Inverse coefficient problems for nonlinear parabolic differential equations. (English) Zbl 1157.35402

Summary: This paper is devoted to a class of inverse problems for a nonlinear parabolic differential equation. The unknown coefficient of the equation depends on the gradient of the solution and belongs to a set of admissible coefficients. It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence. Based on this result the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.


35K55 Nonlinear parabolic equations
49J20 Existence theories for optimal control problems involving partial differential equations
65K10 Numerical optimization and variational techniques
35R30 Inverse problems for PDEs
Full Text: DOI


[1] Tikhonov, A., Arsenin, V.: Solutions of ill-posed problems, New York, Wiley 1977 · Zbl 0354.65028
[2] Ackleh, A. S., Ke, L.: Existence-uniqueness and long time behaviour for a class of nonlocal nonlinear parabolic evolution equations. Proceedings of the American Mathematical Society, 128, 3483–3492 (2000) · Zbl 0959.35086 · doi:10.1090/S0002-9939-00-05912-8
[3] DuChateau, P.: Monotonicity and invertibility of coefficient-to-data mappings for parabolic inverse problems. SIAM J. Math. Anal., 26, 1473–1487 (1995) · Zbl 0849.35146 · doi:10.1137/S0036141093259257
[4] DuChateau, P., Thelwell, R., Butters, G.: Analysis of an adjoint problem approach to the identification of an unknown diffusion coeficient. Inverse Problems, 20, 601–625 (2004) · Zbl 1054.35125 · doi:10.1088/0266-5611/20/2/019
[5] Hanke, M., Scherzer, O.: Error analysis of an equation error method for the identification of the diffusion coefficient in a quasilinear parabolic differential equation. SIAM J. Math. Anal., 59, 1012–1027 (1999) · Zbl 0928.35198
[6] Liu, Z. H.: On the identification of coefficients of semilinear parabolic equations. Acta Math. Appl. Sinica, 10, 356–367 (1994) · Zbl 0822.35150 · doi:10.1007/BF02016326
[7] Liu, Z. H.: Identification of parameters in semilinear parabolic equations. Acta Mathematica Scientia, English Series, 19, 175–180 (1999) · Zbl 0932.35204
[8] Liu, Z. H.: Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities. Inverse Problems, 21, 13–20 (2005) · Zbl 1078.49006 · doi:10.1088/0266-5611/21/1/002
[9] Liu, Z. H., Zou, J. Z.: Strong convergence results for hemivariational inequalities. Science in China, Series A, Mathematics, 49(7), 893–901 (2006) · Zbl 1149.49012 · doi:10.1007/s11425-006-2002-8
[10] Liu, Z. H., Li, J., Li, Z. W.: Regularization method with two parameters for nonlinear ill-posed problems. Science in China, Series A, Mathematics, 51(1), 70–78 (2008) · Zbl 1141.65040 · doi:10.1007/s11425-007-0131-3
[11] Hasanov, A.: Inverse coefficient problems for monotone potential operators. Inverse Problems, 13, 1265–1278 (1997) · Zbl 0883.35128 · doi:10.1088/0266-5611/13/5/011
[12] Hasanov, A.: Inverse coefficient problems for elliptic variational inequalities with a nonlinear monotone operator. Inverse Problems, 14, 1151–1169 (1998) · Zbl 0912.35157 · doi:10.1088/0266-5611/14/5/005
[13] Hasanov, A.: Computational material dagnostics based on limited boundary measurements: An inversion method for identification of elastoplastic properties from indentation measurements, In Book: System Modeling and Simulation: Theory and Applications, Asian Simulation Conference 2006, Springer, Tokyo, pp. 11–15, 2006
[14] Kachanov, L. M.: Fundamentals of the Theory of Plasticity Dover Books on Engineering, Dower Publications, New York, 2004
[15] Zeidler, E.: Nonlinear Functional Analysis and Its Applications II A/B, Springer, New York, 1990 · Zbl 0684.47028
[16] Liu, Z. H.: On the solvability of degenerate quasilinear parabolic equations of second order. Acta Mathematica Sinica, English Series, 16, 313–324 (1999) · Zbl 0955.35046 · doi:10.1007/s101140000052
[17] Liu, Z. H.: On doubly degenerate quasilinear parabolic equations of higher order. Acta Mathematica Sinica, English Series, 21(1), 197–208 (2005) · Zbl 1084.35036 · doi:10.1007/s10114-004-0415-2
[18] Ladyzhenskaya, O. A.: Boundary Value Problems in Mathematical Physics, Springer, New York, 1985 · Zbl 0588.35003
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.