The inverse problem of recovering the source in a parabolic equation under a condition of nonlocal observation

A. B. Kostin

Sbornik. Mathematics, Tome 204 (2013) no. 10, pp. 1391-1434

Résumé

We study the inverse problem for a parabolic equation of recovering the source, that is, the right-hand side $F(x,t)=h(x,t)f(x)$, where the function $f(x)$ is unknown. To find $f(x)$, along with the initial and boundary conditions, we also introduce an additional condition of nonlocal observation of the form $\displaystyle\int_{0}^{T}u(x,t)\,d\mu(t)=\chi(x)$. We prove the Fredholm property for the problem stated in this way, and obtain sufficient conditions for the existence and uniqueness of a solution. These conditions are of the form of readily verifiable inequalities and put no restrictions on the value of $T>0$ or the diameter of the domain $\Omega$ under consideration. The proof uses a priori estimates and the qualitative properties of solutions of initial-boundary value problems for parabolic equations. Bibliography: 40 titles.

Détail

Keywords: inverse problems, nonlocal overdetermination.
Mots-clés : parabolic equations

Bibliographie

[1] A. I. Prilepko, “Reports delivered at a general meeting of the mathematics division, Academy of Sciences of the USSR”, Math. Notes, 14:5 (1973), 990–996 | DOI | MR

[2] V. M. Isakov, “On a class of inverse problems for parabolic equations”, Soviet Math. Dokl., 25:2 (1982), 519–521 | MR | Zbl

[3] A. I. Prilepko, V. V. Solov'ev, “Solvability theorems and Rothe's method for inverse problems for a parabolic equation. II”, Differential Equations, 23:11 (1987), 1341–1349 | MR | Zbl

[4] V. V. Solov'ev, “Solvability of the inverse problem of finding a source, using overdetermination on the upper base for a parabolic equation”, Differential Equations, 25:9 (1989), 1114–1119 | MR | Zbl

[5] Yu. S. Eidelman, “Edinstvennost resheniya obratnoi zadachi dlya differentsialnogo uravneniya v banakhovom prostranstve”, Differents. uravneniya, 23:9 (1987), 1647–1649 | MR

[6] D. G. Orlovskiǐ, “Determination of a parameter of an evolution equation”, Differential Equations, 26:9 (1990), 1201–1207 | MR | Zbl

[7] V. Isakov, “Inverse parabolic problems with the final overdetermination”, Comm. Pure Appl. Math., 44:2 (1991), 185–209 | DOI | MR | Zbl

[8] A. I. Prilepko, A. B. Kostin, “On certain inverse problems for parabolic equations with final and integral observation”, Russian Acad. Sci. Sb. Math., 75:2 (1993), 473–490 | DOI | MR | Zbl

[9] A. I. Prilepko, A. B. Kostin, “Estimation of the spectral radius of an operator and the solvability of inverse problems for evolution equations”, Math. Notes, 53:1 (1993), 63–66 | DOI | MR | Zbl

[10] A. I. Prilepko, I. V. Tikhonov, “Recovery of the nonhomogeneous term in an abstract evolution equation”, Russian Acad. Sci. Izv. Math., 44:2 (1995), 373–394 | DOI | MR | Zbl

[11] I. V. Tikhonov, Obratnye zadachi dlya evolyutsionnykh uravnenii i prilozheniya k uravneniyu perenosa, Dis. $\dots$ kand. fiz.-matem. nauk, 1993

[12] A. I. Prilepko, I. V. Tikhonov, “The principle of the positiveness of a solution to a linear inverse problem and its application to the coefficient heat conduction problem”, Dokl. Math., 59:1 (1999), 14–16 | MR | Zbl

[13] A. I. Prilepko, D. G. Orlovsky, I. A. Vasin, Methods for solving inverse problems in mathematical physics, Monogr. Textbooks Pure Appl. Math., 231, Marcel Dekker, New York, 2000 | MR | Zbl

[14] A. I. Prilepko, D. S. Tkachenko, “The Fredholm property and the well-posedness of the inverse source problem with integral overdetermination”, Comput. Math. Math. Phys., 43:9 (2003), 1338–1347 | MR | Zbl

[15] V. L. Kamynin, “On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition”, Math. Notes, 77:4 (2005), 482–493 | DOI | MR | Zbl

[16] A. I. Prilepko, “The semigroup method for inverse, nonlocal, and nonclassical problems. Prediction-control and prediction-observation for evolution equations. I”, Differ. Equ., 41:11 (2005), 1635–1646 | DOI | MR | Zbl

[17] A. I. Prilepko, A. B. Kostin, “Inverse problems of the determination of the coefficient in parabolic equations. I”, Siberian Math. J., 33:3 (1992), 489–496 | DOI | MR | Zbl

[18] A. I. Prilepko, A. B. Kostin, “Inverse initial-boundary-value problems for linearized nonstationary Navier–Stokes equations”, Differential Equations, 25:1 (1989), 85–92 | MR | Zbl

[19] A. I. Kozhanov, R. R. Safiullova, “Linear inverse problems for parabolic and hyperbolic equations”, J. Inverse Ill-Posed Probl., 18:1 (2010), 1–24 | DOI | MR

[20] A. B. Kostin, “Basis property of a system of functions related to the inverse problem of finding the source”, Differential Equations, 44:2 (2008), 256–266 | DOI | MR | Zbl

[21] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural'tseva, Linear and quasi-linear equations of parabolic type, Transl. Math. Monogr., 23, Amer. Math. Soc., Providence, RI, 1968 | MR | MR | Zbl | Zbl

[22] N. Dunford, J. T. Schwartz, Linear operators. I. General theory., Pure Appl. Math., 7, Intersci. Publ., New York–London, 1958 | MR | MR | Zbl

[23] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, Lan, SPb., 1999 ; I. P. Natanson, Theory of functions of a real variable, Frederick Ungar Publ. Co., New York, 1955 | MR | Zbl | MR | Zbl

[24] G. M. Lieberman, Second order parabolic differential equations, World Scientific, Singapore, 2005 | MR | Zbl

[25] O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and quasilinear elliptic equations, Math. Sci. Engrg., 46, Academic Press, New York–London, 1968 | MR | MR | Zbl | Zbl

[26] V. P. Mikhajlov, Partial differential equations, Mir, Moscow, 1978 | MR | MR | Zbl | Zbl

[27] N. S. Trudinger, “Pointwise estimates and quasilinear parabolic equations”, Comm. Pure Appl. Math., 21 (1968), 205–226 | DOI | MR | Zbl

[28] M. G. Krein, M. A. Rutman, “Lineinye operatory, ostavlyayuschie invariantnym konus v prostranstve Banakha”, UMN, 3:1(23) (1948), 3–95 | MR | Zbl

[29] G. F. Votruba, L. F. Boron (eds.), Functional analysis, Wolters-Noordhoff Publ., Groningen, 1972 | MR | MR | Zbl

[30] M. Chicco, “Principio di massimo per soluzioni di problemi al contorno misti per equazioni ellittiche di tipo variazionale”, Boll. Un. Mat. Ital. (4), 3 (1970), 384–394 | MR | Zbl

[31] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss., 224, Springer-Verlag, Berlin, 1983 | MR | MR | Zbl | Zbl

[32] A. B. Kostin, “Razreshimost odnoi problemy momentov i ee svyaz s parabolicheskoi obratnoi zadachei”, Vestn. Mosk. un-ta. Ser. 15. Vychislit. matem. i kibern., 1995, no. 1, 28–33 | MR

[33] D. G. Orlovskii, “Determination of a parameter of a parabolic equation in Hilbert's structure”, Math. Notes, 55:3 (1994), 312–317 | DOI | MR | Zbl

[34] A. I. Prilepko, I. V. Tikhonov, “An inverse problem with final time additional condition for an abstract evolution equation in an ordered Banach space”, Funct. Anal. Appl., 27:1 (1993), 68–70 | DOI | MR | Zbl

[35] A. B. Kostin, A. I. Prilepko, “On some problems of restoration of a boundary condition for a parabolic equation. I”, Differential Equations, 32:1 (1996), 113–122 | MR | Zbl

[36] V. P. Mikhailov, A. K. Guschin, “Dopolnitelnye glavy kursa «Uravneniya matematicheskoi fiziki»”, Lekts. kursy NOTs, 7, MIAN, M., 2007, 3–144 | DOI

[37] A. M. Denisov, Elements of the theory of inverse problems, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 1999 | MR | Zbl

[38] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 ; I. C. Gohberg, M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, RI, 1969 | MR | MR | Zbl

[39] T. Carleman, “Über die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen”, Ber. Verh. Sachs. Akad. Leipzig, 88 (1936), 119–132 | Zbl

[40] M. Sh. Birman, M. Z. Solomyak, “Asymptotic behavior of the spectrum of differential equations”, J. Soviet Math., 12:3 (1979), 247–283 | DOI | MR | Zbl | Zbl