Saitoh, Saburou; Tuan, Vu Kim; Yamamoto, Masahiro Reverse convolution inequalities and applications to inverse heat source problems. (English) Zbl 1029.44002 JIPAM, J. Inequal. Pure Appl. Math. 3, No. 5, Paper No. 80, 11 p. (2002). Authors’ 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 \(f(t)\), \(0< t< T\), in the heat source of the heat equation \[ \partial_t u(x, t)=\Delta u(x, t)+ f(t)\varphi(x),\quad x\in\mathbb{R}^n,\quad t> 0 \] from the observation \(u(x_0, t)\), \(0< t< T\), at a remote point \(x_0\) away from the support of \(\varphi\). Under an a priori assumption that \(f\) changes the signs at most \(N\)-times, we give a conditional stability of Hölder type, as an example of applications. Reviewer: Robert G.Buschman (Langlois) Cited in 22 Documents MSC: 44A35 Convolution as an integral transform 26D20 Other analytical inequalities 35K05 Heat equation 35R30 Inverse problems for PDEs Keywords:convolution; heat source; weighted convolution inequalities; Young’s inequality; Hölder 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 PDFBibTeX XMLCite \textit{S. Saitoh} et al., JIPAM, J. Inequal. Pure Appl. Math. 3, No. 5, Paper No. 80, 11 p. (2002; Zbl 1029.44002) Full Text: EuDML