Global uniqueness and stability for an inverse wave source problem for less regular data. (English) Zbl 0988.35172
Summary: By means of a Carleman estimate, we obtain the global uniqueness and stability of the solution of the inverse problem of determining a wave source \(f\in H^{-1} (\Omega)\) in the problem
\[
y_{tt} (t,x)- \Delta y(t,x)= a_0(x)y+a_1 (x)y_t+ \bigl\langle a_2(x),\nabla y\bigr \rangle+ \mu(t)f(x),\;(t,x)\in Q,
\]
\[ y(t,x)=0,\;(t,x)\in\Sigma,\qquad y(0,x)= y_t(0,x)=0,\;x\in\Omega, \] where \(Q=(0,T) \times\Omega\), \(\Sigma=(0,T) \times \partial \Omega\), \(T>0\), \(\langle \cdot, \cdot \rangle\) is the scalar product in \(\mathbb{R}^n\), and \(\Omega\subset\mathbb{R}^n\) is a bounded domain which is either convex or of the class \(C^{1,1}\), from the boundary observation \({\partial y \over \partial \nu}|_{(0,T) \times\Gamma_0}\) or the interior observation \(y |_{(0,T) \times G}\), where \(\Gamma_0\) and \(G\) are some given open subsets of \(\Gamma\) and \(\Omega\), respectively.
\[ y(t,x)=0,\;(t,x)\in\Sigma,\qquad y(0,x)= y_t(0,x)=0,\;x\in\Omega, \] where \(Q=(0,T) \times\Omega\), \(\Sigma=(0,T) \times \partial \Omega\), \(T>0\), \(\langle \cdot, \cdot \rangle\) is the scalar product in \(\mathbb{R}^n\), and \(\Omega\subset\mathbb{R}^n\) is a bounded domain which is either convex or of the class \(C^{1,1}\), from the boundary observation \({\partial y \over \partial \nu}|_{(0,T) \times\Gamma_0}\) or the interior observation \(y |_{(0,T) \times G}\), where \(\Gamma_0\) and \(G\) are some given open subsets of \(\Gamma\) and \(\Omega\), respectively.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
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