Existence in the Sense of Sequences of Stationary Solutions for Some Non-Fredholm Integro-Differential Equations

We establish the existence in the sense of sequences of stationary solutions for some reaction-diffusion type equations in appropriate H2 spaces. It is shown that, under reasonable technical conditions, the convergence in L1 of the integral kernels implies the existence and convergence in H2 of solutions. The nonlocal elliptic equations involve second order differential operators with and without the Fredholm property. Bibliography: 21 titles.

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References

  1. 1.

    V. Volpert, Elliptic Partial Differential Equations. I. Fredholm Theory of Elliptic Problems in Unbounded Domains, Birkhäuser, Basel (2011).

    Google Scholar 

  2. 2.

    V. Vougalter and V. Volpert, “Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems,” Anal. Math. Phys. 2, No. 4, 473–496 (2012).

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    V. Volpert, B. Kazmierczak, M. Massot, and Z. Peradzynski, “Solvability conditions for elliptic problems with non-Fredholm operators,” Appl. Math. 29, No. 2, 219–238 (2002).

    MathSciNet  MATH  Google Scholar 

  4. 4.

    V. Vougalter and V. Volpert, “Solvability conditions for some non-Fredholm operators,” Proc. Edinb. Math. Soc., II. Ser. 54, No. 1, 249–271 (2011).

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    V. Vougalter and V. Volpert, “On the solvability conditions for some non-Fredholm operators,” Int. J. Pure Appl. Math. 60 No. 2, 169–191 (2010).

    MathSciNet  MATH  Google Scholar 

  6. 6.

    V. Vougalter and V. Volpert, “On the solvability conditions for the diffusion equation with convection terms,” Commun. Pure Appl. Anal. 11, No. 1, 365–373 (2012).

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    V. Vougalter and V. Volpert, “Solvability relations for some non-Fredholm operators,” Int. Electron. J. Pure Appl. Math. 2, No. 1 75–83 (2010).

    MATH  Google Scholar 

  8. 8.

    V. Vougalter and V. Volpert, “On the solvability conditions for a linearized Cahn-Hilliard equation,” Rend. Ist. Mat. Univ. Trieste 43, 1–9 (2011).

    MathSciNet  MATH  Google Scholar 

  9. 9.

    V. Vougalter and V. Volpert, “Solvability conditions for some systems with non-Fredholm operators,” Int. Electron. J. Pure Appl. Math. 2, No. 3 183–187 (2010).

    MATH  Google Scholar 

  10. 10.

    V. Vougalter and V. Volpert, “On the existence of stationary solutions for some non-Fredholm integro-differential equations,” Documenta Math. 16, 561–580 (2011).

    MathSciNet  MATH  Google Scholar 

  11. 11.

    A. Ducrot, M. Marion and V. Volpert, “Reaction-diffusion systems without the Fredholm property” [in French], C.R., Math., Acad. Sci. Paris 340, No. 9, 659–664 (2005).

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    A. Ducrot, M. Marion and V. Volpert, “Reaction-diffusion problems with non-Fredholm operators,” Adv. Differ. Equ. 13, No. 11-12, 1151–1192 (2008).

    MathSciNet  MATH  Google Scholar 

  13. 13.

    A. Ducrot, M. Marion and V. Volpert, Reaction-Diffusion Waves. With the Lewis Number Different from 1, Publibook, Paris (2009).

    Google Scholar 

  14. 14.

    V. Vougalter and V. Volpert, “Existence of stationary solutions for some nonlocal reactiondiffusion equations,” Dyn. Partial Differ. Equ. 12, No. 1, 43–51 (2015).

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    N. Apreutesei, N. Bessonov, V. Volpert, and V. Vougalter, “Spatial structures and generalized travelling waves for an integro-differential equation,” Discrete Contin. Dyn. Syst., Ser. B 13, No. 3, 537–557 (2010).

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    H. Berestycki, G. Nadin, B. Perthame, and L. Ryzhik, “The non-local Fisher-KPP equation: travelling waves and steady states,” Nonlinearity 22, No. 12, 2813–2844 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    S. Genieys, V. Volpert, and P. Auger, “Pattern and waves for a model in population dynamics with nonlocal consumption of resources,” Math. Model. Nat. Phenom. 1, No. 1, 63–80 (2006).

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    V. Volpert and V. Vougalter, “Solvability in the sense of sequences to some non-Fredholm operators,” Electron. J. Differ. Equ. 2013, Paper No. 160, 16 p. (2013).

  19. 19.

    V. Vougalter and V. Volpert, “On the solvability in the sense of sequences for some non-Fredholm operators,” Dyn. Partial Differ. Equ. 11, No. 2, 109–124 (2014).

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    D. E. Pelinovsky and J. Yang, “A normal form for nonlinear resonance of embedded solitons,” Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 458 No. 2022, 1469–1497 (2002).

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory with Applications to Schrödinger Operators, Springer, New York (1996)

    Google Scholar 

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Correspondence to V. Vougalter.

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Translated from Problemy Matematicheskogo Analiza 90, January 2018, pp. 3-28.

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Vougalter, V., Volpert, V. Existence in the Sense of Sequences of Stationary Solutions for Some Non-Fredholm Integro-Differential Equations. J Math Sci 228, 601–632 (2018). https://doi.org/10.1007/s10958-017-3650-7

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