On the backward problem for parabolic equations with memory

References

• Nunziato JW. On heat conduction in materials with memory. Quart Appl Math. 1971;29:187–204. doi: 10.1090/qam/295683
   Web of Science ®Google Scholar
• Yanik EG, Fairweather G. Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal. 1988;12(8):785–809. doi: 10.1016/0362-546X(88)90039-9
   Web of Science ®Google Scholar
• Habetler GJ, Schiffman RL. A finite difference method for analyzing the compression of poro-viscoelastic media. Computing (Arch Elektron Rechnen). 1970;6:342–348.
   Google Scholar
• Pachpatte BG. On a nonlinear diffusion system arising in reactor dynamics. J Math Anal Appl. 1983;94(2):501–508. doi: 10.1016/0022-247X(83)90078-1
   Web of Science ®Google Scholar
• Zhang NY. On fully discrete Galerkin approximations for partial integro-differential equations of parabolic type. Math. Comp. 1993;60(201):133–166. doi: 10.1090/S0025-5718-1993-1149295-4
   Web of Science ®Google Scholar
• Guerrero S, Imanuvilov OY. Remarks on non controllability of the heat equation with memory. ESAIM Control Optim Calc Var. 2013;19(1):288–300. doi: 10.1051/cocv/2012013
   Web of Science ®Google Scholar
• Tao Q, Gao H. On the null controllability of heat equation with memory. J Math Anal Appl. 2016;440(1):1–13. doi: 10.1016/j.jmaa.2016.03.036
   Web of Science ®Google Scholar
• Tao Q, Gao H, Zhang B. Approximate controllability of a parabolic equation with memory. Nonlinear Anal Hybrid Syst. 2012;6(2):839–845. doi: 10.1016/j.nahs.2011.11.001
   Web of Science ®Google Scholar
• Tao Q, Gao H, Zhang B, Yao Z. Approximate controllability of a parabolic integrodifferential equation. Math Methods Appl Sci. 2014;37(15):2236–2244. doi: 10.1002/mma.2970
   Web of Science ®Google Scholar
• Zhou X, Gao H. Interior approximate and null controllability of the heat equation with memory. Comput Math Appl. 2014;67(3):602–613. doi: 10.1016/j.camwa.2013.12.005
   Web of Science ®Google Scholar
• Larsson S, Thomée V, Wahlbin LB. Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method. Math Comp. 1998;67(221):45–71. doi: 10.1090/S0025-5718-98-00883-7
   Web of Science ®Google Scholar
• Li X, Xu Q, Zhu A. Weak galerkin mixed finite element methods for parabolic equations with memory. Discrete Cont Dyn Syst Ser S. 2019;12(3):513–531.
   Web of Science ®Google Scholar
• Loreti P, Sforza D, Yamamoto M. Carleman estimates for integro-differential parabolic equations with singular memory kernels. J Elliptic Parabol Equ. 2017;3(1–2):53–64. doi: 10.1007/s41808-017-0004-z
   Web of Science ®Google Scholar
• Li L, Zhou X, Gao H. The stability and exponential stabilization of the heat equation with memory. J Math Anal Appl. 2018;466(1):199–214. doi: 10.1016/j.jmaa.2018.05.078
   Web of Science ®Google Scholar
• Sakthivel K, Balachandran K, Nagaraj BR. On a class of non-linear parabolic control systems with memory effects. Internat J Control. 2008;81(5):764–777. doi: 10.1080/00207170701447114
   Web of Science ®Google Scholar
• Cheng J, Liu JJ. A quasi Tikhonov regularization for a two-dimensional backward heat problem by a fundamental solution. Inverse Probl. 2008;24(6):065012, 18. doi: 10.1088/0266-5611/24/6/065012
   Web of Science ®Google Scholar
• Jana A, Thamban Nair M. Quasi-reversibility method for an ill-posed nonhomogeneous parabolic problem. Numer Funct Anal Optim. 2016;37(12):1529–1550. doi: 10.1080/01630563.2016.1216448
   Web of Science ®Google Scholar
• Latt R, Lions JL. Methode de quasi-reversibility et applications. Paris: Dunod; 1967.
   Google Scholar
• Miller K. Stabilized quasi-reversibility and other nearly-best-possible methods for non-well-posed problems. Vol. 316, Lecture Notes in Math.; 1973. p. 161–176.
   Google Scholar
• Showalter RE. Quasi-reversibility of first and second order parabolic evolution equations. Res. Notes in Math., No. 1; 1975. p. 76–84.
   Google Scholar
• Hào DN, Van Duc N. A non-local boundary value problem method for semi-linear parabolic equations backward in time. Appl Anal. 2015;94(3):446–463. doi: 10.1080/00036811.2014.970537
   Web of Science ®Google Scholar
• Hào DN, Van Duc N, Lesnic D. Regularization of parabolic equations backward in time by a non-local boundary value problem method. IMA J Appl Math. 2010;75(2):291–315. doi: 10.1093/imamat/hxp026
   Web of Science ®Google Scholar
• Hào DN, Van Duc N, Sahli H. A non-local boundary value problem method for parabolic equations backward in time. J Math Anal Appl. 2008;345(2):805–815. doi: 10.1016/j.jmaa.2008.04.064
   Web of Science ®Google Scholar
• Khanh TQ, Van Hoa N. On the axisymmetric backward heat equation with non-zero right hand side: regularization and error estimates. J Comput Appl Math. 2018;335:156–167. doi: 10.1016/j.cam.2017.11.036
   Web of Science ®Google Scholar
• Showalter RE. The final value problem for evolution equations. J Math Anal Appl. 1974;47:563–572. doi: 10.1016/0022-247X(74)90008-0
   Web of Science ®Google Scholar
• Trong DD, Duy BT, Minh MN. Backward heat equations with locally Lipschitz source. Appl Anal. 2015;94(10):2023–2036. doi: 10.1080/00036811.2014.963063
   Web of Science ®Google Scholar
• Jana A, Thamban Nair M. Truncated spectral regularization for an ill-posed nonhomogeneous parabolic problem. J Math Anal Appl. 2016;438(1):351–372. doi: 10.1016/j.jmaa.2016.01.069
   Web of Science ®Google Scholar
• Nam PT. An approximate solution for nonlinear backward parabolic equations. J Math Anal Appl. 2010;367(2):337–349. doi: 10.1016/j.jmaa.2010.01.020
   Web of Science ®Google Scholar
• Zhang Y-X, Fu C-L, Ma Y-J. An a posteriori parameter choice rule for the truncation regularization method for solving backward parabolic problems. J Comput Appl Math. 2014;255:150–160. doi: 10.1016/j.cam.2013.04.046
   Web of Science ®Google Scholar
• Xiong X-T, Fu C-L, Qian Z. Two numerical methods for solving a backward heat conduction problem. Appl Math Comput. 2006;179(1):370–377.
   Web of Science ®Google Scholar
• Khanh TQ, Khieu TT, Van Hoa N. Stabilizing the nonlinear spherically symmetric backward heat equation via two-parameter regularization method. J Fixed Point Theory Appl. 2017;19(4):2461–2481. doi: 10.1007/s11784-017-0429-x
   Web of Science ®Google Scholar
• Qin H-H, Wei T. Some filter regularization methods for a backward heat conduction problem. Appl Math Comput. 2011;217(24):10317–10327.
   Web of Science ®Google Scholar
• Evans LC. Partial differential equations. Providence (RI): American Mathematical Society; 1998.
   Google Scholar
• Jin B, Rundell W. A tutorial on inverse problems for anomalous diffusion processes. Inverse Probl. 2015;31(3):035003, 40. doi: 10.1088/0266-5611/31/3/035003
   Web of Science ®Google Scholar
• Engl HW, Hanke M, Neubauer A. Regularization of inverse problems. Berlin: Springer Science & Business Media; 1996.
   Google Scholar