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- W14533060 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: $$ {{left| f right|}_{{{{L}^{2}}}}}_{{(Omega )}} = 0left( {{{{left( {log frac{1}{eta }} right)}}^{{ - beta }}}} right) $$ 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 Classification35K0535R2535R3035B30Key words and phrasesConditional stabilitya priori boundbiorthogonal functionboundary observationdensity of heat source" @default.
- W14533060 created "2016-06-24" @default.
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- W14533060 date "1994-01-01" @default.
- W14533060 modified "2023-10-16" @default.
- W14533060 title "Conditional Stability in Determination of Densities of Heat Sources in a Bounded Domain" @default.
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- W14533060 doi "https://doi.org/10.1007/978-3-0348-8530-0_20" @default.
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