Uniqueness for an ill-posed problem (Q913118)

Consider the ill-posed problem for the wave equation \[ u_{tt}- (u_{xx}+u_{yy})+au=F,\quad u|_{x=0}=u_ 0,\quad u_ x|_{x=0}=u_ 1. \] Suppose that \(u_{tt}- (u_{xx}+u_{yy})+au=0\), \(u|_{x<0}=0\), supp \(u\cap \{x=0\}\) compact, then \(u\equiv 0\) (near \(\{x=0\})\). If the perturbation a depends only on the variables x,y, that is if \[ u_{tt}- (u_{xx}+u_{yy})+a(x,y)u=0,\quad u|_{x<0}=0 \] does it follow that \(u=0?\) This paper gives a positive answer to the question.