A multidimensional inverse boundary value problem for a linear hyperbolic equation in a bounded domain (Q1874857)

This paper considers an inverse boundary value problem for a linear hyperbolic equation. As a particular result of the paper we have that: For a sufficiently small \(T> 0\), there exists a unique solution \(u(x,t)\in C^1(\overline D_T)\) and \(c(t)\in C[0,T]\) of the problem \[ u_{tt}- u_{xx}= c(t) u_t,\quad (x,t)\in\overline D_T= [0,1]\times [0,T], \] \[ u(x,0)= \phi(x)\in W^3_2(0, 1),\quad x\in[0,1],\quad\phi(0)= \phi''(0)= \phi(1)= \phi''(1)= 0, \] \[ u_t(x,0)= \psi(x)\in W^2_2(0, 1),\quad x\in [0,1],\quad \psi(0)= \psi(1)= 0, \] \[ u(0, t)= u(1, t)= 0,\quad t\in [0,T], \] \[ 0\leq u(x_0, t)= h(t)\in C^2[0,T],\;h(0)= \phi(x_1),\;h'(0)= \psi(x_1),\;h(t)\neq 0,\;h'(t)\neq 0,\;t\in [0,T]. \] This existence and uniqueness result is extended to higher dimensions \(n\) for the three coefficients \(a(t)\), \(c(t)\) and \(f(t)\) involved in the linear hyperbolic equation \[ u_{tt}+ K(x)u- \sum^n_{i,j=1} (\partial/\partial x_i)(a_{ij}(x)\partial u/\partial x_j)= c(t) d(x,t) u_t+ a(t) b(x, t)u+ f(t) F(x,t), \] \[ (x,t)\in\overline D_T:= \overline\Omega\times [0,T] \] subject to initial and boundary conditions and three internal measurements of \(u(x^i, t)= h_i(t)\), \(t\in [0,T]\), \(i= 1,2,3\), where \(d\), \(b\) and \(F\) are given functions satisfying certain properties.