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Convolution inequalities and inverse problems play an important role in mathematical analysis. This paper studies inverse inequalities for the Fourier cosine convolution and the backward heat problem. | JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci. 2014 Vol. 59 No. 7 pp. 9-20 This paper is available online at http stdb.hnue.edu.vn REVERSE INEQUALITIES FOR THE FOURIER COSINE CONVOLUTION AND APPLICATIONS TO INVERSE HEAT SOURCE PROBLEMS Nguyen Xuan Thao and Bui Minh Khoi School of Applied Mathematics and Informatics Hanoi University of Science and Technology Abstract. Convolution inequalities and inverse problems play an important role in mathematical analysis. This paper studies inverse inequalities for the Fourier cosine convolution and the backward heat problem. We give new reverse inequalities for the Fourier cosine convolution and their applications to inverse heat source problems ut uxx f x t 0 lt x lt t gt 0 our purpose is to evaluate the stability of non-negative heat source f x t from any observations u x t0 where 0 lt t0 is a constant t0 T δ T gt 0 δ gt 0 0 lt x lt X X gt 0 or u x0 t where 0 lt x0 is a constant 0 lt t lt T . Applying a new inverse inequality allow us to evaluate heat source through some initial observations with space or time variables. Keywords Reverse inequalities Fourier cosine convolution one-dimensional inverse heat source. 1. Introduction Inverse problems have developed robustly and attracted the attention of many mathematicians in recent decades. Two examples are inverse problems for partial differential equations 2 4-7 and inverse problems for heat equations 1 3 8 9 . Of these the problem in 9 is recent research on the reverse heat source problem using reverse inequality for Laplace convolution. We show the following multidimensional heat source equation see 9 t u x t u x t f t φ x x Rn t gt 0 u x 0 0 x Rn Received March 20 2014. Accepted September 30 2014. Contact Nguyen Xuan Thao e-mail address thao.nguyenxuan@hust.edu.vn 9 Nguyen Xuan Thao and Bui Minh Khoi for φ is a given function and satisfies φ 0 in Rn φ has compact support φ C Rn n 4 φ L2 Rn n 3. We then estimate the stabilization of heat source f t 0 lt t lt T
