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An inverse problem for the heat equation. (English) Zbl 0987.35164

Summary: Let \(u_t=u_{xx}- q(x)u\), \(0\leq x\leq 1\), \(t>0\), \(u(0,t) =0\), \(u(1,t)= a(t)\), \(u(x,0)=0\), where \(a(t)\) is a given function vanishing for \(t>T\), \(a(t)\not\equiv 0\), \(\int^T_0a(t)dt <\infty\). Suppose one measures the flux \(u_x(0,t):=b_0(t)\) for all \(t>0\). Does this information determine \(q(x)\) uniquely? Do the measurements of the flux \(u_x(1,t): =b(t)\) give more information about \(q(x)\) than \(b_0(t)\) does? These questions are answered in this note.

MSC:

35R30 Inverse problems for PDEs
35K05 Heat equation
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References:

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