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Generic well-posedness in a multidimensional hyperbolic inverse problem. (English) Zbl 0867.35115
Summary: Let $y(f)$ and $u(p,a,h)$ be the solutions, respectively, to the following initial/boundary value problems in a bounded domain $\Omega\subset\bbfR^n$ $(n\ge 1)$ with a smooth boundary $\partial\Omega$: $${\partial^2y\over\partial t^2} (x,t)= \Delta y(x,t)-p(x)y(x,t)-f(x)\lambda(x,t),\quad x\in\Omega,\quad t>0\tag1$$ $$y(x,0)={\partial y\over\partial t} (x,0)=0,\quad x\in\Omega,\quad y(x,t)=0,\quad x\in\partial\Omega,\quad t>0$$ $${\partial^2u\over\partial t^2} (x,t)=\Delta u(x,t)-p(x)u(x,t),\quad x\in\Omega,\quad t>0\tag2$$ $$u(x,0)=a(x),\quad{\partial u\over\partial t} (x,0)=0,\quad x\in\Omega,\quad u(x,t)=h(x,t),\quad \in\partial\Omega,\quad t>0.$$ For a given $\Gamma\subset\partial\Omega$ and a sufficiently large $T<\infty$, by the exact controllability method, we get stability estimates for two inverse problems: (I) Determine $f(x)$ $(x\in\Omega)$ from $(\partial y(f)/\partial n)(x,t)$ ($x\in\Gamma$, $0<t<T$) provided that $p(x)$ and $\lambda(x,t)$ are given functions. (II) Determine $p(x)$ $(x\in\Omega)$ from $(\partial u(p,a,h)/\partial n)(x,t)$ $(x\in\Gamma$, $0<t<T$) provided that $a(x)$ and $h(x,t)$ are given functions.

35R30Inverse problems for PDE
35L15Second order hyperbolic equations, initial value problems
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