Abstract
We study a time-reversed hyperbolic heat conduction problem based upon the Maxwell–Cattaneo model of non-Fourier heat law. This heat and mass diffusion problem is a hyperbolic type equation for thermodynamics systems with thermal memory or with finite time-delayed heat flux, where the Fourier or Fick law is proven to be unsuccessful with experimental data. In this work, we show that our recent variational quasi-reversibility method for the classical time-reversed heat conduction problem, which obeys the Fourier or Fick law, can be adapted to cope with this hyperbolic scenario. We establish a generic regularization scheme in the sense that we perturb both spatial operators involved in the PDE. Driven by a Carleman weight function, we exploit the natural energy method to prove the well-posedness of this regularized scheme. Moreover, we prove the Hölder rate of convergence in the mixed
Dedicated to Professor Michael Victor Klibanov on his 70th birth anniversary
Funding statement: This work was supported by US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044. The work of Vo Anh Khoa was also partly supported by the Research Foundation-Flanders (FWO) in Belgium under the project named “Approximations for forward and inverse reaction-diffusion problems related to cancer models”. The work of Manh-Khang Dao was supported by the Swedish Research Council grant (2016-04086).
Acknowledgements
Vo Anh Khoa would like to thank Professor Michael Victor Klibanov (Charlotte, USA) for his wholehearted guidance during the fellowship at UNCC and for giving a chance to delve into the Carleman estimates and convexification.
References
[1] M. Asch, M. Bocquet and M. Nodet, Data Assimilation, Fundam. Algorithms 11, SIAM, Philadelphia, 2016. 10.1137/1.9781611974546Search in Google Scholar
[2] S. A. Avdonin, S. A. Ivanov and J. M. Wang, Inverse problems for the heat equation with memory, Inverse Probl. Imaging 13 (2019), 31–38. 10.3934/ipi.2019002Search in Google Scholar
[3] E. Bonetti, E. Rocca, R. Scala and G. Schimperna, On the strongly damped wave equation with constraint, Comm. Partial Differential Equations 42 (2017), no. 7, 1042–1064. 10.1080/03605302.2017.1345937Search in Google Scholar
[4] A. S. Carasso, J. G. Sanderson and J. M. Hyman, Digital removal of random media image degradations by solving the diffusion equation backwards in time, SIAM J. Numer. Anal. 15 (1978), no. 2, 344–367. 10.1137/0715023Search in Google Scholar
[5] C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett. 94 (2005), Article ID 154301. 10.1103/PhysRevLett.94.154301Search in Google Scholar PubMed
[6] V. N. Doan, H. T. Nguyen, V. A. Khoa and V. A. Vo, A note on the derivation of filter regularization operators for nonlinear evolution equations, Appl. Anal. 97 (2018), no. 1, 3–12. 10.1080/00036811.2016.1276176Search in Google Scholar
[7] R. E. Ewing, The approximation of certain parabolic equations backward in time by Sobolev equations, SIAM J. Math. Anal. 6 (1975), 283–294. 10.1137/0506029Search in Google Scholar
[8] D. N. Hào and N. V. Duc, Stability results for the heat equation backward in time, J. Math. Anal. Appl. 353 (2009), no. 2, 627–641. 10.1016/j.jmaa.2008.12.018Search in Google Scholar
[9] D. N. Hào, N. V. Duc and N. V. Thang, Backward semi-linear parabolic equations with time-dependent coefficients and local Lipschitz source, Inverse Problems 34 (2018), no. 5, Article ID 055010. 10.1088/1361-6420/aab8cbSearch in Google Scholar
[10] R. Jaroudi, G. Baravdish, F. Å ström and B. T. Johansson, Source localization of reaction-diffusion models for brain tumors, Pattern Recognition, Lecture Notes in Comput. Sci. 9796, Springer, Cham (2016), 414–425. 10.1007/978-3-319-45886-1_34Search in Google Scholar
[11] D. Jou, J. Casas-Vázquez and G. Lebon, Extended Irreversible Thermodynamics, Springer, New York, 2001. 10.1007/978-3-642-56565-6Search in Google Scholar
[12] S. I. Kabanikhin, Definitions and examples of inverse and ill-posed problems, J. Inverse Ill-Posed Probl. 16 (2008), no. 4, 317–357. 10.1515/JIIP.2008.019Search in Google Scholar
[13] B. Kaltenbacher and W. Rundell, Regularization of a backwards parabolic equation by fractional operators, Inverse Probl. Imaging 13 (2019), 401–430. 10.3934/ipi.2019020Search in Google Scholar
[14] M. V. Klibanov, Carleman estimates for the regularization of ill-posed Cauchy problems, Appl. Numer. Math. 94 (2015), 46–74. 10.1016/j.apnum.2015.02.003Search in Google Scholar
[15] M. V. Klibanov and A. G. Yagola, Convergent numerical methods for parabolic equations with reversed time via a new Carleman estimate, Inverse Problems 35 (2019), no. 11, Article ID 115012. 10.1088/1361-6420/ab2777Search in Google Scholar
[16] R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications, Trav. Rech. Math. 15, Dunod, Paris, 1967. Search in Google Scholar
[17] N. T. Long and A. P. N. Dinh, Approximation of a parabolic non-linear evolution equation backwards in time, Inverse Problems 10 (1994), no. 4, 905–914. 10.1088/0266-5611/10/4/010Search in Google Scholar
[18] N. T. Long and A. Pham Ngoc Dinh, Note on a regularization of a parabolic nonlinear evolution equation backwards in time, Inverse Problems 12 (1996), no. 4, 455–462. 10.1088/0266-5611/12/4/008Search in Google Scholar
[19] T. N. Luan and T. Q. Khanh, On the backward problem for parabolic equations with memory, Appl. Anal. (2019), 10.1080/00036811.2019.1643013. 10.1080/00036811.2019.1643013Search in Google Scholar
[20] V. Méndez and J. Camacho, Dynamics and thermodynamics of delayed population growth, Phys. Rev. E 55 (1997), 6476–6482. 10.1103/PhysRevE.55.6476Search in Google Scholar
[21] H. T. Nguyen, V. A. Khoa and V. A. Vo, Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements, SIAM J. Math. Anal. 51 (2019), no. 1, 60–85. 10.1137/18M1174064Search in Google Scholar
[22] S. B. M. Sambatti, H. F. de Campos Velho and L. D. Chiwiacowsky, Epidemic genetic algorithm for solving inverse problems: Parallel algorithms, Integral Methods in Science and Engineering, Birkhäuser/Springer, Cham (2019), 381–394. 10.1007/978-3-030-16077-7_30Search in Google Scholar
[23] R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal. 1 (1970), 1–26. 10.1137/0501001Search in Google Scholar
[24] N. H. Tuan, V. V. Au, V. A. Khoa and D. Lesnic, Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction, Inverse Problems 33 (2017), no. 5, Article ID 055019. 10.1088/1361-6420/aa635fSearch in Google Scholar
[25] N. H. Tuan, V. A. Khoa, M. T. N. Truong, T. T. Hung and M. N. Minh, Application of the cut-off projection to solve a backward heat conduction problem in a two-slab composite system, Inverse Probl. Sci. Eng. 27 (2019), no. 4, 460–483. 10.1080/17415977.2018.1470623Search in Google Scholar
[26] N. H. Tuan, D. V. Nguyen, V. V. Au and D. Lesnic, Recovering the initial distribution for strongly damped wave equation, Appl. Math. Lett. 73 (2017), 69–77. 10.1016/j.aml.2017.04.014Search in Google Scholar
[27] J. J. Vadasz, S. Govender and P. Vadasz, Heat transfer enhancement in nano-fluids suspensions: Possible mechanisms and explanations, Int. J. Heat Mass Transf. 48 (2005), 2673–2683. 10.1016/j.ijheatmasstransfer.2005.01.023Search in Google Scholar
[28] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems 25 (2009), no. 12, Article ID 123013. 10.1088/0266-5611/25/12/123013Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
