Reverse Convolution Inequalities and Applications to Inverse Heat Source Problems

Volume 3, Issue 5, 2002

Article 80

REVERSE CONVOLUTION INEQUALITIES AND APPLICATIONS TO INVERSE HEAT SOURCE PROBLEMS

SABUROU SAITOH, VU KIM TUAN AND MASAHIRO YAMAMOTO

DEPARTMENT OF MATHEMATICS,
FACULTY OF ENGINEERING,
GUNMA UNIVERSITY,
KIRYU 376-8515 JAPAN.
E-Mail: ssaitoh@math.sci.gunma-u.ac.jp

DEPARTMENT OF MATHEMATICS,
FACULTY OF SCIENCES,
KUWAIT UNIVERSITY,
P.O. BOX 5969,
KUWAIT SAFAT 13060
E-Mail: vu@sci.kuniv.edu.kw

GRADUATE SCHOOL OF MATHEMATICAL SCIENCES,
THE UNIVERSITY OF TOKYO,
3-8-1 KOMABA MEGURO,
TOKYO 153-8914 JAPAN.
E-Mail: myama@ms.u-tokyo.ac.jp

Received 02 April, 2001; Accepted 30 October, 2002.
Communicated by: S.S. Dragomir

ABSTRACT.

We introduce reverse convolution inequalities obtained recently and at the same time, we give new type reverse convolution inequalities and their important applications to inverse source problems. We consider the inverse problem of determining , , in the heat source of the heat equation $\partial_t u(x,t) = \Delta u(x,t) + f(t)\varphi(x)$ , $x\in {\mathbb{R}}^n$ , from the observation , , at a remote point away from the support of $\varphi$ . Under an a priori assumption that changes the signs at most -times, we give a conditional stability of Hölder type, as an example of applications.

Key words:
Convolution, Heat source, Weighted convolution inequalities, Young's inequality, Hölder's inequality, Reverse Hölder's inequality, Green's function, Stability in inverse problems, Volterra's equation, Conditional stability of Hölder type, Analytic semigroup, Interpolation inequality, Sobolev inequality.

2000 Mathematics Subject Classification:
Primary 44A35; Secondary 26D20.

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