×

zbMATH — the first resource for mathematics

Identification of a diffusion coefficient in strongly degenerate parabolic equations with interior degeneracy. (English) Zbl 1325.35278
The authors study two identification problems for a parabolic equation with a second-order differential degenerate operator both for homogeneous Dirichlet boundary conditions and for homogeneous Dirichlet-Neumann boundary conditions. Both problems are treated as nonlinear optimization problems by approaching them as optimal control problems in coefficients. The solution to the equation is assumed to be known on the whole time period for the first problem and at the final moment for the second problem. The authors first prove the existence and uniqueness of solutions to the considered problems by a variational technique, and then, they prove the existence of a minimizer constructed as a limit of a minimizing sequence. In order to get a clear representation of the control, the authors introduce an approximating control problem by regularizing the state equation.

MSC:
35R30 Inverse problems for PDEs
35K65 Degenerate parabolic equations
49N45 Inverse problems in optimal control
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alabau-Boussouira, F.; Cannarsa, P.; Fragnelli, G., Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6, 161-204, (2006) · Zbl 1103.35052
[2] Barbu V.: Mathematical Methods in Optimization of Differential Systems. Dordrecht, Kluwer Academic Publishers (1994) · Zbl 0819.49002
[3] Barbu V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, New York (2010) · Zbl 1197.35002
[4] Barbu, V.; Friedman, A., Optimal design of domains with free boundary problems, SIAM J. Control Optimiz., 29, 623-627, (1991) · Zbl 0732.49003
[5] Belmiloudi, A., Nonlinear optimal control problems of degenerate parabolic equations with logistic time-varying delays of convolution type, Nonlinear Anal., 63, 1126-1152, (2005) · Zbl 1079.49022
[6] Buchot J.M., Raymond J.P.: A linearized model for boundary layer equations in Optimal control of complex structures, Oberwolfach 2000. Internat. Ser. Numer. Math. 139, Basel: Birkhä user 31-42 (2002) · Zbl 1029.35029
[7] Caldiroli, P.; Musina, R., On a variational degenerate elliptic problem, NoDEA Nonlinear Differ. Equ. Appl., 7, 187-199, (2000) · Zbl 0960.35039
[8] Campiti, M.; Metafune, G.; Pallara, D., Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, 57, 1-36, (1998) · Zbl 0915.47029
[9] Cannarsa, P.; Tort, J.; Yamamoto, M., Determination of source terms in a degenerate parabolic equation, Inverse Problems, 26, 1-20, (2010) · Zbl 1200.35319
[10] Favini, A.; Marinoschi, G., Identification of the time derivative coefficient in a fast diffusion degenerate equation, J. Optim. Theory Appl., 145, 249-269, (2010) · Zbl 1201.49042
[11] Favini A., Yagi A.: Degenerate Differential Equations in Banach Spaces. New York, Marcel Dekker Inc. (1999) · Zbl 0913.34001
[12] Fleming W.H., Viot M.: Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28, 817-843 (1979) · Zbl 0444.60064
[13] Fragnelli G., Ruiz Goldstein G., Goldstein J.A., Romanelli S.: Generators with interior degeneracy on spaces of \(L\)\^{2} type. Electron. J. Differential Equations 2012, 1-30 (2012) · Zbl 1301.47065
[14] Fragnelli G., Mugnai D.: Carleman estimates and observability inequalities for parabolic equations with interior degeneracy. Adv. Nonlinear Anal. 2, 339-378 (2013) · Zbl 1282.35101
[15] Goldstein, J.A.; Lin, A.Y., An \(L\)\^{\(p\)} semigroup approach to degenerate parabolic boundary value problems, Ann. Mat. Pura Appl., 4, 211-227, (1991) · Zbl 0786.35083
[16] Lenhart, S.; Yong, J.M., Optimal control for degenerate parabolic equations with logistic growth, Nonlinear Anal., 25, 681-698, (1995) · Zbl 0851.49002
[17] Lions J.L.: Quelques méthodes de resolution des problèmes aux limites non linéaires. Paris, Dunod (1969) · Zbl 1103.35052
[18] Marinoschi, G., Optimal control of metabolite transport across cell membranes driven by the membrane potential, Nonlinear Anal. Real World Appl., 10, 1276-1298, (2009) · Zbl 1160.93024
[19] Mininni R.M., Romanelli S.: Martingale estimating functions for Feller diffusion processes generated by degenerate elliptic operators. J. Concr. Appl. Math. 1, 191-216 (2003) · Zbl 1082.47036
[20] Stahel, A., Degenerate semilinear parabolic equations, Differential Integral Equations, 5, 683-691, (1992) · Zbl 0766.35022
[21] Shimakura N.: Partial Differential Operators of elliptic type. Translations of Mathematical Monographs 99, (Providence: American Mathematical Society) (1992) · Zbl 0851.49002
[22] Tort, J.; Vancostenoble, J., Determination of the insolation function in the nonlinear sellers climate model, Ann. I. H. Poincaré-AN, 29, 683-713, (2012) · Zbl 1270.35283
[23] Vespri V.: Analytic semigroups, degenerate elliptic operators and applications to nonlinear Cauchy problems. Ann. Mat. Pura Appl. (IV) CLV, 353-388 (1989) · Zbl 0709.35065
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.