AIMS Mathematics, 2020, 5(5): 4529-4545. doi: 10.3934/math.2020291 Research article

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Lipschitz stability of an inverse problem for the Kawahara equation with damping

Arivazhagan Anbu, Sakthivel Kumarasamy, Barani Balan Natesan

1 Department of Mathematics, Central University of Tamil Nadu, Thiruvarur 610 005, INDIA
2 Department of Mathematics, Indian Institute of Space Science and Technology (IIST), Trivandrum-695 547, INDIA

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The aim of this paper is to establish a stability result regarding the inverse problem of retrieving the damping coefficient in Kawahara equation. We first establish an internal Carleman estimate for the linearized problem with the help of Dirichlet-Neumann type boundary conditions. Using the obtained Carleman estimate and the regularity of solutions for the Kawahara equation, we prove the Lipschitz type stability and uniqueness of the considered inverse problems.

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1. F. D. Araruna, R. A. Capistrano-Filho, G. Doronin, Energy decay for the modified Kawahara equation posed in a bounded domain, J. Math. Anal. Appl., 385 (2012), 743-756.

2. L. Baudouin, E. Cerpa, E. Crepeau, et al. On the determination of the principal coefficient from boundary measurements in a KdV equation, J. Inverse Ill-Posed Probl., 22 (2013), 819-845.

3. L. Baudouin, E. Cerpa, E. Crepeau, et al. Lipschitz stability in an inverse problem for the KuramotoSivashinsky equation, Appl. Anal., 92 (2013), 2084-2102.

4. A. L. Bukhgeim, M. V. Klibanov, Uniqueness in the large class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247.

5. A. L. Bukhgeim, Carleman estimates for Volterra operators and uniqueness of inverse problems, Siberian Math. J., 25 (1984), 43-50.

6. R. A. Capistrano-Filho, A. F. Pazoto, L. Rosier, Internal controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM: COCV, 21 (2015), 1076-1107.

7. M. Chen, Internal controllability of the Kawahara equation on a bounded domain, Nonlinear Anal., 185 (2019), 356-373.

8. O. Glass, S. Guerrero, On the controllability of the fifth-order Korteweg-de Vries equation, Ann. I. H. Poincaré - AN., 26 (2009), 2181-2209.

9. P. Gao, A new global Carleman estimate for the one-dimensional Kuramoto-Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem, Nonlinear Anal., 117 (2015), 133-147.

10. P. Gao, Global Carleman Estimate for the Kawahara equation and its applications, Commun. Pure Appl. Anal., 17 (2018), 1853-1874.

11. O. Yu. Imanuvilov, M. Yamamoto, Lipschitz stability in inverse problems by the Carleman estimates, Inverse Probl., 14 (1998), 1229-1245.

12. T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), 260-264.

13. P. G. Meléndez, Lipschitz stability in an inverse problem for the mian coefficient of a KuramotoSivashinsky type equation, J. Math. Anal. Appl., 408 (2013), 275-290.

14. M. Renardy, B. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, New York Inc, 2004.

15. C. F. Vasconcellos, P. N. Da Silva, Stabilization of the Kawahara Equation with localized damping, ESAIM: COCV, 17 (2011), 102-116.

16. M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Probl., 25 (2009), 123013.

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