| Title:
|
Identification problems for degenerate parabolic equations (English) |
| Author:
|
Awawdeh, Fadi |
| Author:
|
Obiedat, Hamed M. |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
58 |
| Issue:
|
4 |
| Year:
|
2013 |
| Pages:
|
389-404 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
identification problem |
| Keyword:
|
perturbation theory for linear operators |
| Keyword:
|
degenerate differential equation |
| MSC:
|
34A55 |
| MSC:
|
34G10 |
| MSC:
|
34G25 |
| MSC:
|
34G99 |
| MSC:
|
35K65 |
| MSC:
|
35R30 |
| MSC:
|
47A55 |
| idZBL:
|
Zbl 06221237 |
| idMR:
|
MR3083520 |
| DOI:
|
10.1007/s10492-013-0019-1 |
| . |
| Date available:
|
2013-07-18T15:15:37Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143337 |
| . |
| Reference:
|
[1] Horani, M. Al: Projection method for solving degenerate first-order identification problem.J. Math. Anal. Appl. 364 (2010), 204-208. Zbl 1193.34021, MR 2576064, 10.1016/j.jmaa.2009.10.033 |
| Reference:
|
[2] Horani, M. Al, Favini, A.: An identification problem for first-order degenerate differential equations.J. Optim. Theory Appl. 130 (2006), 41-60. Zbl 1129.65044, MR 2275353, 10.1007/s10957-006-9083-y |
| Reference:
|
[3] Horani, M. Al, Favini, A., Lorenzi, A.: Second-order degenerate identification differential problems.J. Optim. Theory Appl. 141 (2009), 13-36. Zbl 1165.49013, MR 2495916, 10.1007/s10957-008-9497-9 |
| Reference:
|
[4] Awawdeh, F., Obiedat, H. M.: Source identification problem for degenerate differential equations.Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar. 73 (2011), 61-72. Zbl 1249.35341, MR 2871880 |
| Reference:
|
[5] Awawdeh, F.: Perturbation method for abstract second-order inverse problems.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 1379-1386. Zbl 1186.34020, MR 2577538, 10.1016/j.na.2009.08.021 |
| Reference:
|
[6] Awawdeh, F.: Ordinary Differential Equations in Banach Spaces with Applications, PhD thesis.Jordan University (2006). Zbl 1062.22046 |
| Reference:
|
[7] Cannarsa, P., Martinez, P., Vancostenoble, J.: Carleman estimates for a class of degenerate parabolic operators.SIAM J. Control Optim. 47 (2008), 1-19. Zbl 1168.35025, MR 2373460, 10.1137/04062062X |
| Reference:
|
[8] Cannarsa, P., Tort, J., Yamamoto, M.: Determination of source terms in a degenerate parabolic equation.Inverse Probl. 26 (2010), Article ID 105003, pp. 20. Zbl 1200.35319, MR 2679467 |
| Reference:
|
[9] Desh, W., Schappacher, W.: On relatively bounded perturbations of linear $C_0$-semigroups.Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11 (1984), 327-341. MR 0764949 |
| Reference:
|
[10] DuChateau, P., Thelwell, R., Butters, G.: Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient.Inverse Probl. 20 (2004), 601-625. Zbl 1054.35125, MR 2065443 |
| Reference:
|
[11] Favini, A., Yagi, A.: Degenerate Differential Equations in Banach Spaces.Pure and Applied Mathematics, Marcel Dekker New York (1999). Zbl 0913.34001, MR 1654663 |
| Reference:
|
[12] Imanuvilov, O. Y., Yamamoto, M.: Lipschitz stability in inverse parabolic problems by the Carleman estimate.Inverse Probl. 14 (1998), 1229-1245. Zbl 0992.35110, MR 1654631 |
| Reference:
|
[13] Klibanov, M. V., Timonov, A. A.: Carleman Estimates for Coefficient Inverse Problems and Numerical Applications.Inverse and Ill-Posed Problems Series VSP, Utrecht (2004). Zbl 1069.65106, MR 2126149 |
| Reference:
|
[14] Ling, L., Yamamoto, M., Hon, Y. C., Takeuchi, T.: Identification of source locations in two-dimensional heat equations.Inverse Probl. 22 (2006), 1289-1305. Zbl 1112.35147, MR 2249466 |
| Reference:
|
[15] Lorenzi, A., Paparoni, E.: Direct and inverse problems in the theory of materials with memory.Rend. Semin. Mat. Univ. Padova 87 (1992), 105-138. Zbl 0757.73018, MR 1183905 |
| Reference:
|
[16] Lorenzi, A.: An Introduction to Identification Problems, Via Functional Analysis.VSP Utrecht (2001). |
| Reference:
|
[17] Lorenzi, A.: An inverse problem for a semilinear parabolic equation.Ann. Mat. Pura Appl., IV. Ser. 131 (1982), 145-166. Zbl 0493.35078, MR 0681561, 10.1007/BF01765150 |
| Reference:
|
[18] Lorenzi, A., Vrabie, I. I.: Identification for a semilinear evolution equation in a Banach space.Inverse Probl. 26 (2010), Article ID 085009, pp. 16. Zbl 1205.34072, MR 2658826 |
| Reference:
|
[19] Orlovskii, D. G.: An inverse problem for a second-order differential equation in a Banach space.Differ. Uravn. 25 (1989), 1000-1009 Russian; Differ. Equations 25 (1989), 730-738 English. MR 1008642 |
| Reference:
|
[20] Orlovskij, D. G.: Weak and strong solutions of inverse problems for differential equations in a Banach space.Differ. Uravn. 27 (1991), 867-874 Russian; Differ. Equations 27 (1991), 611-617 English. MR 1117116 |
| Reference:
|
[21] Pavlov, G. A.: Uniqueness of a solution of an abstract inverse problem.Differ. Uravn. 24 (1988), 1402-1406 Russian; Differ. Equations 24 917-920 English. Zbl 0672.31009, MR 0964736 |
| Reference:
|
[22] Pilant, M., Rundell, W.: An inverse problem for a nonlinear elliptic differential equation.SIAM J. Math. Anal. 18 (1987), 1801-1809. Zbl 0647.35081, MR 0911664, 10.1137/0518127 |
| Reference:
|
[23] Prilepko, A. I., Orlovskij, D. G., Vasin, I. A.: Methods for Solving Inverse Problems in Mathematical Physics, Pure and Applied Mathematics.Marcel Dekker New York (2000). MR 1748236 |
| . |
