Unique Continuation for the Linearized Benjamin-Bona-Mahony Equation with Space-Dependent Potential

Abstract

In this paper, we analyze the unique continuation problem for the linearized Benjamin-Bona-Mahony equation with space-dependent potential in a bounded interval with Dirichlet boundary conditions. We prove two unique continuation results by means of spectral analysis and the (generalized) eigenvector expansion of the solution, instead of the usual Holmgren-type method or Carleman-type estimates. It is found that the unique continuation property depends very strongly on the nature of the potential and, in particular, on its zero set, and not only on its boundedness or integrability properties.