## 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].

### MSC:

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