Abstract
Degenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading coefficient make the PDE non-uniformly parabolic, and new theories need to be developed in the context of practical applications of such rather unstudied mathematical models arising in porous media, population dynamics, financial mathematics, etc. With this new challenge in mind, this paper considers investigating newly formulated direct and inverse problems associated with non-uniform parabolic PDEs where the leading space- and time-dependent coefficient is allowed to vanish on a non-empty, but zero measure, kernel set. In the context of inverse analysis, we consider the linear but ill-posed identification of a space-dependent source from a time-integral observation of the weighted main dependent variable. For both, this inverse source problem as well as its corresponding direct formulation, we rigorously investigate the question of well-posedness. We also give examples of inverse problems for which sufficient conditions guaranteeing the unique solvability are fulfilled, and present the results of numerical simulations. It is hoped that the analysis initiated in this study will open up new avenues for research in the field of direct and inverse problems for degenerate parabolic equations with applications.
Funding source: Engineering and Physical Sciences Research Council
Award Identifier / Grant number: EP/P005985
Funding statement: V. L. Kamynin and A. B. Kostin were partially supported by the Programme of Competitiveness Increase of the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute); contract no 02.a03.21.0005, 27.08.2013. D. Lesnic would like to acknowledge some small financial support received from the EPSRC funded research network on inverse problems EP/P005985/1 for a week research visit of Professor V. L. Kamynin to Leeds in July 2017.
References
[1] T. Arbogast and A. L. Taicher, A linear degenerate elliptic equation arising from two-phase mixtures, SIAM J. Numer. Anal. 54 (2016), no. 5, 3105–3122. 10.1137/16M1067846Search in Google Scholar
[2] M. S. Birman, N. Y. Vilenkin, E. A. Gorin, P. P. Zabreĭko, I. S. Iokhvidov, M. U. Kadets’, A. G. Kostyuchenko, M. A. Krasnosel’skiĭ, S. G. Kreĭn, B. S. Mityagin, Y. I. Petunin, Y. B. Rutitskiĭ, E. M. Semenov, V. I. Sobolev, V. Y. Stetsenko, L. D. Faddeev and E. S. Tsitlanadze, Functional Analysis (in Russian), 2nd ed., Izdat. “Nauka”, Moscow, 1972; English translation of the first edition: G. F. Votruba and L. F. Boron, Functional Analysis, Wolters-Noordhoff, Groningen, 1972. Search in Google Scholar
[3] I. Bouchouev and V. Isakov, Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets, Inverse Problems 15 (1999), no. 3, R95–R116. 10.1088/0266-5611/15/3/201Search in Google Scholar
[4] P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications, Mem. Amer. Math. Soc. 239 (2016), no. 1133. 10.1090/memo/1133Search in Google Scholar
[5] P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Problems 26 (2010), no. 10, Article ID 105003. 10.1088/0266-5611/26/10/105003Search in Google Scholar
[6] T. F. Coleman and Y. Li, An interior trust region approach for nonlinear minimization subject to bounds, SIAM J. Optim. 6 (1996), no. 2, 418–445. 10.1137/0806023Search in Google Scholar
[7] Z. Deng and L. Yang, An inverse problem of identifying the radiative coefficient in a degenerate parabolic equation, Chin. Ann. Math. Ser. B 35 (2014), no. 3, 355–382. 10.1007/s11401-014-0836-xSearch in Google Scholar
[8] Z.-C. Deng, K. Qian, X.-B. Rao, L. Yang and G.-W. Luo, An inverse problem of identifying the source coefficient in a degenerate heat equation, Inverse Probl. Sci. Eng. 23 (2015), no. 3, 498–517. 10.1080/17415977.2014.922079Search in Google Scholar
[9] Z.-C. Deng and L. Yang, An inverse problem of identifying the coefficient of first-order in a degenerate parabolic equation, J. Comput. Appl. Math. 235 (2011), no. 15, 4404–4417. 10.1016/j.cam.2011.04.006Search in Google Scholar
[10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1998. Search in Google Scholar
[11] N. Huzyk, Inverse problem of determining the coefficients in a degenerate parabolic equation, Electron. J. Differential Equations 2014 (2014), Paper No. 172. Search in Google Scholar
[12] M. Ivanchov and N. Saldina, An inverse problem for strongly degenerate heat equation, J. Inverse Ill-Posed Probl. 14 (2006), no. 5, 465–480. 10.1515/156939406778247598Search in Google Scholar
[13] M. I. Ivanchov and N. V. Saldina, Inverse problem for the heat equation with degeneration, Ukrainian Math. J. 57 (2005), no. 11, 1825–1835. 10.1007/s11253-006-0032-6Search in Google Scholar
[14] M. I. Ivanchov and N. V. Saldina, Inverse problem for a parabolic equation with strong power degeneration, Ukrainian Math. J. 58 (2006), no. 11, 1685–1703. 10.1007/s11253-006-0162-xSearch in Google Scholar
[15] V. L. Kamynin, On the well-posed solvability of the inverse problem for determining the right-hand side of a degenerate parabolic equation with an integral observation condition, Mat. Zametki 98 (2015), no. 5, 710–724. Search in Google Scholar
[16] V. L. Kamynin, Inverse problem of determining the right-hand side in a degenerating parabolic equation with unbounded coefficients, Comput. Math. Math. Phys. 57 (2017), no. 5, 833–842. 10.1134/S0965542517050049Search in Google Scholar
[17] A. Kawamoto, Inverse problems for linear degenerate parabolic equations by “time-like” Carleman estimate, J. Inverse Ill-Posed Probl. 23 (2015), no. 1, 1–21. 10.1515/jiip-2013-0027Search in Google Scholar
[18] M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space (in Russian), Uspehi Matem. Nauk (N. S.) 3 (1948), no. 1(23), 3–95. Search in Google Scholar
[19] S. N. Kružkov, Quasilinear parabolic equations and systems with two independent variables (in Russian), Trudy Sem. Petrovsk. (1979), no. 5, 217–272. Search in Google Scholar
[20] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type (in Russian), Izdat. “Nauka”, Moscow, 1967; English translation: Transl. Math. Monogr. 23, American Mathematical Society, Providence, 1968. Search in Google Scholar
[21] O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Appl. Math. Sci. 49, Springer, New York, 1985. 10.1007/978-1-4757-4317-3Search in Google Scholar
[22] L. A. Lyusternik and V. I. Sobolev, A Short Course in Functional Analysis (in Russian), “Vyssh. Shkola”, Moscow, 1982. Search in Google Scholar
[23] V. P. Mikhaĭlov, Partial Differential Equations (in Russian), 2nd ed., “Nauka”, Moscow, 1983. Search in Google Scholar
[24] V. A. Morozov, On the solution of functional equations by the method of regularization, Soviet Math. Dokl. 7 (1966), 414–417. Search in Google Scholar
[25] A. I. Prilepko, V. L. Kamynin and A. B. Kostin, Inverse source problem for parabolic equation with the condition of integral observation in time, J. Inverse Ill-Posed Probl. 26 (2018), no. 4, 523–539. 10.1515/jiip-2017-0049Search in Google Scholar
[26] A. I. Prilepko and A. B. Kostin, Some inverse problems for parabolic equations with final and integral observation, Sb. Math. 75 (1993), 473–490. 10.1070/SM1993v075n02ABEH003394Search in Google Scholar
[27] 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
[28] A. I. Prilepko and I. V. Tikhonov, Reconstruction of the inhomogeneous term in an abstract evolution equation, Izv. Math. 44 (1995),373–394. 10.1070/IM1995v044n02ABEH001602Search in Google Scholar
[29] Documentation Optimization Toolbox, https://www.mathworks.com/, 2012. Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
