Solvability of Some Systems of Non-Fredholm Integro-Differential Equations with Mixed Diffusion

Springer Science and Business Media LLC - 2022
Messoud Efendiev1,2, Vitali Vougalter3
1Department of Mathematics, Marmara University, Istanbul, Turkey
2Helmholtz Zentrum München, Institut für Computational Biology, Neuherberg, Germany
3Department of Mathematics, University of Toronto, Toronto, Canada

Tóm tắt

We prove the existence in the sense of sequences of solutions for some system of integro-differential type equations in two dimensions containing the normal diffusion in one direction and the anomalous diffusion in the other direction in $$H^{2}({\mathbb R}^{2}, {{\mathbb {R}}}^{N})$$ using the fixed point technique. The system of elliptic equations contains second order differential operators without the Fredholm property. It is established that, under the reasonable technical assumptions, the convergence in $$L^{1}({{\mathbb {R}}}^{2})$$ of the integral kernels yields the existence and convergence in $$H^{2}({{\mathbb {R}}}^{2}, {\mathbb R}^{N})$$ of the solutions. We emphasize that the study of the systems is more difficult than of the scalar case and requires to overcome more cumbersome technicalities.

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Tài liệu tham khảo

Agranovich, M.S.: Elliptic boundary problems. Encyclopaedia Math. Sci., 79, Partial Differential Equations, IX, Springer, Berlin, 1–144 (1997)

Apreutesei, N., Bessonov, N., Volpert, V., Vougalter, V.: Spatial structures and generalized travelling waves for an integro-differential equation. Discrete Contin. Dyn. Syst. Ser. B 13(3), 537–557 (2010)

Berestycki, H., Nadin, G., Perthame, B., Ryzhik, L.: The non-local Fisher-KPP equation: traveling waves and steady states. Nonlinearity 22(12), 2813–2844 (2009)

Brezis, H., Oswald, L.: Remarks on sublinear elliptic equations. Nonlinear Anal. 10(1), 55–64 (1986)

Ducrot, A., Marion, M., Volpert, V.: Reaction-diffusion problems with non-Fredholm operators. Adv. Differ. Equ. 13(11–12), 1151–1192 (2008)

Efendiev, M.A.: Fredholm structures, topological invariants and applications. AIMS Series on Differential Equations & Dynamical Systems, 3. American Institute of Mathematical Sciences (AIMS), Springfield, MO, p. 205 (2009)

Efendiev, M.A.: Finite and infinite dimensional attractors for evolution equations of mathematical physics. GAKUTO International Series. Mathematical Sciences and Applications, 33. \(Gakk{{\bar{o}}}tosho \ Co., \ Ltd., \ Tokyo\), p. 239 (2010)

Efendiev, M.A., Peletier, L.A.: On the large time behavior of solutions of fourth order parabolic equations and \(\varepsilon \)- entropy of their attractors. C. R. Math. Acad. Sci. Paris 344(2), 93–96 (2007)

Efendiev, M.A., Vougalter, V.: Solvability of some integro-differential equations with drift. Osaka J. Math. 57(2), 247–265 (2020)

Efendiev, M.A., Vougalter, V.: Verification of biomedical processes with anomalous diffusion, transport and interaction of species. Nonlinear dynamics, chaos, and complexity – in memory of Professor Valentin Afraimovich, Nonlinear Phys. Sci., pp. 65–74. Springer, Singapore (2021)

Efendiev, M.A., Vougalter, V.: On the necessary conditions for preserving the nonnegative cone: mixed diffusion. The many facets of complexity science – in memory of Professor Valentin Afraimovich, Nonlinear Phys. Sci., pp. 185–192. Higher Ed. Press, Beijing (2021)

Efendiev, M.A., Vougalter, V.: Solvability in the sense of sequences for some fourth order non-Fredholm operators. J. Differ. Equ. 271, 280–300 (2021)

Efendiev, M.A., Vougalter, V.: Existence of solutions for some non-Fredholm integro-differental equations with mixed diffusion. J. Differ. Equ. 284, 83–101 (2021)

Efendiev, M.A., Vougalter, V.: Linear and nonlinear non-Fredholm operators and their applications. Electron. Res. Arch. 30(2), 515–534 (2022)

Efendiev, M.A., Vougalter, V.: Solvability of some integro-differential equations with drift and superdiffusion. J. Dynam. Differ. Equ. (2022). https://doi.org/10.1007/s10884-022-10147-0

Efendiev, M.A., Zelik, S.V.: The attractor for a nonlinear reaction-diffusion system in an unbounded domain. Comm. Pure Appl. Math. 54(6), 625–688 (2001)

Gebran, H.G., Stuart, C.A.: Fredholm and properness properties of quasilinear elliptic systems of second order. Proc. Edinb. Math. Soc. (2) 48(1), 91–124 (2005)

Gebran, H.G., Stuart, C.A.: Exponential decay and Fredholm properties in second-order quasilinear elliptic systems. J. Differ. Equ. 249(1), 94–117 (2010)

Krasnosel’skii, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations, p. 395. Pergamon Press Book The Macmillan Company, New York (1964)

Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. 1, 17th edn., p. 372. (French) Travaux et Recherches Mathematiques, Dunod, Paris (1968)

Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 77 (2000)

Rabier, P.J., Stuart, C.A.: Fredholm and properness properties of quasilinear elliptic operators on \({{\mathbb{R} }}^{N}\). Math. Nachr. 231, 129–168 (2001)

Volevich, L.R.: Solubility of boundary value problems for general elliptic systems (Russian). Mat. Sb. (N.S.) 68(110), 373–416 (1965)

Volpert, V.: Elliptic partial differential equations. Volume 1: Fredholm theory of elliptic problems in unbounded domains. Monographs in Mathematics, 101. Birkhäuser/Springer Basel AG, p. 639 Basel (2011)

Volpert, V., Kazmierczak, B., Massot, M., Peradzynski, Z.: Solvability conditions for elliptic problems with non-Fredholm operators. Appl. Math. (Warsaw) 29(2), 219–238 (2002)

Volpert, V., Vougalter, V.: On the existence of stationary solutions for some systems of non-Fredholm integro-differential equations. Disc., Nonlin., and Complex 1(2), 197–209 (2012)

Volpert, V., Vougalter, V.: Solvability in the sense of sequences to some non-Fredholm operators. Electron. J. Differ. Equ. 160, 16 (2013)

Vougalter, V., Volpert, V.: Solvability conditions for some non-Fredholm operators. Proc. Edinb. Math. Soc. (2) 54(1), 249–271 (2011)

Vougalter, V., Volpert, V.: Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems. Anal. Math. Phys. 2(4), 473–496 (2012)

Vougalter, V., Volpert, V.: Existence of stationary solutions for some systems of integro-differential equations with superdiffusion. Rocky Mountain J. Math. 47(3), 955–970 (2017)

Vougalter, V., Volpert, V.: On the existence of stationary solutions for some systems of non-Fredholm integro-differential equations with superdiffusion. Disc., Nonlin., and Complex 6(1), 75–86 (2017)

Vougalter, V., Volpert, V.: On the existence in the sense of sequences of stationary solutions for some systems of non-Fredholm integro-differential equations, Mediterr. J. Math., 15(5), Paper No. 205, 19 (2018)

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