Nonlinear inverse problems for elliptic equations (Q2777925)

Let \(D\) be a bounded domain in \(\mathbb{R}^n\) with boundary \(\Gamma\), \(Q=D\times (0,T)\), \(T>0\) and let \(f(x,t)\), \(c(x,t)\) and \(u_1(x)\) be given functions for \(x\in\overline D\), \(t\in[0,T]\). The author proves that under certain conditions, we have existence and uniqueness for the functions \(u(x,t)\) and \(q(x)\), satisfying the problem: NEWLINE\[NEWLINEq(x)(u_{tt}+ \Delta_xu)-c(x,t)u> f(x,t)\text{ on }Q,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(x,0)=0,\;u(x,T)=u_1(x), u_t(x,T)=0 \text{ for }x\in D,\;u|_S=0,NEWLINE\]NEWLINE where \(S=\Gamma \times(0,T)\). The approach is based on passing from an inverse problem to a special direct problem for a nonlinear equation of composite type.