ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
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

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

Discontinuity, Nonlinearity, and Complexity 1(2) (2012) 197--209 | DOI:10.5890/DNC.2012.05.003

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

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

$^{2}$ Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch 7701, South Africa.

Abstract

We prove the existence of stationary solutions for certain systems of reaction-diffusion type equations in the corresponding H2 spaces. Our method relies on the fixed point theorem when the elliptic problem involves second order differential operators with and without Fredholm property.

References

1.  [1] Agranovich, M.S. (1997), Elliptic boundary problems,Encyclopaedia Math. Sci., 79, Partial Differential Equations, IX, Springer, Berlin, 1-144.
2.  [2] 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.
3.  [3] Beck, M., Ghazaryan, A. and Sandstede, B. (2009), Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities, J. Differential Equations, 246 (11), 4371-4390.
4.  [4] Berestycki, H., Nadin, G., Perthame, B., Ryzhik, L. (2009), The non-local Fisher-KPP equation: traveling waves and steady states, Nonlinearity, 22, 12, 2813-2844.
5.  [5] Ducrot, A., Marion, M. and Volpert, V. (2005), Systemes de réaction-diffusion sans propriété de Fredholm, CRAS, 340(9), 659-664.
6.  [6] Ducrot, A., Marion, M. and Volpert V. (2008), Reaction-diffusion problems with non Fredholm operators, Advances Diff. Equations , 13 (11-12), 1151-1192.
7.  [7] Ducrot, A., Marion, M. and Volpert, V. (2009), Reaction-diffusion waves (with the Lewis number different from 1), Publibook, Paris.
8.  [8] Genieys, S., Volpert, V., 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.
9.  [9] 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.
10.  [10] Hislop, P.D. and Sigal, I.M. (1996), Introduction to spectral theory with applications to Schrödinger operators, Springer.
11.  [11] Lions, J.L. and Magenes, E. (1968), Problemes aux limites non homogenes et applications. Volume 1. Dunod, Paris.
12.  [12] 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.
13.  [13] Volevich, L.R. (1968), Solubility of boundary problems for general elliptic systems, Mat. Sbor., 68, (1965), 373- 416; English translation: Amer. Math. Soc. Transl., 67 , Ser. 2, 182-225.
14.  [14] Volpert, V. (2011), Elliptic partial differential equations. Volume I. Fredholm theory of elliptic problems in unbounded domains. Birkhäuser.
15.  [15] Volpert, V., Kazmierczak, B., Massot. M., Peradzynski, Z.(2002), Solvability conditions for elliptic problems with non-Fredholm operators, Appl. Math., 29(2), 219-238.
16.  [16] Vougalter, V. and Volpert, V. (2011), Solvability conditions for some non Fredholm operators, Proc. Edinb. Math. Soc. (2), 54(1), 249-271.
17.  [17] Vougalter, V. and Volpert, V.(2010), On the solvability conditions for some non Fredholm operators, Int. J. Pure Appl. Math., 60(2), 169-191.
18.  [18] Vougalter, V. andVolpert, V.(2012), On the solvability conditions for the diffusion equation with convection terms, Commun. Pure Appl. Anal., 11 (1), 365-373.
19.  [19] Vougalter, V. and Volpert, V. (2010), Solvability relations for some non Fredholm operators, Int. Electron. J. Pure Appl.Math., 2(1), 75-83.
20.  [20] Volpert, V. and Vougalter, V.(2011), On the solvability conditions for a linearized Cahn-Hilliard equation, Rend. Istit. Mat. Univ. Trieste, 43, 1-9.
21.  [21] 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.
22.  [22] 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.
23.  [23] Vougalter, V. and Volpert, V. (2011), On the existence of stationary solutions for some non-Fredholm integrodifferential equations, Doc. Math., 16, 561-580.
24.  [24] Volpert, V. and Vougalter, V. (Preprint 2011) Emergence and propagation of patterns in nonlocal reactiondiffusion equations arising in the theory of speciation.
25.  [25] Vougalter,V. and Volpert, V., Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems. To appear in Anal. Math. Phys.