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Inverse problems for time-dependent singular heat conductivities: multi-dimensional case. (English) Zbl 1327.35438

The paper examines an inverse boundary value problem for the heat equation \( \partial_t u = \text{div} (\gamma \nabla_x u) \) in \((0,T) \times \Omega\), \(u=f\) on \((0,T) \times \partial \Omega\), \(u|_{t=0}=u_0\), in a bounded domain \(\Omega \in \mathbb R^n\), \(n \geq 2\), where the heat conductivity \(\gamma(t,x)\) is piecewise constant and the surface of discontinuity depends on time, that is, \(\gamma(t,x)=k^2\), \(x \in D(t)\), \(\gamma(t,x)=1\), \(x \in \Omega \setminus D(t)\). Fix a direction \(\mathbf{e}^* \in \mathrm{S}^{n-1}\) arbitrarily. Assuming that \(\partial D(t)\) is strictly convex for \(0 \leq t \leq T\), the authors show that \(k\) and \(\sup \{ \mathbf{e}^* \cdot x : x \in D(t) \} \), in particular, \(D(t)\) itself, are determined from the Dirichlet-to-Neumann map \( f \to \partial_{\nu} u(x,t)|_{(0,T) \times \partial \Omega}\), where \(\nu\) is the outer normal to \(\partial \Omega\). The knowledge of the initial data is not used in the proof. If \( \min _{0 \leq t \leq T}( \sup \mathbf{e}^* \cdot x )\) is known, the same conclusion can be made from the local Dirichlet-to-Neumann map. The present paper is different from related works (see, for example, Y. Daido et al. [Inverse Probl. 23, No. 5, 1787–1800 (2007; Zbl 1126.35092)]) in the use of the asymptotic heat flow and energy inequalities.
Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. For example, consider a physical body consisting of a homogeneous material with constant heat conductivity except for a moving inclusion with different heat conductivity. Then, the location and the shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.

MSC:

35R30 Inverse problems for PDEs
35K05 Heat equation

Citations:

Zbl 1126.35092
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References:

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