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].


35K05 Heat equation
35R25 Ill-posed problems for PDEs
35R30 Inverse problems for PDEs
93B30 System identification