Conditional stability in determination of force terms of heat equations in a rectangle

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Abstract

We consider a heat equation in a rectangle: ∂u ∂t(x1,2,t)=Δu(x1,x2,t)+σ(t)f(x1,x2), (x1x2)∈(0,1)×(0,1),0<t<T with the zero initial value condition and the zero Neumann boundary condition. Assuming that σ is a known function with σ(0) ≠ 0 and depends only on time t, we prove: (1) f(x1,x2) (0 < x1,x2 < 1) can be uniquely determined from the base boundary data u(x1,0,t) (0 < x1 < 1, 0 < t < T). (2) If f is restricted to a compact set in the Sobolev spaces, then we get an estimate:{norm of matrix}f{norm of matrix}L2=Olog 1 η-β as η≡{norm of matrix}u(·,0,·{norm of matrix})1((0,T);L2(0,1))↓0. Here the exponent β is given by the order of the Sobolev space which is assumed to contain the set of f's. © 1993.

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Yamamoto, M. (1993). Conditional stability in determination of force terms of heat equations in a rectangle. Mathematical and Computer Modelling, 18(1), 79–88. https://doi.org/10.1016/0895-7177(93)90081-9

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