Identification problems for degenerate parabolic equations
Tóm tắt
This paper deals with multivalued identification problems for parabolic equations. The problem consists of recovering a source term from the knowledge of an additional observation of the solution by exploiting some accessible measurements. Semigroup approach and perturbation theory for linear operators are used to treat the solvability in the strong sense of the problem. As an important application we derive the corresponding existence, uniqueness, and continuous dependence results for different degenerate identification problems. Applications to identification problems for the Stokes system, Poisson-heat equation, and Maxwell system are given to illustrate the theory.
Tài liệu tham khảo
M. Al Horani: Projection method for solving degenerate first-order identification problem. J. Math. Anal. Appl. 364 (2010), 204–208.
M. Al Horani, A. Favini: An identification problem for first-order degenerate differential equations. J. Optim. Theory Appl. 130 (2006), 41–60.
M. Al Horani, A. Favini, A. Lorenzi: Second-order degenerate identification differential problems. J. Optim. Theory Appl. 141 (2009), 13–36.
F. Awawdeh, H. M. Obiedat: Source identification problem for degenerate differential equations. Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar. 73 (2011), 61–72.
F. Awawdeh: Perturbation method for abstract second-order inverse problems. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 1379–1386.
F. Awawdeh: Ordinary Differential Equations in Banach Spaces with Applications, PhD thesis. Jordan University, 2006.
P. Cannarsa, P. Martinez, J. Vancostenoble: Carleman estimates for a class of degenerate parabolic operators. SIAM J. Control Optim. 47 (2008), 1–19.
P. Cannarsa, J. Tort, M. Yamamoto: Determination of source terms in a degenerate parabolic equation. Inverse Probl. 26 (2010), Article ID 105003, pp. 20.
W. Desh, W. Schappacher: On relatively bounded perturbations of linear C 0-semigroups. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11 (1984), 327–341.
P. DuChateau, R. Thelwell, G. Butters: Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient. Inverse Probl. 20 (2004), 601–625.
A. Favini, A. Yagi: Degenerate Differential Equations in Banach Spaces. Pure and Applied Mathematics, Marcel Dekker, New York, 1999.
O. Y. Imanuvilov, M. Yamamoto: Lipschitz stability in inverse parabolic problems by the Carleman estimate. Inverse Probl. 14 (1998), 1229–1245.
M. V. Klibanov, A. A. Timonov: Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. Inverse and Ill-Posed Problems Series, VSP, Utrecht, 2004.
L. Ling, M. Yamamoto, Y. C. Hon, T. Takeuchi: Identification of source locations in two-dimensional heat equations. Inverse Probl. 22 (2006), 1289–1305.
A. Lorenzi, E. Paparoni: Direct and inverse problems in the theory of materials with memory. Rend. Semin. Mat. Univ. Padova 87 (1992), 105–138.
A. Lorenzi: An Introduction to Identification Problems, Via Functional Analysis. VSP, Utrecht, 2001.
A. Lorenzi: An inverse problem for a semilinear parabolic equation. Ann. Mat. Pura Appl., IV. Ser. 131 (1982), 145–166.
A. Lorenzi, I. I. Vrabie: Identification for a semilinear evolution equation in a Banach space. Inverse Probl. 26 (2010), Article ID 085009, pp. 16.
D.G. Orlovskii: An inverse problem for a second-order differential equation in a Banach space. Differ. Uravn. 25 (1989), 1000–1009 (In Russian.); Differ. Equations 25 (1989), 730–738. (In English.)
D.G. Orlovskij: Weak and strong solutions of inverse problems for differential equations in a Banach space. Differ. Uravn. 27 (1991), 867–874 (In Russian.); Differ. Equations 27 (1991), 611–617. (In English.)
G. A. Pavlov: Uniqueness of a solution of an abstract inverse problem. Differ. Uravn. 24 (1988), 1402–1406 (In Russian.); Differ. Equations 24, 917–920. (In English.)
M. Pilant, W. Rundell: An inverse problem for a nonlinear elliptic differential equation. SIAM J. Math. Anal. 18 (1987), 1801–1809.
A. I. Prilepko, D.G. Orlovskij, I. A. Vasin: Methods for Solving Inverse Problems in Mathematical Physics, Pure and Applied Mathematics. Marcel Dekker, New York, 2000.