Uniqueness of a solution of the inverse problem for the evolution equation and application to the transport equation. (English. Russian original) Zbl 0792.34011
Math. Notes 51, No. 2, 158-165 (1992); translation from Mat. Zametki 51, No. 2, 77-87 (1992).
The paper deals with the linear evolution equation (1) \(u'(t)=Au+\Phi (t) f\) \((0 \leq t \leq T)\) in the Banach space \(E=L_ p (\Omega)\), where \(\Phi\) is known and \(f\) is unknown. The inverse problem of determining \(f\) and \(u\) from (1) and conditions (2) \(u(0)=0\) and (3) \(u(T)=0\) is analyzed with respect to uniqueness. A uniqueness theorem is formulated, where \(A\) generates a \(C_ 0\)-semigroup of operators. There is also given a modification to the case when \(A\) is dissipative. Furthermore, the results are applied to the transport equation. The paper is completed by a comparison of presented results with previous works.
Reviewer: B.Hofmann (Chemnitz)
MSC:
| 34A55 | Inverse problems involving ordinary differential equations |
References:
| [1] | S. G. Krein, Linear Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1967). · Zbl 0193.09302 |
| [2] | A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations. Applied Math. Sciences,44, Springer-Verlag, New York (1983). · Zbl 0516.47023 |
| [3] | V. S. Vladimirov, ”Mathematical problem of single-velocity theory of transport of particles,” Trudy Mat. Inst., Akad. Nauk SSSR, Nauka, Moscow,61, 3–158 (1961). |
| [4] | K. Jorgens, ”An asymptotic expansion in the theory of neutron transport,” Commun. Pure Appl. Math.,11, No. 2, 219–242 (1958). · Zbl 0081.44105 · doi:10.1002/cpa.3160110206 |
| [5] | I. Vidav, ”Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator,” J. Math. Anal. Appl.,22, No. 1, 144–155 (1968). · Zbl 0155.19203 · doi:10.1016/0022-247X(68)90166-2 |
| [6] | A. I. Prilepko, ”On the inverse problems in potential theory,” Differents. Uravn.,3, No. 1, 30–44 (1967). · Zbl 0168.09502 |
| [7] | A. I. Prilepko and V. V. Solov’ev, ”Solvability theorems and Rote’s method in inverse problems for equations of the parabolic type. II,” Differents. Uravn.,23, No. 11, 1971–1980 (1987). |
| [8] | D. G. Orlovskii, ”The inverse Cauchy problem for evolution equations in a Banach space,” in: Analiz. Matemat. Model. Fizich. Protsessov, Énergoatomizdat, Moscow (1988), pp. 1614–1621. |
| [9] | D. G. Orlovskii, ”On a problem of determining a parameter of an evolution equation,” Differents. Uravn.,26, No. 9, 1614–1621 (1990). · Zbl 0712.34016 |
| [10] | A. D. Iskenderov and R. G. Tagiev, ”An inverse problem of determination of the r.h.s. of evolution equations in a Banach space,” in: Voprosy Prikl. Mat. Kibern., Transact, of the Azerbaidzhan Univ., No. 1, 51–56 (1979). · Zbl 0438.35062 |
| [11] | Yu. S. Éidel’man, ”The uniqueness of a solution of the inverse problem for a differential equation in a Banach space,” Differents. Uravn.,23, No. 9, 1647–1649 (1987). · Zbl 0646.34020 |
| [12] | A. I. Prilepko and A. L. Ivankov, ”Inverse problems of determining a coefficient, dispersion index and the right-hand side of a nonstationary multivelocity transport equation,” Differents. Uravn.,21, No. 5, 870–885 (1985). · Zbl 0572.45013 |
| [13] | A. I. Prilepko and N. P. Volkov, ”Inverse problems of determining parameters of a nonstationary kinetic transport equation from the additional information on traces of the unknown function,” Differents. Uravn.,24, No. 1, 136–146 (1988). · Zbl 0664.35077 |
| [14] | W. Rundell, ”Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data,” Appl. Anal.,10, No. 3, 231–242 (1980). · Zbl 0454.35045 · doi:10.1080/00036818008839304 |
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