A Source Identification Problem in a Bi-parabolic Equation: Convergence Rates and Some Optimal Results

Abstract

This paper is concerned with identification of a spatial source function from final time observation in a bi-parabolic equation, where the full source function is assumed to be a product of time dependent and a space dependent function. Due to the ill-posedness of the problem, recently some authors have employed different regularization method and analysed the convergence rates. But, to the best of our knowledge, the quasi-reversibility method is not explored yet, and thus we study that in this paper. As an important implication, the Hölder rates for the apriori and aposteriori error estimates obtained in this paper improve upon the rates obtained in earlier works. Also, in some cases we show that the rates obtained are of optimal order. Further, this work seems to be the first one that has broaden the applicability of the problem by allowing the time dependent component of the source function to change sign. To the best of our knowledge, the earlier known work assumed the fixed sign of the time dependent component by assuming some bounded below condition.

Keywords:

• Apriori and aposteriori parameter choice
• bi-parabolic equation
• optimal rate
• quasi-reversibility method
• source identification

2010 Mathematics Subject Classification:

• 35R25
• 47A52
• 65M30
• 41A25

Acknowledgments

The authors sincerely thank the anonymous referees for providing valuable comments and suggestions which led to this improved version. The first author, Subhankar Mondal, is supported by the postdoctoral fellowship of TIFR Centre for Applicable Mathematics, Bangalore, and the second author, M. Thamban Nair, gratefully acknowledges the support received from BITS Pilani, K.K. Birla Goa Campus, where he is a Visiting Professor.