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Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation. (English) Zbl 1091.35112
Summary: We study the global stability in determination of the coefficient $$a(x)$$ in the acoustic equation $\partial_t^2 u(x,t)-\text{div}(a(x) \nabla u(x,t))=0$ from data of the solution in a subboundary $$\Gamma_1$$ over a time interval. Providing regular initial data and values of coefficients in a neighbourhood of the boundary, without any assumption on the observation subboundary $$\Gamma_1\subset \partial\Omega$$, we prove a logarithmic stability estimate in the inverse problem with a single measurement. Moreover the exponent in the stability estimate depends on the regularity of initial data.

MSC:
 35R30 Inverse problems for PDEs 35L15 Initial value problems for second-order hyperbolic equations 35B35 Stability in context of PDEs
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