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On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition. (English. Russian original) Zbl 1075.35106
Math. Notes 77, No. 4, 482-493 (2005); translation from Mat. Zametki 77, No. 4, 522-534 (2005).
Summary: We study the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation \[ \rho(t,x)u_t-\Delta u=f(x)g(t,x)+h(t,x) \] whose leading coefficient depends on both the time and the spatial variable under an integral overdetermination condition with respect to time. We obtain two types of condition sufficient for the local solvability of the inverse problem as well as study the so-called Fredholm solvability of the inverse problem under consideration.

MSC:
35R30 Inverse problems for PDEs
35K15 Initial value problems for second-order parabolic equations
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[1] A. I. Prilepko and A. B. Kostin, ”On inverse problems for parabolic equations with final and integral overdetermination,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 183 (1992), no. 4, 49–68. · Zbl 0802.35158
[2] A. I. Prilepko and A. B. Kostin, ”On inverse problems of determining a coeflicient in a parabolic equation, II,” Sibirsk. Mat. Zh. [Siberian Math. J.], 34 (1993), no. 5, 147–162. · Zbl 0807.35164
[3] A. I. Prilepko and I. V. Tikhonov, ”Reconstruction of an inhomogeneous summand in an abstract evolution equation,” Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 58 (1994), no. 2, 167–188. · Zbl 0826.47028
[4] I. V. Tikhonov and Yu. S. Éidel’man, ”Well-posedness of direct and inverse problems for an evolution equation of special form” Mat. Zametki [Math. Notes], 56 (1994), no. 2, 99–113.
[5] A. I. Prilepko and I. V. Tikhonov, ”The positivity principle for a solution in a linear inverse problem and its application to the coefficient heat problem,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 364 (1999), no. 1, 21–23. · Zbl 0961.35179
[6] A. Zegnal, ”Un résultat d’existence pour problème inverse parabolique quasi lineaire, ” C. R. Acad. Sci. Paris. Ser. I, 332 (2001), 909–912.
[7] A. I. Prilepko and D. S. Tkachenko, ”Properties of solutions of a parabolic equation and the uniqueness of the solution of the inverse problem on a source with integral overdetermination,” Zh. Vychisl. Mat. i Mat. Fiz. [Comput. Math. and Math. Phys.], 43 (2003), no. 4, 562–570. · Zbl 1078.35138
[8] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow, 1967. · Zbl 0164.12302
[9] Yu. A. Alkhutov and I. T. Mamedov, ”First boundary-value problem for nondivergent parabolic equations of second order with discontinuous coefficients,” Mat. Sb. [Math. USSR-Sb.], 173 (1986), no. 12, 477–500.
[10] V. L. Kamynin, ”On the unique solvability of an inverse problem for parabolic equations under a final overdetermination condition,” Mat. Zametki [Math. Notes], 73 (2003), no. 2, 217–227. · Zbl 1033.35138
[11] O. A. Ladyzhenskaya, Mathematical Questions of the Dynamics of a Viscous Incompressible Liquid [in Russian], Nauka, Moscow, 1970. · Zbl 0215.29004
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