stationary Solutions for Some Systems of Non-Fredholm Integro-Differential Equations with Superdiffusion

Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 75--86 | DOI:10.5890/DNC.2017.03.007

Vitali Vougalter$^{1}$, Vitaly Volpert$^{2}$

$^{1}$ Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

$^{2}$ Institute Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622, France

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Abstract

We establish the existence of stationary solutions for certain systems of reaction-diffusion equations with superdiffusion. The corresponding elliptic problem involves the operators with or without Fredholm property. The fixed point technique in appropriate H2 spaces of vector functions is employed.

References

  1. [1]  Agranovich, M.S. (1997), Elliptic boundary problems, Encyclopaedia Math. Sci., Partial Differential Equations, IX, Springer, Berlin, 79, 1-144.
  2. [2]  Lions, J.L. and Magenes, E. (1968), Problemes aux limites non homogenes et applications. Dunod, Paris, (1), 372.
  3. [3]  Volevich, L.R. (1965), Solubility of boundary value problems for general elliptic systems, Mat. Sb., 68(110), 373-416; (1968), English translation: Amer. Math. Soc. Transl., 67(2), 182-225.
  4. [4]  Volpert, V. (2011), Elliptic partial differential equations. Volume I. Fredholm theory of elliptic problems in unbounded domains. Birkh¨auser, 639.
  5. [5]  Vougalter, V. and Volpert, V. (2012), Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems, Anal. Math. Phys., 2(4), 473-496.
  6. [6]  Vougalter, V. and Volpert, V. (2011), Solvability conditions for some non Fredholm operators, Proc. Edinb.Math. Soc., (2), 54(1), 249-271.
  7. [7]  Volpert, V., Kazmierczak, B., Massot, M., and Peradzynski, Z. (2002), Solvability conditions for elliptic problems with non-Fredholm operators, Appl. Math., 29(2), 219-238.
  8. [8]  Vougalter, V. and Volpert, V. (2010), On the solvability conditions for some non Fredholm operators, Int. J. Pure Appl. Math., 60(2), 169-191.
  9. [9]  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.
  10. [10]  Vougalter, V. and Volpert, V. (2010), Solvability relations for some non Fredholm operators, Int. Electron. J. Pure Appl.Math., 2(1), 75-83.
  11. [11]  Volpert, V. and Vougalter, V. (2011), On the solvability conditions for a linearized Cahn-Hilliard equation, Rend. Istit. Mat. Univ. Trieste, 43, 1-9.
  12. [12]  Vougalter, V. and Volpert, V. (2010), Solvability conditions for some systems with non Fredholm operators, Int. Electron. J. Pure Appl. Math., 2(3), 183-187.
  13. [13]  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.
  14. [14]  Ducrot, A., Marion M., and Volpert, V. (2005), Systemes de réaction-diffusion sans propriété de Fredholm, CRAS, 340(9), 659-664.
  15. [15]  Ducrot, A., Marion M., and Volpert V. (2008), Reaction-diffusion problems with non Fredholm operators, Advances Diff. Equations , 13(11-12), 1151-1192.
  16. [16]  Ducrot, A., Marion, M., and Volpert, V. (2009), Reaction-diffusion waves (with the Lewis number different from 1). Publibook, Paris, 113.
  17. [17]  Vougalter, V. and Volpert, V. (2011), On the existence of stationary solutions for some non-Fredholm integrodifferential equations, Doc. Math., 16, 561-580.
  18. [18]  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.
  19. [19]  Solomon, T., Weeks, E. and Swinney, H. (1993), Observation of anomalous diffusion and Levy flights in a twodimensional rotating flow, Phys. Rev. Lett., 71, 3975-3978.
  20. [20]  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.
  21. [21]  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.
  22. [22]  Scher, H. and Montroll, E. (1975), Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B, 12, 2455- 2477.
  23. [23]  Metzler, R. and Klafter, J. (2000), The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77.
  24. [24]  Apreutesei, N., Bessonov, N., Volpert, V., and Vougalter. V. (2010), Spatial Structures and Generalized Travelling Waves for an Integro-Differential Equation, Discrete Contin. Dyn. Syst. Ser. B, 13(3), 537-557.
  25. [25]  Berestycki, H., Nadin, G., Perthame, B., and Ryzhik L. (2009), The non-local Fisher-KPP equation: travelling waves and steady states, Nonlinearity, 22(12), 2813-2844.
  26. [26]  Genieys, S., Volpert, V., and Auger, P. (2006), Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1(1), 63-80.
  27. [27]  Beck, M., Ghazaryan, A., and Sandstede, B. (2009), Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities, J. Differential Equations, 246, 4371-4390.
  28. [28]  Ghazaryan, A. and Sandstede, B. (2007), Nonlinear convective instability of Turing-unstable fronts near onset: a case study, SIAM J. Appl. Dyn. Syst. 6(2), 319-347.
  29. [29]  Vougalter, V. and Volpert, V. (2016), Existence of stationary solutions for some non-Fredholm integro-differential equations with superdiffusion, Preprint.
  30. [30]  Shen, W. and Zhang, A. (2010), Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249(4), 747-795.
  31. [31]  Hislop, P.D. and Sigal, I.M. (1996), Introduction to spectral theory. With applications to Schr¨odinger operators. Springer, 337.
  32. [32]  Bessonov, N., Reinberg, N., and Volpert, V. (2014), Mathematics of Darwins Diagram, Math. Model. Nat. Phenom., 9(3), 5-25.