Linear and nonlinear non-Fredholm operators and their applications

Messoud Efendiev; Vitali Vougalter; Messoud Efendiev; Vitali Vougalter

doi:10.3934/era.2022027

Electronic Research Archive

2022, Volume 30, Issue 2: 515-534. doi: 10.3934/era.2022027

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Research article Special Issues

Linear and nonlinear non-Fredholm operators and their applications

Messoud Efendiev ^{1,2
,
,},
Vitali Vougalter ³

1.
Helmholtz Zentrum München, Institut für Computational Biology, Germany
2.
Department of Mathematics, Marmara University, Istanbul, Turkey
3.
Department of Mathematics, University of Toronto, Canada

Received: 13 August 2021 Revised: 13 December 2021 Accepted: 14 December 2021 Published: 10 February 2022

In this survey we discuss the recent results on the existence in the sense of sequences of solutions for certain elliptic problems containing the non-Fredholm operators. First of all, we deal with the solvability in the sense of sequences for some fourth order non-Fredholm operators, such that the methods of the spectral and scattering theory for Schrödinger type operators are used for the analysis. Moreover, we present the easily verifiable necessary condition of the preservation of the nonnegativity of the solutions of a system of parabolic equations in the case of the anomalous diffusion with the negative Laplacian in a fractional power in one dimension, which imposes the necessary form of such system of equations that must be studied mathematically. This class of systems of PDEs has a wide range of applications. We conclude the survey with several new results nowhere published concerning the solvability in the sense of sequences for the generalized Poisson type equation with a scalar potential.
- solvability conditions,
- non-Fredholm operators,
- anomalous diffusion,
- nonnegativity of solutions
Citation: Messoud Efendiev, Vitali Vougalter. Linear and nonlinear non-Fredholm operators and their applications[J]. Electronic Research Archive, 2022, 30(2): 515-534. doi: 10.3934/era.2022027

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Abstract

In this survey we discuss the recent results on the existence in the sense of sequences of solutions for certain elliptic problems containing the non-Fredholm operators. First of all, we deal with the solvability in the sense of sequences for some fourth order non-Fredholm operators, such that the methods of the spectral and scattering theory for Schrödinger type operators are used for the analysis. Moreover, we present the easily verifiable necessary condition of the preservation of the nonnegativity of the solutions of a system of parabolic equations in the case of the anomalous diffusion with the negative Laplacian in a fractional power in one dimension, which imposes the necessary form of such system of equations that must be studied mathematically. This class of systems of PDEs has a wide range of applications. We conclude the survey with several new results nowhere published concerning the solvability in the sense of sequences for the generalized Poisson type equation with a scalar potential.

References

[1]	M. S. Agranovich, Elliptic boundary problems, Encyclopaedia Math. Sci., vol. 79, Partial Differential Equations, IX, Springer, Berlin, 1997, 1–144. https://doi.org/10.1007/9783662067215 1
[2]	M. A. Efendiev, Fredholm structures, topological invariants and applications, American Institute of Mathematical Sciences, 2009,205 pp.
[3]	J. L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, Volume 1. Dunod, Paris, 1968.
[4]	L. R. Volevich, Solubility of boundary problems for general elliptic systems, Mat. Sbor., 68 (1965), 373–416; English translation: Amer. Math. Soc. Transl., 67 (1968), 182–225.
[5]	V. Volpert, Elliptic partial differential equations, Volume I. Fredholm theory of elliptic problems in unbounded domains, Birkhäuser, Basel, 2011. https://doi.org/10.1007/978-3-0346-0537-3
[6]	H. G. Gebran, C. A. Stuart, Fredholm and properness properties of quasilinear elliptic systems of second order, Proc. Edinb. Math. Soc., 48 (2005), 91–124. https://doi.org/10.1017/S0013091504000550 doi: 10.1017/S0013091504000550
[7]	P. J. Rabier, C. A. Stuart, Fredholm and properness properties of quasilinear elliptic operators on ${\mathbb R}^{N}$, Math. Nachr., 231 (2001), 129–168. https://doi.org/10.1002/1522-2616(200111)231:1<129::AID-MANA129>3.0.CO;2-V
[8]	H. G. Gebran, C. A. Stuart, Exponential decay and Fredholm properties in second-order quasilinear elliptic systems, J. Differ. Equ., 249 (2010), 94–117. https://doi.org/10.1016/j.jde.2010.03.001 doi: 10.1016/j.jde.2010.03.001
[9]	M. A. Efendiev, Symmetrization and stabilization of solutions of nonlinear elliptic equations, Fields Institute Monographs 36 Fields Institute for Research in Mathematical Sciences, Toronto, ON; Springer, Cham (2018), 258 pp. https://doi.org/10.1007/978-3-319-98407-0
[10]	E. N. Dancer, Y. Du, M. A. Efendiev, Quasilinear elliptic equations on half -and quarter-spaces, Adv. Nonlinear Stud., 13 (2013), 115–136. https://doi.org/10.1515/ans-2013-0107 doi: 10.1515/ans-2013-0107
[11]	Y. Du, M. A. Efendiev, Existence and exact multiplicity for quasilinear elliptic equations in quarter-spaces, Int. Conf. Patterns of Dynamics, Springer, Cham (2017), 128–137. https://doi.org/10.1007/978-3-319-64173-7_8
[12]	M. A. Efendiev, Finite and infinite dimensional attractors for evolution equations of mathematical physics, GAKUTO International Series. Mathematical Sciences and Applications, $Gakk{\bar o}tosho Co., Ltd., Tokyo$, 2010,239 pp.
[13]	M. A. Efendiev, S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625–688. https://doi.org/10.1002/cpa.1011 doi: 10.1002/cpa.1011
[14]	V. Vougalter, V. Volpert, Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems, Anal. Math. Phys., 2 (2012), 473–496. https://doi.org/10.1007/s13324-012-0046-1 doi: 10.1007/s13324-012-0046-1
[15]	V. Vougalter, V. Volpert, Solvability conditions for some non Fredholm operators, Proc. Edinb. Math. Soc., 54 (2011), 249–271. https://doi.org/10.1017/S0013091509000236 doi: 10.1017/S0013091509000236
[16]	M. A. Efendiev, V. Vougalter, Solvability in the sense of sequences for some fourth order non-Fredholm operators, J. Differ. Equ., 271 (2021), 280–300. https://doi.org/10.1016/j.jde.2020.08.032 doi: 10.1016/j.jde.2020.08.032
[17]	V. Volpert, B. Kazmierczak, M. Massot, Z.Peradzynski, Solvability conditions for elliptic problems with non-Fredholm operators, Appl. Math., 29 (2002), 219–238. https://doi.org/10.4064/am29-2-7 doi: 10.4064/am29-2-7
[18]	V. Volpert, V. Vougalter, Solvability in the sense of sequences to some non-Fredholm operators, Electron. J. Differ. Equ., 160 (2013), 16 pp.
[19]	A. Ducrot, M. Marion, V. Volpert, Reaction-diffusion problems with non Fredholm operators, Adv. Diff. Equ., 13 (2008), 1151–1192.
[20]	M. A. Efendiev, V. Vougalter, Solvability of some integro-differential equations with drift, Osaka J. Math., 57 (2020), 247–265.
[21]	V. Vougalter, V. Volpert, On the existence of stationary solutions for some non-Fredholm integro-differential equations, Doc. Math., 16 (2011), 561–580.
[22]	V. Vougalter, V. Volpert, Existence in the sense of sequences of stationary solutions for some non-Fredholm integro- differential equations, J. Math. Sci. (N.Y.) 228, Problems in mathematical analysis, 90 (Russian) (2018), 601–632. https://doi.org/10.1007/s10958-017-3650-7
[23]	M. A. Efendiev, V. Vougalter, Existence of solutions for some non-Fredholm integro-differential equations with mixed diffusion, J. Differ. Equ., 284 (2021), 83–101. https://doi.org/10.1016/j.jde.2021.03.002 doi: 10.1016/j.jde.2021.03.002
[24]	N. Apreutesei, N. Bessonov, V. Volpert, V. Vougalter, Spatial Structures and Generalized Travelling Waves for an Integro- Differential Equation, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 537–557. https://doi.org/10.3934/dcdsb.2010.13.537 doi: 10.3934/dcdsb.2010.13.537
[25]	H. Berestycki, G. Nadin, B. Perthame, L. Ryzhik, The non-local Fisher-KPP equation: traveling waves and steady states, Nonlinearity, 22 (2009), 2813–2844. https://doi.org/10.1088/0951-7715/22/12/002 doi: 10.1088/0951-7715/22/12/002
[26]	M. A. Efendiev, V. Vougalter, Verification of Biomedical Processes with Anomalous Diffusion, Transport and Interaction of Species, Nonlinear Dynamics, Chaos, and Complexity, Springer, 2021. https://doi.org/10.1007/978-981-15-9034-4
[27]	V. Vougalter, V. Volpert, Existence of stationary solutions for some integro-differential equations with anomalous diffusion, J. Pseudo-Differ. Oper. Appl., 6 (2015), 487–501. https://doi.org/10.1007/s11868-015-0128-6 doi: 10.1007/s11868-015-0128-6
[28]	V. Vougalter, V. Volpert. Existence of stationary solutions for some non-Fredholm integro-differential equations with superdiffusion, J. Pseudo-Differ. Oper. Appl., 9 (2018), 1–24. https://doi.org/10.1007/s11868-016-0173-9 doi: 10.1007/s11868-016-0173-9
[29]	R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1–77. https://doi.org/10.1016/S0370-1573(00)00070-3 doi: 10.1016/S0370-1573(00)00070-3
[30]	V. G. Maz'ja, M. Otelbaev, Imbedding theorems and the spectrum of a certain pseudodifferential operator, (Russian) Sibirsk. Mat. Z., 18 (1977), 1073–1087, 1206.
[31]	M. A. Efendiev, L. A. Peletier, 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 (2007), 93–96. https://doi.org/10.1016/j.crma.2006.10.028 doi: 10.1016/j.crma.2006.10.028
[32]	V. Vougalter, V. Volpert, On the solvability in the sense of sequences for some non-Fredholm operators, Dyn. Partial Differ. Equ., 11 (2014), 109–124. https://doi.org/10.4310/DPDE.2014.v11.n2.a1 doi: 10.4310/DPDE.2014.v11.n2.a1
[33]	E. H. Lieb, M. Loss, Graduate Studies in Mathematics, Analysis, 14 (2001), 278 pp.
[34]	T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1965/1966), 258–279. https://doi.org/10.1007/BF01360915 doi: 10.1007/BF01360915
[35]	I. Rodnianski, W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004), 451–513. https://doi.org/10.1007/s00222-003-0325-4 doi: 10.1007/s00222-003-0325-4
[36]	M. Reed, B. Simon, Methods of modern mathematical physics. III. Scattering theory, Academic Press, 1979,463 pp.
[37]	H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, IX, Springer-Verlag, Berlin, 1987. https://doi.org/10.1007/978-3-540-77522-5
[38]	E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729–743. https://doi.org/10.1090/S0002-9947-1984-0743741-4 doi: 10.1090/S0002-9947-1984-0743741-4
[39]	E. N. Dancer, On positive solutions of some pairs of differential equations. II, J. Differ. Equ., 60 (1985), 236–258. https://doi.org/10.1016/0022-0396(85)90115-9 doi: 10.1016/0022-0396(85)90115-9
[40]	M. A. Efendiev, Evolution equations arising in the modelling of life sciences, International Series of Numerical Mathematics, 163 (2013), 217 pp. Birkhäuser/Springer Basel AG, Basel. https://doi.org/10.1007/978-3-0348-0615-2
[41]	N. Guerngar, E. Nane, R. Tinaztepe, S. Ulusoy, H. W. Van Wyk, Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation $ {{\partial}_{t}^{\beta}u = -(-\Delta)^{\frac{\alpha}{2}}u-(-\Delta)^{\frac{\gamma}{2}}u}$, Fract. Calc. Appl. Anal., 24 (2021), 818–847. https://doi.org/10.1515/fca-2021-0035 doi: 10.1515/fca-2021-0035
[42]	V. Vougalter, V. Volpert, Solvability of some integro-differential equations with anomalous diffusion, Regularity and stochasticity of nonlinear dynamical systems, Nonlinear Syst. Complex., 21, Springer, Cham (2018), 1–17. https://doi.org/10.1007/978-3-319-58062-3_1

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