Abstract
In this paper, we are interested in the nonlinear transport equation in the class of essentially bounded functions. More precisely, we prove the existence and uniqueness of the inverse problem solution. Moreover, we give the size of the neighborhood from which we can bring the function of the overdetermination condition so that the solution of the inverse problem is unique. The method is based on known properties of solutions of the (direct and inverse) linear problems and uses twice the inverse function theorem in the corresponding function spaces. Then we use the refined inverse function theorem to obtain the sufficient conditions for the unique solvability of the initial nonlinear inverse problem in terms of the size of the neighborhood from which we can bring the function of the overdetermination condition.
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References
N. Volkov, Sufficient solvability conditions of inverse problems (control problems) for mass-transport processes, Inverse problems for math models of physical processes (Moscow State Institute of Engineering Physics, 1991), p. 16.
A. Prilepko and N. Volkov, Inverse problems of determining parameters of the nonstationary kinetic transport equation from the additional information concerning the traces of a sought function, Differ. Equations 24(1), 136 (1988).
A. Prilepko, D. Orlovsky, and I. Vasin, Methods for solving inverse problems in mathematical physics (New York: Marcel Dekker, 2000).
M. Sukhinin, Selected chapters of nonlinear analysis (Moscow: RPFU Press, 1992).
A. Prilepko and N. Volkov, Inverse problems of finding parameters of a nonstationary transport equation from integral overdeterminations, Differ. Equations 23(1), 91 (1987).
Hans G. Kaper, C. G. Lekkerkerker, and J. Hejtmanek, Spectral methods in linear transport theory (Birkhauser Basel, 1982).
G. Greiner, Spectral properties and asymptotic behavior of the linear transport equation, Mathematische Zeitschrift 185(2), 167 (1984).
D. Orlovsky, Solution of one inverse problem for transport equation with integral overdetermination, Inverse problems formathmodels of physical processes (Moscow State Institute of Engineering Physics, 1991), p. 71.
J. Voigt, Spectral properties of the neutron transport equation, J. Math. Anal. Appl. 106(1), 140 (1985).
N. Volkov, Determination of characteristics of nonstationary transport processes from final-state information, Physical processes math models analysis (Moscow: Energoatomizdat, 1988), p. 11.
N. Volkov, Optimal sources control in some neutron transport processes, Theoretical-functional methods in math physics problems (Moscow: Energoatomizdat, 1986), p. 22.
I. Tikhonov, Well-posedness of an inverse problem with final overdetermination for the nonstationary transport equation, Mosc. Univ. Comput. Math. Cybern. 1, 51 (1995).
O. Ladyzhenskaya, The boundary value problems of mathematical physics (Springer-Verlag, 1985).
A. Kirillov and A. Gvishiani, Teoremy i zadachi funktsionalnogo analiza (Moscow: Nauka, 1988).
A. Rozanova, Controllability for a nonlinear abstract evolution equation, Mathematical Notes 76(4), 511 (2004).
M. Krasnoselskii, P. Zabreiko, E. Pustylnik, and P. Sobolevskii, Integral operators in spaces of summable functions (Noordhoff International Publishing, 1976).
L. Kantorovich and G. Akilov, Functional analysis in normed spaces Ed. Robertson (A.P. Pergamon press, 1964).
M. Sukhinin, Solvability of nonlinear stationary transfer equation, Theoretical and Mathematical Physics 103(1), 366 (1995).
M. Mokhtar Kharroubi,Mathematical topics in neutron transport theory: New Aspects (World Scientific Publishing Co. Inc. River Edge. NJ. 1998).
N. Hamdi, An inverse problem for a nonlinear transport equation with an integral overdetermination, VESTNIC Bulletin Math Series 1(10), 109 (2003).
M. Klibanov and M. Yamamoto, Exact controllability for the time dependent transport equation, SIAM J. Control Optim. 46(6), 2071 (2007).
A. Rozanova, Controllability in a nonlinear parabolic problem with integral overdetermination, Differential Equations, 40(6), 853 (2004).
A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Boundary layer and homogenization of transport processes (RIMS Kyoto Univ, 1979), p. 53.
N. J. McCormick, Recent developments in inverse scattering transport methods, Transport Theory and Statist. Phys. 13, 15 (1984).
N. J. McCormick, Methods for solving inverse problems for radiation transport methods, Transport Theory and Statist. Phys. 15, 759 (1986).
V. Protopopescu and L. Thevenot, Diffusion approximation of a neutron transport equation with multiplying boundary conditions (HYKE 2003-024, 2003).
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Hamdi, N. An inverse problem for a nonlinear transport equation with final overdetermination. Lobachevskii J Math 29, 230–244 (2008). https://doi.org/10.1134/S1995080208040057
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DOI: https://doi.org/10.1134/S1995080208040057