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Stability results for backward heat equations with time-dependent coefficient in the Banach space \(L_p (\mathbb{R})\). (English) Zbl 1485.35082

Summary: In this paper, we investigate the problem of backward heat equations with time-dependent coefficient in the Banach space \(L_p (\mathbb{R}),\; (1 < p < \infty)\). For this problem, we first prove the stability estimates of Hölder type. After that the Tikhonov-type regularization is applied to solve the problem. A priori and a posteriori parameter choice rules are investigated, which yield error estimates of Hölder type. Numerical implementations are presented to show the validity of the proposed scheme.

MSC:

35B45 A priori estimates in context of PDEs
35K10 Second-order parabolic equations
35R25 Ill-posed problems for PDEs
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[1] Ames, Karen A.; Hughes, Rhonda J., Structural Stability for Ill-Posed Problems in Banach Space, Semigroup Forum, vol. 70, 127-145 (2005), Springer · Zbl 1109.34041
[2] Chen, De-Han; Hofmann, Bernd; Zou, Jun, Regularization and convergence for ill-posed backward evolution equations in Banach spaces, J. Differ. Equ., 265, 8, 3533-3566 (2018) · Zbl 06901111
[3] Duc, Nguyen Van; Thang, Nguyen Van; Minh, Luong Duy Nhat; Thành, Nguyen Trung, Identifying an unknown source term of a parabolic equation in Banach spaces, Appl. Anal. (2020) · Zbl 1475.65112
[4] Fletcher, Roger, Practical Methods of Optimization (2013), John Wiley & Sons · Zbl 0988.65043
[5] Fury, Matthew A., Nonautonomous ill-posed evolution problems with strongly elliptic differential operators, Electron. J. Differ. Equ., 2013, 92, Article 1 pp. (2013) · Zbl 1293.34072
[6] Fury, Matthew A., Logarithmic well-posed approximation of the backward heat equation in Banach space, J. Math. Anal. Appl., 475, 2, 1367-1384 (2019) · Zbl 1423.47004
[7] Gilding, Brian H.; Tesei, Alberto, The Riemann problem for a forward-backward parabolic equation, Phys. D, Nonlinear Phenom., 239, 6, 291-311 (2010) · Zbl 1190.35122
[8] Grunau, Hans-Christoph; Miyake, Nobuhito; Okabe, Shinya, Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations, Adv. Nonlinear Anal., 10, 1, 353-370 (2021) · Zbl 1447.35179
[9] Hào, Dinh Nho, A mollification method for ill-posed problems, Numer. Math., 68, 469-506 (1994) · Zbl 0817.65041
[10] Hào, Dinh Nho; Van Duc, Nguyen, Stability results for the heat equation backward in time, J. Math. Anal. Appl., 353, 2, 627-641 (2009) · Zbl 1170.35097
[11] Hào, Dinh Nho; Van Duc, Nguyen, Regularization of backward parabolic equations in Banach spaces, J. Inverse Ill-Posed Probl., 20, 5-6, 745-763 (2012) · Zbl 1279.35105
[12] Nikol’skii, Sergei Mihailovic, Approximation of Functions of Several Variables and Imbedding Theorems, vol. 205 (2012), Springer Science & Business Media
[13] Smarrazzo, Flavia; Tesei, Alberto, Degenerate regularization of forward-backward parabolic equations: the regularized problem, Arch. Ration. Mech. Anal., 204, 1, 85-139 (2012) · Zbl 1255.35149
[14] Smarrazzo, Flavia; Tesei, Alberto, Degenerate regularization of forward-backward parabolic equations: the vanishing viscosity limit, Math. Ann., 355, 2, 551-584 (2013) · Zbl 1264.35275
[15] Thanh, Bui Le Trong; Smarrazzo, Flavia; Tesei, Alberto, Sobolev regularization of a class of forward-backward parabolic equations, J. Differ. Equ., 257, 5, 1403-1456 (2014) · Zbl 1295.35277
[16] Van Duc, Nguyen, An a posteriori mollification method for the heat equation backward in time, J. Inverse Ill-Posed Probl., 25, 4, 403-422 (2017) · Zbl 1370.35152
[17] Wang, Xingchang; Xu, Runzhang, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10, 1, 261-288 (2021) · Zbl 1447.35078
[18] Yagi, Atsushi, Abstract Parabolic Evolution Equations and Their Applications (2009), Springer Science & Business Media · Zbl 1190.35004
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