Conditional Stability in Determination of Densities of Heat Sources in a Bounded Domain

Abstract

We consider the heat equation in a bounded domain Ω ⊂ ℝ^r :

$$ \frac{{\partial u}}{{\partial t}}(x,t) = \Delta u(x,t) + \sigma (t)f(x) (x \in \Omega ,0 < t < T)$$

$$ u(x,0) = 0 (x \in \Omega ), \frac{{\partial u}}{{\partial n}}(x,t) = 0 (x \in \partial \Omega ,0 < t < T)$$

Assuming that σ is a known function with σ (0) ≠ 0, we prove :(1) ƒ(x) (x ∊ Ω) can be uniquely determined from the boundary data u(x, t) (x ∊ ∂Ω, 0 < t < T). (2) If ƒ is restricted to a compact set in a Sobolev space, then we get an estimate:

$$ {{\left\| f \right\|}_{{{{L}^{2}}}}}_{{(\Omega )}} = 0\left( {{{{\left( {\log \frac{1}{\eta }} \right)}}^{{ - \beta }}}} \right) $$

as

$$ \eta \equiv {{\left\| {u(,)} \right\|}_{{{{H}^{1}}}}}_{{(0 T {{L}^{2}}(\partial \Omega ))}} \downarrow 0 $$

Here the exponent ß is given by the order of the Sobolev space which is assumed to contain the set of ƒ’s.