Inverse problem for integro-differential Kelvin–Voigt equations

References

[1] U. U. Abylkairov and K. Khompysh, An inverse problem of identifying the coefficient in Kelvin–Voight equations, Appl. Math. Sci. 9 (2015), 5079–5089. 10.12988/ams.2015.57464Search in Google Scholar

[2] A. B. Al’shin, M. O. Korpusov and A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, De Gruyter Ser. Nonlinear Anal. Appl. 15, Walter de Gruyter, Berlin, 2011. 10.1515/9783110255294Search in Google Scholar

[3] S. N. Antontsev, H. B. de Oliveira and K. Khompysh, Generalized Kelvin–Voigt equations for nonhomogeneous and incompressible fluids, Commun. Math. Sci. 17 (2019), no. 7, 1915–1948. 10.4310/CMS.2019.v17.n7.a7Search in Google Scholar

[4] S. N. Antontsev, H. B. de Oliveira and K. Khompysh, Kelvin–Voigt equations perturbed by anisotropic relaxation, diffusion and damping, J. Math. Anal. Appl. 473 (2019), no. 2, 1122–1154. 10.1016/j.jmaa.2019.01.011Search in Google Scholar

[5] S. N. Antontsev, H. B. de Oliveira and K. Khompysh, Kelvin–Voigt equations with anisotropic diffusion, relaxation and damping: Blow-up and large time behavior, Asymptot. Anal. 121 (2021), no. 2, 125–157. 10.3233/ASY-201597Search in Google Scholar

[6] S. N. Antontsev and K. Khompysh, Generalized Kelvin–Voigt equations with p-Laplacian and source/absorption terms, J. Math. Anal. Appl. 456 (2017), no. 1, 99–116. 10.1016/j.jmaa.2017.06.056Search in Google Scholar

[7] S. N. Antontsev and K. Khompysh, Kelvin–Voight equation with p-Laplacian and damping term: existence, uniqueness and blow-up, J. Math. Anal. Appl. 446 (2017), no. 2, 1255–1273. 10.1016/j.jmaa.2016.09.023Search in Google Scholar

[8] A. Asanov and E. R. Atamanov, Nonclassical and Inverse Problems for Pseudoparabolic Equations, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 1997. 10.1515/9783110900149Search in Google Scholar

[9] H. A. Barnes, A Handbook of Elementary Rheology, University of Wales, Cardiff, 2000. Search in Google Scholar

[10] A. Y. Chebotarëv, Determination of the right-hand side of the Navier–Stokes system and inverse problems for thermal convection equations, Comp. Math. Math. Phys. 51 (2011), 2146–2154. 10.1134/S0965542511120098Search in Google Scholar

[11] J. Cheng and J. Liu, An inverse source problem for parabolic equations with local measurements, Appl. Math. Lett. 103 (2020), Article ID 106213. 10.1016/j.aml.2020.106213Search in Google Scholar

[12] J. Fan and G. Nakamura, Well-posedness of an inverse problem of Navier–Stokes equations with the final overdetermination, J. Inverse Ill-Posed Probl. 17 (2009), no. 6, 565–584. 10.1515/JIIP.2009.035Search in Google Scholar

[13] V. E. Fedorov and N. D. Ivanova, Inverse problem for Oskolkov’s system of equations, Math. Methods Appl. Sci. 40 (2017), no. 17, 6123–6126. 10.1002/mma.3807Search in Google Scholar

[14] 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

[15] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, Springer, New York, 2011. 10.1007/978-0-387-09620-9Search in Google Scholar

[16] O. Y. Imanuvilov and M. Yamamoto, Remark on boundary data for inverse boundary value problems for the Navier–Stokes equations [Addendum to MR3319370], Inverse Problems 31 (2015), no. 10, Article Id 109401. 10.1088/0266-5611/31/10/109401Search in Google Scholar

[17] V. Isakov, Inverse Problems for Partial Differential Equations, Appl. Math. Sci. 127, Springer, New York, 2006. Search in Google Scholar

[18] S. I. Kabanikhin, Inverse and Ill-Posed Problems. Theory and Applications, Inverse Ill-posed Probl. Ser. 55, De Gruyter, Berlin, 2011. 10.1515/9783110224016Search in Google Scholar

[19] N. A. Karazeeva, Solvability of initial boundary value problems for equations describing motions of linear viscoelastic fluids, J. Appl. Math. (2005), no. 1, 59–80. 10.1155/JAM.2005.59Search in Google Scholar

[20] K. Khompysh, Inverse Problem with integral overdetermination for system of equations of Kelvin–Voight fluids, Adv. Mat. Res. 705 (2013), 15–20. 10.4028/www.scientific.net/AMR.705.15Search in Google Scholar

[21] M. O. Korpusov and A. G. Sveshnikov, Blow-up of Oskolkov’s system of equations, Sb. Math. 200 (2009), no. 4, 549–572. 10.1070/SM2009v200n04ABEH004008Search in Google Scholar

[22] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Nauka, Moscow, 1970. Search in Google Scholar

[23] S. Lu, N. Chen, B. Hu and J. Cheng, On the inverse problems for the coupled continuum pipe flow model for flows in karst aquifers, Inverse Problems 28 (2012), no. 6, Article ID 065003. 10.1088/0266-5611/28/6/065003Search in Google Scholar

[24] A. S. Lyubanova and A. Tani, An inverse problem for pseudoparabolic equation of filtration: The existence, uniqueness and regularity, Appl. Anal. 90 (2011), no. 10, 1557–1571. 10.1080/00036811.2010.530258Search in Google Scholar

[25] A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 98–136. Search in Google Scholar

[26] A. P. Oskolkov, Some nonstationary linear and quasilinear systems that arise in the study of the motion of viscous fluids, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 59 (1976), 133–177. Search in Google Scholar

[27] A. P. Oskolkov, Theory of nonstationary flows of Kelvin–Voigt fluids, J. Math. Sci. 28 (1985), 751–758. 10.1007/BF02112340Search in Google Scholar

[28] A. P. Oskolkov, Initial-boundary value problems for equations of motion of Kelvin–Voight fluids and Oldroyd fluids, Proc. Steklov Inst. Math. 179 (1989), 137–182. Search in Google Scholar

[29] A. P. Oskolkov, Nonlocal problems for the equations of motion of the Kelvin–Voight fluids, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 197 (1992), 120–158. Search in Google Scholar

[30] A. P. Oskolkov, Nonlocal problems for the equations of motion of the Kelvin–Voight fluids, J. Math. Sci. 75 (1995), no. 6, 2058–2077. 10.1007/BF02362946Search in Google Scholar

[31] A. P. Oskolkov and R. D. Shadiev, Nonlocal problems in the theory of equations of motion for Kelvin–Voigt fluids, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 181 (1990), 146–185. Search in Google Scholar

[32] V. A. Pavlovsky, On the theoretical description of weak water solutions of polymers, Dokl. Akad. Nauk SSSR. 200 (1971), no. 4, 809–812. Search in Google Scholar

[33] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 2000. Search in Google Scholar

[34] L. N. Pyatnitsky, Turbulence Nature and the Inverse Problem, Fluid Mech. Appl. 89, Springer, New York, 2009. 10.1007/978-90-481-2251-6Search in Google Scholar

[35] G. A. Sviridyuk and A. S. Shipilov, On the stability of solutions of the Oskolkov equations on a graph, Differ. Equ. 46 (2010), 742–747. 10.1134/S0012266110050137Search in Google Scholar

[36] Q. Wu, A new type of the Gronwall–Bellman inequality and its application to fractional stochastic differential equations, Cogent Math. 4 (2017), Article ID 1279781. 10.1080/23311835.2017.1279781Search in Google Scholar

[37] V. G. Zvyagin and V. P. Orlov, On weak solutions of the equations of motion of a viscoelastic medium with variable boundary, Bound. Value Probl. 3 (2005), 215–245. 10.1155/BVP.2005.215Search in Google Scholar

[38] V. G. Zvyagin and M. V. Turbin, Investigation of initial-boundary value problems for mathematical models of the motion of Kelvin–Voigt fluids, J. Math. Sci. 168 (2010), 157–308. 10.1007/s10958-010-9981-2Search in Google Scholar