×

Inverse problem for the nonlinear heat equation with the final overdetermination. (English) Zbl 0835.35157

Summary: An inverse problem for the one-dimensional heat equation with a nonlinear term and an unknown space-dependent coefficient is considered when, in addition to initial and boundary conditions, a solution at the final moment of time is given. Existence and uniqueness results are proven, and a numerical method for the explicit determination of the solution to the said inverse problem is also developed.

MSC:

35R30 Inverse problems for PDEs
35K55 Nonlinear parabolic equations
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amirov, A. Kh., On the question of solvability of inverse problems, Soviet. Math. Dokl., 34, 258-259 (1987) · Zbl 0657.35130
[2] Prilepko, A. I.; Solov’ev, V. V., Solvability theorems and Rothe’s method for inverse problems for parabolic equation II, Differential Equations, 23, 11, 1341-1349 (1987) · Zbl 0683.35090
[3] Solov’ev, V. V., On solvability of the inverse problem of finding the source with overdetermination on upper boundary for a parabolic equation, Differential Equations, 25, 9, 1114-1119 (1989) · Zbl 0695.35215
[4] Bazdyreva, M. B., The inverse problem for the second and fourth order differential equations, Ph.D. Dissertation in Phys.-Math. Sciences (1990), Novosibirsk, (Russian) · Zbl 0816.35148
[5] Prilepko, A. I.; Vasin, I. A., On inverse initial-boundary problem for nonlinear Navier-Stokes system in the final overdeterminations case, Differential Equations, 25, 12, 2164-2177 (1989), (Russian) · Zbl 0850.76119
[6] Anikonov, Yu. E.; Bubnov, V. A., Control and inverse problems, Soviet Phys. Dokl., 34, 2, 13-14 (1989) · Zbl 0682.93033
[7] Rundel, W., The determination of a parabolic equation from initial and final data, (Proc. Amer. Math. Soc., 99 (1987)), 637-642, (4) · Zbl 0644.35093
[8] Isakov, V., Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math., 44, 185-209 (1991) · Zbl 0729.35146
[9] Prilepko, A. I.; Kostin, A. B., On inverse problems of determination of a coefficient in the parabolic equation I, Sib. Math. J., 33, 3, 146-155 (1992), (Russian) · Zbl 0784.35122
[10] Ladyzenskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., (Trans. Math. Monographs, Vol. 23 (1968)), English translation in · Zbl 0174.15403
[11] Lions, J.-L., Quelques Methodes de Resolution des Problems aux Limites non Lineaires (1969), Dunod, Gauthier-Villars: Dunod, Gauthier-Villars Paris · Zbl 0189.40603
[12] Shelukhin, V. V., On the solution of linear evolution equations by a variational method, Soviet Math. Dokl., 43, 3, 735-737 (1991) · Zbl 0782.35030
[13] Shelukhin, V. V., A problem with time averaged data for nonlinear parabolic equations, Siberian Math. J., 32, 2, 309-320 (1991) · Zbl 0764.35049
[14] Bergh, J.; Löfström, J., Interpolation spaces-An Introduction, (Grundlehren Math. Wissensch. 223 (1976), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0344.46071
[15] Lions, J.-L.; Magenes, E., (Problemes aux Limites Non Homogenes et Applications, Vol. 1 et 2 (1968), Dunod: Dunod Paris) · Zbl 0165.10801
[16] Nikolskij, S. M., On theorems imbedding continuation and approximation for differentiable functions of several variables, Uspekhi Matem. Nauk., 16, 63-114 (1961), (in Russian) · Zbl 0117.29101
[17] Shelukhin, V. V., Nonlocal with respect to time problems for the equations of hydrodynamics and variational principles, Ph.D. Dissertation in Phys.-Math. Sciences (1992), Novosibirsk
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.