Yamamoto, Masahiro Conditional stability in determination of densities of heat sources in a bounded domain. (English) Zbl 0810.35032 Desch, W. (ed.) et al., Control and estimation of distributed parameter systems: nonlinear phenomena. International conference in Vorau (Austria), July 18-24, 1993. Basel: Birkhäuser. ISNM, Int. Ser. Numer. Math. 118, 359-370 (1994). Summary: We consider the heat equation in a bounded domain \(\Omega\subset \mathbb{R}^ r\): \[ {{\partial u}\over {\partial t}} (x,t)= \Delta u(x,t)+ \sigma(t) f(x) \qquad (x\in\Omega,\;0<t<T), \]\[ u(x,0)=0 \quad (x\in\Omega), \qquad {{\partial u}\over {\partial n}} (x,t)=0 \quad (x\in \partial\Omega,\;0<t<T). \] Assuming that \(\sigma\) is a known function with \(\sigma(0)\neq 0\), we prove: (1) \(f(x)\) (\(x\in\Omega\)) can be uniquely determined from the boundary data \(u(x,t)\) (\(x\in \partial\Omega\), \(0<t<T\)). (2) If \(f\) is restricted to a compact set in a Sobolev space, then we get an estimate: \[ \| f\|_{L^ 2(\Omega)}= O\Bigl( \bigl(\log {\textstyle {1\over \eta}} \bigr)^{-\beta} \Bigr) \quad \text{ as }\quad \eta\equiv | u(\cdot, \cdot) \|_{H^ 1 (0,T; L^ 2( \partial\Omega))} \downarrow 0. \] Here the exponent \(\beta\) is given by the order of the Sobolev space which is assumed to contain the set of \(f\)’s.For the entire collection see [Zbl 0801.00053]. Cited in 15 Documents MSC: 35K05 Heat equation 35R25 Ill-posed problems for PDEs 35R30 Inverse problems for PDEs 93B30 System identification Keywords:conditional stability; boundary observation; density of heat source PDF BibTeX XML Cite \textit{M. Yamamoto}, ISNM, Int. Ser. Numer. Math. 118, 359--370 (1994; Zbl 0810.35032) OpenURL