An initial value problem is studied for a class of evolutionary equations with a weak degeneration, which are nonlinear with respect to lower order fractional Gerasimov – Caputo derivatives. The linear part of the equations contains a respectively bounded pair of operators. Unique local solvability is proved in the case of a nonlinear operator depending on elements of the degeneration space only. Examples of an equation and a system of partial differential equations are given, the initial-boundary value problems for which are reduced to the initial problem for an equation in a Banach space of the studied class.

Guzel Baybulatova, Chelyabinsk State University, 129, Br.Kashirins str., Chelyabinsk, 454021, Russian Federation, tel.: (351)799-72-35, email: baybulatova g d@mail.ru

Marina Plekhanova, Dr. Sci. (Phys.– Math.), South Ural State University (National Research University), 76, Lenin Av., Chelyabinsk, 454080, Russian Federation; Chelyabinsk State University, 129, Br.Kashirins str., Chelyabinsk, 454021, Russian Federation, tel.: (351)799-72-35, email: mariner79@mail.ru

Baybulatova G.D., Plekhanova M.V. An Initial Problem for a Class of Weakly Degenerate Semilinear Equations with Lower Order Fractional Derivatives. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 35, pp. 34-48. https://doi.org/10.26516/1997-7670.2021.35.34

1. Bajlekova E.G. Fractional evolution equations in Banach spaces. PhD thesis. Eindhoven,Eindhoven University of Technology, University Press Facilities, 2001. https://doi.org/10.6100/IR549476
2. Debbouche A., Torres D.F.M. Sobolev type fractional dynamic equations and optimal multi-integral controls with fractional nonlocal conditions. Fractional Calculus and Applied Analysis, 2015, vol. 18, pp. 95–121. https://doi.org/10.1515/fca-2015-0007
3. Demidenko G.V., Uspensky S.V. Equations and systems not resolved with respect to the highest derivative. Novosibirsk, Scientific book Publ., 1998. 438+xviii p. (in Russian)
4. Falaleev M.V. On Solvability in the Class of Distributions of Degenerate Integro-Differential Equations in Banach Spaces. The Bulletin of Irkutsk State University. Series Mathematics, 2020, vol. 34, pp. 77–92. (in Russian) https://doi.org/10.26516/1997-7670.2020.34.77
5. Fedorov V.E., Gordievskikh D.M. Resolving operators of degenerate evolution equations with fractional derivative with respect to time. Russian Mathematics, 2015, vol. 59, pp. 60–70. https://doi.org/10.3103/S1066369X15010065
6. Fedorov V.E., Gordievskikh D.M., Plekhanova M.V. Equations in Banach spaces with a degenerate operator under a fractional derivative. Differential Equations, 2015, vol. 51, pp. 1360-1368. https://doi.org/10.1134/S0012266115100110
7. Fedorov V.E., Phuong T.D., Kien B.T., Boyko K.V., Izhberdeeva E.M. A class of distributed order semilinear equations Chelyabinsk Physical and Mathematical Journal, 2020, vol. 5, iss. 3, pp. 343–351. (in Russian) https://doi.org/10.47475/2500-0101-2020-15308
8. Fedorov V.E., Plekhanova M.V., Nazhimov R.R. Degenerate linear evolution equationswith the Riemann – Liouville fractional derivative. Siberian Math. J., 2018, vol. 59, no. 1, pp. 136–146. https://doi.org/10.17377/smzh.2018.59.115
9. Fedorov V.E., Romanova E.A., Debbouche A. Analytic in a sector resolving families of operators for degenerate evolution fractional equations. Journal of Mathematical Sciences, 2018, vol. 228, no. 4, pp. 380–394. https://doi.org/10.1007/s10958-017-3629-4
10. Glushak A.V., Avad Kh.K. On the solvability of an abstract differential equation of fractional order with a variable operator. Sovrem. maths. Foundation. Directions, 2013, vol. 47, pp. 18–32. (in Russian)
11. Hassard B.D., Kazarinoff N.D., Wan Y.-H. Theory and applications of hopf bifurcation. Cambridge University Press, Cambridge, London, 1981.
12. Kosti´c M. Abstract Volterra integro-differential equations. Boca Raton, FL, CRC Press, 2015, 484 p.
13. Mainardi F., Paradisi F. Fractional diffusive waves. Journal of Computational Acoustics, 2001, vol. 9, no. 4, pp. 1417–1436. https://doi.org/10.1142/S0218396X01000826
14. Mainardi F., Spada G. Creep. Relaxation and viscosity properties for basic fractional models in rheology. The European Physical Journal Special Topics, 2011, vol. 193, pp. 133–160. https://doi.org/10.1140/epjst/e2011-01387-1
15. Plekhanova M.V. Nonlinear equations with degenerate operator at fractional Caputo derivative. Mathematical Methods in the Applied Sciences, 2016, vol. 40, pp.41–44. https://doi.org/10.1002/mma.3830
16. Plekhanova M.V. Sobolev type equations of time-fractional order with periodical boundary conditions. AIP Conf. Proc., 2016, vol. 1759, pp. 020101-1–020101-4. https://doi.org/10.1063/1.4959715
17. Plekhanova M.V., Baybulatova G.D. A class of semilinear degenerate equations with fractional lower order derivatives. Stability, Control, Differential Games(SCDG2019), Yekaterinburg, Russia, 16–20 September 2019, eds.: T.F. Filippova, V.T. Maksimov, A.M. Tarasyev. Proceedings of the International Conference devoted to the 95th anniversary of Academician N.N. Krasovskii, 2019, pp. 444-448. https://doi.org/10.1007/978-3-030-42831-0 18
18. Plekhanova M.V., Baybulatova G.D. Semilinear equations in Banach spaces with lower fractional derivatives. Nonlinear Analysis and Boundary Value Problems(NABVP 2018), Santiago de Compostela, Spain, September 4–7, eds: I. Area, A. Cabada, J.A. Cid, etc., Springer Proceedings in Mathematics and Statistics. 2019. Vol. 292, xii+298. Cham, Springer Nature Switzerland AG, 2019, pp. 81–93. https://doi.org/10.1007/978-3-030-26987-6_6
19. Pruss J. Evolutionary Integral Equations and Applications. Basel, Birkhauser-Verlag, 1993, 366 p.
20. Sidorov N.A., Loginov B.V., Sinitsyn A.V., Falaleev M.V. Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications. Dordrecht, Kluwer Academic Publishers, 2002.https://doi.org/10.1007/978-94-017-2122-6
21. Sveshnikov A.G., Alshin A.B., Korpusov M.O., Pletner Yu.D. Linear and nonlinear equations of Sobolev type. Moscow, Fizmatlit Publ., 2007, 736 p. (in Russian)
22. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, VSP, 2003, 216 p. https://doi.org/10.1515/9783110915501
23. Uchaikin V.V. Fractional phenomenology of cosmic ray anomalous diffusion. Physics-Uspekhi, 2013, vol. 56, no. 11, pp. 1074–1119. https://doi.org/10.3367/UFNr.0183.201311b.1175