Abstract
In this paper, we consider the problem of finding the initial distribution for the linear inhomogeneous and nonlinear biparabolic equation. The problem is severely ill-posed in the sense of Hadamard. First, we apply a general filter method to regularize the linear nonhomogeneous problem. Then, we also give a regularized solution and consider the convergence between the regularized solution and the sought solution. Under the a priori assumption on the exact solution belonging to a Gevrey space, we consider a generalized nonlinear problem by using the Fourier truncation method to obtain rigorous convergence estimates in the norms on Hilbert space and Hilbert scale space.
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Fichera, G.: Is the Fourier theory of heat propagation paradoxical? Rend. Circ. Mat. Palermo 41, 5–28 (1992)
Joseph, L.P., Preziosi, D.D.: Heat waves. Rev. Mod. Phys. 61, 41–73 (1989)
Fushchich, V.L., Galitsyn, A.S., Polubinskii, A.S.: A new mathematical model of heat conduction processes. Ukr. Math. J. 42, 210–216 (1990)
Ames, K.A., Straughan, B.: Non-Standard and Improperly Posed Problems. Academic Press, New York (1997)
Cao, C., Rammaha, M.A., Titi, E.S.: The NavierStokes equations on the rotating 2-D sphere: Gevrey regularity and asymptotic degrees of freedom. Z. Angew. Math. Phys. 50, 341–360 (1999)
Carasso, A.S.: Bochner subordination, logarithmic diffusion equations, and blind deconvolution of Hubble space telescope imagery and other scientific data. SIAM J. Imaging Sci. 3(4), 954–980 (2010)
Nair, M.T., Pereverzev, S.V., Tautenhahn, U.: Regularization in Hilbert scales under general smoothing conditions. Inverse Probl. 21(6), 1851–1869 (2005)
Payne, L.E.: On a proposed model for heat conduction. IMA J. Appl. Math. 71, 590–599 (2006)
Tuan, N.H., Kirane, M., Long, L.D., Thinh, N.V.: Filter regularization for an inverse parabolic problem in several variables. Electron. J. Differ. Equ. 24, 13 (2016)
Tuan, N.H., Quan, P.H.: Some extended results on a nonlinear ill-posed heat equation and remarks on a general case of nonlinear terms. Nonlinear Anal. Real World Appl. 12(6), 2973–2984 (2011)
Wang, L., Zhou, X., Wei, X.: Heat Conduction: Mathematical Models and Analytical Solutions. Springer, Berlin (2008)
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Nguyen, H.T., Kirane, M., Quoc, N.D.H. et al. Approximation of an Inverse Initial Problem for a Biparabolic Equation. Mediterr. J. Math. 15, 18 (2018). https://doi.org/10.1007/s00009-017-1053-0
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DOI: https://doi.org/10.1007/s00009-017-1053-0