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

  • Masahiro Yamamoto
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 118)

Abstract

We consider the heat equation in a bounded domain Ω ⊂ ℝr :
$$ \frac{{\partial u}}{{\partial t}}(x,t) = \Delta u(x,t) + \sigma (t)f(x) (x \in \Omega ,0 < t < T)$$
$$ u(x,0) = 0 (x \in \Omega ), \frac{{\partial u}}{{\partial n}}(x,t) = 0 (x \in \partial \Omega ,0 < t < T)$$
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.

1991 Mathematics Subject Classification

35K05 35R25 35R30 35B30 

Key words and phrases

Conditional stability a priori bound biorthogonal function boundary observation density of heat source 

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Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • Masahiro Yamamoto
    • 1
  1. 1.Department of Mathematical SciencesUniversity of TokyoMeguro, TokyoJapan

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