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On some inverse problem for bi-parabolic equation with observed data in Lp spaces

Tuan Nguyen Huy

Opuscula Mathematica

2022

Vol. 42, no. 2

305--335

The bi-parabolic equation has many practical significance in the field of heat transfer. The objective of the paper is to provide a regularized problem for bi-parabolic equation when the observed data are obtained in Lp. We are interested in looking at three types of inverse problems. Regularization results in the L2 space appears in many related papers, but the survey results are rare in Lp, p ̸= 2. The first problem related to the inverse source problem when the source function has split form. For this problem, we introduce the error between the Fourier regularized solution and the exact solution in Lp spaces. For the inverse initial problem for both linear and nonlinear cases, we applied the Fourier series truncation method. Under the terminal input data observed in Lp, we obtain the approximated solution also in the space Lp. Under some reasonable smoothness assumptions of the exact solution, the error between the the regularized solution and the exact solution are derived in the space Lp. This paper seems to generalize to previous results for bi-parabolic equation on this direction.