×

An inverse problem for a higher order parabolic equation. (English. Russian original) Zbl 0937.35184

Math. Notes 64, No. 5, 590-599 (1998); translation from Mat. Zametki 64, No. 5, 680-691 (1998).
Summary: We prove existence and uniqueness theorems for the inverse problem of finding the right-hand side of a higher-order parabolic equation with two independent variables and an additional condition in the form of integral overdetermination. The results obtained are used to study the passage to the limit in a sequence of such inverse problems with weakly convergent coefficients.

MSC:

35R30 Inverse problems for PDEs
35K25 Higher-order parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. I. Prilepko and D. G. Orlovskii, ”Determination of a parameter in the evolution equation and in inverse problems of mathematical physics. I,”Differentsial’nye Uravneniya [Differential Equations],21, No. 1, 119–129 (1985).
[2] A. I. Prilepko and D. G. Orlovskii, ”Determination of a parameter in the evolution equation and in inverse problems of mathematical physics. II,”Differentsial’nye Uravneniya [Differential Equations],21, No. 4, 694–700 (1985).
[3] J. R. Cannon and Y. Lin, ”Determination of a parameterp(t) in some quasilinear parabolic differential equations,”Inverse Problems,4, No. 1, 35–45 (1988). · Zbl 0697.35162 · doi:10.1088/0266-5611/4/1/006
[4] J. R. Cannon and Y. Lin, ”Determination of a parameterp(t) in Hölder classes for some semilinear parabolic equations,”Inverse Problems,4, No. 3, 595–606 (1988). · Zbl 0688.35104 · doi:10.1088/0266-5611/4/3/005
[5] A. B. Kostin, ”An inverse problem for the heat equation with integral overdetermination,” in:Inverse Problems for Mathematical Modeling of Physical Processes [in Russian], MIFI, Moscow (1991), pp. 45–49.
[6] V. L. Kamynin, ”On convergence of the solutions of inverse problems for parabolic equations with weakly converging coefficients,” in:Elliptic and Parabolic Problems (Pont-à-Mousson, 1994), Vol. 325, Pitman Res. Notes Math. Ser, Longman Sci. Tech., London (1995), pp. 130–151.
[7] I. A. Vasin, ”On several inverse problems of viscous incompressible fluid dynamics in the case of integral overdetermination,”Zh. Vychisl. Mat. i Mat. Fiz. [Comput. Math. and Math. Phys.],31, No. 7, 1071–1079 (1992). · Zbl 0777.76026
[8] V. L. Kamynin and I. A. Vasin, ”Inverse problems for linearized Navier-Stokes equations with integral overdetermination. Unique solvability and passage to the limit,”Ann. Univ. Ferrara. Sez. VII (N.S.),38, 229–247 (1992). · Zbl 0831.35127
[9] S. N. Kruzhkov, ”Quasilinear parabolic equations and systems with two independent variables,”Trudy Sem. Petrovsk.,5, 217–272 (1979).
[10] S. N. Kruzhkov and V. L. Kamynin, ”Convergence of solutions of quasilinear parabolic equations with weakly convergent coefficients,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],270, No. 3, 533–536 (1983). · Zbl 0538.35047
[11] V. L. Kamynin, ”On the passage to the limit in quasilinear higher-order parabolic equations,” in:Continuous Medium Dynamics [in Russian], Vol. No. 63, Siberian Division of the Academy of Sciences of the USSR, Novosibirsk (1983), pp. 122–128. · Zbl 0569.35043
[12] V. L. Kamynin, ”On the passage to the limit in quasilinear elliptic equations with many independent variables,”Mat. Sb. [Math. USSR-Sb.],174, No. 1, 45–63 (1987).
[13] V. L. Kamynin, ”Passage to the limit in quasilinear parabolic equations with weakly convergent coefficients and the asymptotic behavior of the solution of the Cauchy problem,”Mat. Sb. [Math. USSR-Sb.],181, No. 8, 1031–1047 (1990).
[14] V. L. Kamynin, ”Passage to the limit in the inverse problem for nondivergent parabolic equations with final overdetermination,”Differentsial’nye Uravneniya [Differential Equations],28, No. 2, 247–253 (1992). · Zbl 0807.35161
[15] A. Bensoussan, J.-L. Lions, and G. Papanicolaou,Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978). · Zbl 0404.35001
[16] V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik,Averaging of Differential Operators [in Russian], Nauka, Moscow (1993).
[17] Maso G. Dal,An Introduction to {\(\Gamma\)}-Convergence, Birkhäuser, Boston (1993). · Zbl 0816.49001
[18] G. Buttazzo and L. Freddi, ”Sequences of optimal control problems with measures as controls,”Adv. Math. Sci. Appl.,2, 215–230 (1993). · Zbl 0798.49022
[19] O. V. Besov, V. P. Il’in, and S. M. Nikol’skii,Integral Representations of Functions and Embedding Theorems [in Russian], Nauka, Moscow (1975).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.