The problem of determining source term in a kinetic equation in an unbounded domain

Özlem Kaytmaz; Özlem Kaytmaz

doi:10.3934/math.2024447

AIMS Mathematics

2024, Volume 9, Issue 4: 9184-9194. doi: 10.3934/math.2024447

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The problem of determining source term in a kinetic equation in an unbounded domain

Özlem Kaytmaz ^,

Department of Mathematics, Faculty of Science, Zonguldak Bülent Ecevit University, Zonguldak 67100, Türkiye

Received: 12 December 2023 Revised: 26 January 2024 Accepted: 22 February 2024 Published: 06 March 2024
MSC : 35A23, 35R30

In this paper, we deal with an inverse problem of determining the source function in a kinetic equation that is considered in an unbounded domain with Cauchy data. We prove the uniqueness of the solution of an inverse problem by means of a pointwise Carleman estimate. In recent years, kinetic equations have occurred in a variety of important fields and applications, such as aerospace engineering, semi-conductor technology, nuclear engineering, chemotaxis, and immunology.
- kinetic equation,
- inverse problem,
- unbounded domain,
- uniqueness,
- pointwise Carleman estimate
Citation: Özlem Kaytmaz. The problem of determining source term in a kinetic equation in an unbounded domain[J]. AIMS Mathematics, 2024, 9(4): 9184-9194. doi: 10.3934/math.2024447

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Abstract

In this paper, we deal with an inverse problem of determining the source function in a kinetic equation that is considered in an unbounded domain with Cauchy data. We prove the uniqueness of the solution of an inverse problem by means of a pointwise Carleman estimate. In recent years, kinetic equations have occurred in a variety of important fields and applications, such as aerospace engineering, semi-conductor technology, nuclear engineering, chemotaxis, and immunology.

References

[1]	F. Salvarani, Recent advances in kinetic equations and applications, Cham: Springer, 2021. https://doi.org/10.1007/978-3-030-82946-9
[2]	G. Dimarco, L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369–520. https://doi.org/10.1017/S0962492914000063 doi: 10.1017/S0962492914000063
[3]	J. P. Françoise, G. L. Naber, S. T. Tsou, Encyclopedia of mathematical physics, Amsterdam: Elsevier, 2006.
[4]	P. Degond, L. Pareschi, G. Russo, Modeling and computational methods for kinetic equations, Boston: Birkhäuser, 2004. https://doi.org/10.1007/978-0-8176-8200-2
[5]	B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc., 41 (2004), 205–244.
[6]	A. M. Whitman, Thermodynamics: basic principles and engineering applications, Cham: Springer, 2023. https://doi.org/10.1007/978-3-031-19538-9
[7]	V. F. Onyshchenko, L. A. Karachevtseva, K. V. Andrieieva, N. V. Dmytruk, A. Z. Evmenova, Kinetics of charge carriers in bilateral macroporous silicon, Semicond. Phys. Quantum Electron. Optoelectron., 26 (2023), 159–164. https://doi.org/10.15407/spqeo26.02.159 doi: 10.15407/spqeo26.02.159
[8]	L. L. Salas, F. C. Silva, A. S. Martinez, A new point kinetics model for ADS-type reactor using the importance function associated to the fission rate as weight function, Ann. Nucl. Energy, 190 (2023), 109869. https://doi.org/10.1016/j.anucene.2023.109869 doi: 10.1016/j.anucene.2023.109869
[9]	W. H. Shan, P. Zheng, Global boundedness of the immune chemotaxis system with general kinetic functions, Nonlinear Differ. Equ. Appl., 30 (2023), 29. https://doi.org/10.1007/s00030-023-00840-4 doi: 10.1007/s00030-023-00840-4
[10]	A. K. Amirov, Integral geometry and inverse problems for kinetic equations, Berlin, Boston: De Gruyter, 2001. https://doi.org/10.1515/9783110940947
[11]	Yu. E. Anikonov, Inverse problems for kinetic and other evolution equations, Berlin, Boston: De Gruyter, 2001. https://doi.org/10.1515/9783110940909
[12]	M. V. Klibanov, S. E. Pamyatnykh, Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate, J. Math. Anal. Appl., 343 (2008), 352–365. https://doi.org/10.1016/j.jmaa.2008.01.071 doi: 10.1016/j.jmaa.2008.01.071
[13]	F. Golgeleyen, A. Amirov, On the approximate solution of a coefficient inverse problem for the kinetic equation, Math. Commun., 16 (2011), 283–298.
[14]	A. Amirov, Z. Ustaoglu, B. Heydarov, Solvability of a two dimensional coefficient inverse problem for transport equation and a numerical method, Transport Theory Statist. Phys., 40 (2011), 1–22. https://doi.org/10.1080/00411450.2010.529980 doi: 10.1080/00411450.2010.529980
[15]	T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astr. Fys., 26 (1939), 9.
[16]	C. E. Kenig, Carleman estimates, uniform Sobolev inequalities for second-order differential operators, and unique continuation theorems, In: Proceedings of the International Congress of Mathematicians, 1 (1986), 948–960.
[17]	A. P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 16–36. https://doi.org/10.2307/2372819 doi: 10.2307/2372819
[18]	L. Hörmander, Linear partial differential operators, Berlin, Heidelberg: Springer, 1963. https://doi.org/10.1007/978-3-642-46175-0
[19]	A. L. Bukhgeim, M. V. Klibanov, Global uniqueness of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269–272.
[20]	J. P. Puel, M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem, Inverse Probl., 12 (1996), 995. https://doi.org/10.1088/0266-5611/12/6/013 doi: 10.1088/0266-5611/12/6/013
[21]	V. Isakov, M. Yamamoto, Carleman estimate with the Neumann boundary condition and its applications to the observability inequality and inverse hyperbolic problems, Contemp. Math., 268 (2000), 191–225.
[22]	O. Y. Imanuvilov, M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Probl., 17 (2001), 717. https://doi.org/10.1088/0266-5611/17/4/310 doi: 10.1088/0266-5611/17/4/310
[23]	O. Y. Imanuvilov, M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations, Commun. Partial Differ. Equ., 26 (2001), 1409–1425. https://doi.org/10.1081/PDE-100106139 doi: 10.1081/PDE-100106139
[24]	M. Bellassoued, M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation, J. Math. Pures Appl., 85 (2006), 193–224. https://doi.org/10.1016/j.matpur.2005.02.004 doi: 10.1016/j.matpur.2005.02.004
[25]	M. V. Klibanov, M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation, Appl. Anal., 85 (2006), 515–538. https://doi.org/10.1080/00036810500474788 doi: 10.1080/00036810500474788
[26]	M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Probl., 25 (2009), 123013. https://doi.org/10.1088/0266-5611/25/12/123013 doi: 10.1088/0266-5611/25/12/123013
[27]	M. M. Lavrentiev, V. G. Romanov, S. P. Shishatskii, Ill-posed problems of mathematical physics and analysis, American Mathematical Society, 1986.
[28]	V. G. Romanov, Estimate for the solution to the Cauchy problem for an ultrahyperbolic inequality, Dokl. Math., 74 (2006), 751–754. https://doi.org/10.1134/S1064562406050346 doi: 10.1134/S1064562406050346
[29]	F. Gölgeleyen, M. Yamamoto, Stability of inverse problems for ultrahyperbolic equations, Chinese Ann. Math. Ser. B, 35 (2014), 527–556. https://doi.org/10.1007/s11401-014-0848-6 doi: 10.1007/s11401-014-0848-6
[30]	İ. Gö lgeleyen, Ö. Kaytmaz, Conditional stability for a Cauchy problem for the ultrahyperbolic Schrödinger equation, Appl. Anal., 101 (2022), 1505–1516. https://doi.org/10.1080/00036811.2020.1781829 doi: 10.1080/00036811.2020.1781829
[31]	İ. Gölgeleyen, Ö. Kaytmaz, Uniqueness for a Cauchy problem for the generalized Schrödinger equation, AIMS Math., 8 (2023), 5703–5724. https://doi.org/10.3934/math.2023287 doi: 10.3934/math.2023287
[32]	F. Gö lgeleyen, Ö. Kaytmaz, A Hölder stability estimate for inverse problems for the ultrahyperbolic Schrödinger equation, Anal. Math. Phys., 9 (2019), 2171–2199. https://doi.org/10.1007/s13324-019-00326-6 doi: 10.1007/s13324-019-00326-6
[33]	P. Cannarsa, G. Floridia, F. Gölgeleyen, M. Yamamoto, Inverse coefficient problems for a transport equation by local Carleman estimate, Inverse Probl., 35 (2019), 105013. https://doi.org/10.1088/1361-6420/ab1c69 doi: 10.1088/1361-6420/ab1c69
[34]	F. Gölgeleyen, M. Yamamoto, Stability for some inverse problems for transport equations, SIAM J. Math. Anal., 48 (2016), 2319–2344. https://doi.org/10.1137/15M1038128 doi: 10.1137/15M1038128

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