Abstract
The criteria of the well-posedness is obtained for an inverse problem to a class of fractional order in the sense of Caputo degenerate evolution equations with a relatively bounded pair of operators and with the generalized Showalter–Sidorov initial conditions. It is formulated in terms of the relative spectrum of the pair and of the characteristic function of the problem. Sufficient conditions of the unique solvability are obtained for a similar problem with the Cauchy initial condition. For these purposes the unique solvability of the same inverse problem was studied for the equation with a bounded operator near an unknown function, which is solved with respect to the fractional derivative. General results are applied to the inverse problem research for the time fractional system of equations describing the dynamics of a viscoelastic fluid in the weakly degenerate and the strongly degenerate cases.
Funding source: Russian Foundation for Basic Research
Award Identifier / Grant number: 19-41-450001
Funding statement: This work is supported by Act 211 of the Government of the Russian Federation, contract number 02.A03.21.0011, and by the Russian Foundation of Basic Research, grant number 19-41-450001.
References
[1] N. L. Abasheeva, Determination of a right-hand side term in an operator-differential equation of mixed type, J. Inverse Ill-Posed Probl. 10 (2002), no. 6, 547–560. 10.1515/jiip.2002.10.6.547Search in Google Scholar
[2] M. Al Horani and A. Favini, An identification problem for first-order degenerate differential equations, J. Optim. Theory Appl. 130 (2006), no. 1, 41–60. 10.1007/s10957-006-9083-ySearch in Google Scholar
[3] M. Al Horani and A. Favini, Degenerate first-order inverse problems in Banach spaces, Nonlinear Anal. 75 (2012), no. 1, 68–77. 10.1016/j.na.2011.08.001Search in Google Scholar
[4] N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, John Wiley & Sons, New York, 1988. Search in Google Scholar
[5] M. V. Falaleev, Abstract problem of the prediction-control with degeneration in Banach spaces (in Russian), Izv. Irkutskogo Gosuderstvennogo Univ. Ser. Mat. 3 (2010), 126–132. Search in Google Scholar
[6] V. E. Fedorov and D. M. Gordievskikh, Resolving operators of degenerate evolution equations with fractional derivative with respect to time, Russian Math. (Iz. VUZ) 59 (2015), no. 1, 60–70. 10.3103/S1066369X15010065Search in Google Scholar
[7] V. E. Fedorov, D. M. Gordievskikh and M. V. Plekhanova, Equations in Banach spaces with a degenerate operator under a fractional derivative, Differ. Equ. 51 (2015), no. 10, 1360–1368. 10.1134/S0012266115100110Search in Google Scholar
[8] V. E. Fedorov and N. D. Ivanova, Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel, Discrete Contin. Dyn. Syst. Ser. S 9 (2016), no. 3, 687–696. 10.3934/dcdss.2016022Search in Google Scholar
[9] V. E. Fedorov and N. D. Ivanova, Identification problem for degenerate evolution equations of fractional order, Fract. Calc. Appl. Anal. 20 (2017), no. 3, 706–721. 10.1515/fca-2017-0037Search in Google Scholar
[10] V. E. Fedorov and A. V. Urazaeva, An inverse problem for linear Sobolev type equations, J. Inverse Ill-Posed Probl. 12 (2004), no. 4, 387–395. 10.1515/1569394042248210Search in Google Scholar
[11] A. V. Glushak, On an inverse problem for a fractional-order abstract differential equation, Mat. Zametki 87 (2010), no. 5, 684–693. 10.1134/S0001434610050056Search in Google Scholar
[12] A. I. Kozhanov, Composite Type Equations and Inverse Problems, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 1999. 10.1515/9783110943276Search in Google Scholar
[13] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. Search in Google Scholar
[14] D. G. Orlovsky, Parameter determination in a differential equation of fractional order with Riemann–Liouville fractional derivative in a Hilbert space, J. Sib. Federal Univ. Math. Phys. 8 (2015), 55–63. 10.17516/1997-1397-2015-8-1-55-63Search in Google Scholar
[15] A. P. Oskolkov, Initial-boundary value problems for equations of motion of Kelvin–Voight fluids and Oldroyd fluids, Proc. Steklov Inst. Math. 179 (1988), 137–182. Search in Google Scholar
[16] M. V. Plekhanova, Nonlinear equations with degenerate operator at fractional Caputo derivative, Math. Methods Appl. Sci. 40 (2017), no. 17, 6138–6146. 10.1002/mma.3830Search in Google Scholar
[17] M. V. Plekhanova and V. E. Fedorov, Optimal Control for Degenerate Distributed Systems (in Russian), South Ural State University, Chelyabinsk, 2013. Search in Google Scholar
[18] A. I. Prilepko, The semigroup method for solving inverse, nonlocal, and nonclassical problems. Prediction-control and prediction-observation of evolution equations. I, Differ. Equ. 41 (2005), 1635–1646. 10.1007/s10625-005-0323-ySearch in Google Scholar
[19] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Monogr. Textb. Pure Appl. Math. 231, Marcel Dekker, New York, 2000. 10.1201/9781482292985Search in Google Scholar
[20] S. G. Pyatkov and M. L. Samkov, On some classes of coefficient inverse problems for parabolic systems of equations, Mat. Tr. 15 (2012), no. 1, 155–177. 10.3103/S1055134412040050Search in Google Scholar
[21] G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 2003. 10.1515/9783110915501Search in Google Scholar
[22] I. V. Tikhonov, Uniqueness theorems in linear nonlocal problems for abstract differential equations, Izv. Mat. 67 (2003), 333–363. 10.1070/IM2003v067n02ABEH000429Search in Google Scholar
[23] I. V. Tikhonov and Y. S. Eidelman, Questions of the well-posedness of direct and inverse problems for an evolution equation of special type, Math. Notes 56 (1994), 830–839. 10.1007/BF02110743Search in Google Scholar
[24] A. V. Urazaeva and V. E. Fedorov, Prediction-control problems for some systems of equations of fluid dynamics, Differ. Uravn. 44 (2008), no. 8, 1111–1119. 10.1134/S0012266108080120Search in Google Scholar
[25] A. V. Urazaeva and V. E. Fedorov, On the well-posedness of the prediction-control problem for some systems of equations, Mat. Zametki 85 (2009), no. 3, 440–450. 10.4213/mzm4345Search in Google Scholar
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