Osaka Journal of Mathematics

Solvability of some integro-differential equations with drift

Messoud Efendiev and Vitali Vougalter

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Abstract

We prove the existence in the sense of sequences of solutions for some integro-differential type equations involving the drift term in the appropriate $H^{2}$ spaces using the fixed point technique when the elliptic problems contain second order differential operators with and without Fredholm property. It is shown that, under the reasonable technical conditions, the convergence in $L^{1}$ of the integral kernels yields the existence and convergence in $H^{2}$ of solutions.

Article information

Source
Osaka J. Math., Volume 57, Number 2 (2020), 247-265.

Dates
First available in Project Euclid: 6 April 2020

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1586160077

Mathematical Reviews number (MathSciNet)
MR4081731

Zentralblatt MATH identifier
07196677

Subjects
Primary: 35J61: Semilinear elliptic equations 35R09: Integro-partial differential equations [See also 45Kxx] 35K57: Reaction-diffusion equations

Citation

Efendiev, Messoud; Vougalter, Vitali. Solvability of some integro-differential equations with drift. Osaka J. Math. 57 (2020), no. 2, 247--265. https://projecteuclid.org/euclid.ojm/1586160077


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References

  • M.S. Agranovich: Elliptic boundary problems, in Encyclopaedia Math. Sci. 79, Partial Differential Equations, IX, Springer, Berlin, (1997), 1–144.
  • 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 (2010), 537–557.
  • H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik: The non-local Fisher-KPP equation: traveling waves and steady states, Nonlinearity 22 (2009), 2813–2844.
  • H. Berestycki, F. Hamel and N. Nadirashvili: The speed of propagation for KPP type problems. I. Periodic framework, J. Eur. Math. Soc. (JEMS) 7 (2005), 173–213.
  • A. Ducrot, M. Marion and V. Volpert: Reaction-diffusion problems with non Fredholm operators, Adv. Differential. Equations 13 (2008), 1151–1192.
  • M.A. Efendiev: Fredholm structures, topological invariants and applications, American Institute of Mathematical Sciences, Springfield, MO, 2009.
  • P.D. Hislop and I.M. Sigal: Introduction to spectral theory With applications to Schrödinger operators. Springer, New York, 1996.
  • M.A. Krasnosel'skij: Topological methods in the theory of nonlinear integral equations, International Series of Monographs on Pure and Applied Mathematics. Pergamon Press, 1964.
  • J.L. Lions and E. Magenes: Problèmes aux limites non homogènes et applications. Volume 1, Dunod, Paris, 1968.
  • L.R. Volevich: Solubility of boundary problems for general elliptic systems, Mat. Sb. (N.S.) 68 (1965), 373–416; English translation: Amer. Math. Soc. Transl., 67 (1968), Ser. 2, 182–225.
  • V. Volpert: Elliptic partial differential equations. Volume I: Fredholm theory of elliptic problems in unbounded domains, Birkhäuser, Basel, 2011.
  • V. Volpert, B. Kazmierczak, M. Massot and Z. Peradzynski: Solvability conditions for elliptic problems with non-Fredholm operators, Appl. Math (Warsaw), 29 (2002), 219–238.
  • V. Volpert and V. Vougalter: Emergence and propagation of patterns in nonlocal reaction-diffusion equations arising in the theory of speciation; in Dispersal, individual movement and spatial ecology, Lecture Notes in Math. 2071, Springer, Heidelberg, 2013, 331–353.
  • V. Volpert and V. Vougalter: Solvability in the sense of sequences to some non-Fredholm operators, Electron. J. Differential Equations 2013, No. 160, 16pp.
  • V. Vougalter and V. Volpert: Solvability conditions for some non Fredholm operators, Proc. Edinb. Math. Soc. (2), 54 (2011), 249–271.
  • V. Vougalter and V. Volpert: On the existence of stationary solutions for some non-Fredholm integro-differential equations, Doc. Math, 16 (2011), 561–580.
  • V. Vougalter and V. Volpert: On the solvability conditions for the diffusion equation with convection terms, Commun. Pure Appl, Anal. 11 (2012), 365–373.
  • V. Vougalter and V. Volpert: Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems, Anal. Math. Phys, 2 (2012), 473–496.
  • V. Vougalter and V. Volpert: Existence in the sense of sequences of stationary solutions for some non-Fredholm integro- differential equations, J. Math. Sci. (N.Y.) 228, (2018), 601–632.