Stability and global-in-time results for an inverse problem related to a nuclear reactor model

doi:10.1088/0266-5611/25/10/105007

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Inverse Problems Volume 25 Number 10 Create an alert RSS this journal

Fabrizio Colombo 2009 Inverse Problems 25 105007 doi:10.1088/0266-5611/25/10/105007

Stability and global-in-time results for an inverse problem related to a nuclear reactor model

Fabrizio Colombo

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Abstract References

An open bounded set in $\mathbb{R}^3$ is denoted by Ω and let T be a real positive number. We consider the problem of determining the temperature u and the convolution kernel h, under suitable initial-boundary conditions, in the evolution equation (for x Ω and t [0, T])

$\fl D_tu(t,x)=\Delta u(t,x) + \int_0^th(t-s)\Delta u(s,x) \,{\rm d}s +\eta(x)({\rm e}^{{\cal W}(t)}-1)$

where the function ${\cal W}$ is given by

$\frac{{{\rm d}\cal W}(t)}{{\rm d}t}=-\int_\Omega\alpha(x)u(t,x)\,{\rm d}x,\qquad {\cal W}(0)={\cal W}_0.$

To determine simultaneously u and h we assume the following restriction on u:

$\int_{\Omega}\phi(x)u(t,x) \,{\rm d}x=g(t),$

which corresponds to additional measurements on the temperature. The elements ${\cal W}_0, \eta, \alpha, \phi, g$ are given data. The above model describes the dynamics in a nuclear reactor. The convolution kernel h makes finite the heat speed of propagation, this is important when we consider every short intervals of time. We prove stability results and for suitable growth conditions on the nonlinearities we obtain global-in-time existence and uniqueness of the solution for the above inverse problem.

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