Observability of wave equations with time- and space-dependent potentials: a counterexample under GCC and a positive criterion on the disk

Abstract

We show that the geometric control condition (GCC) for the free wave equation does not guarantee internal observability in the presence of a smooth time-dependent potential. On a bounded connected domain with analytic boundary, we construct a nonempty open observation set such that the pair satisfies GCC and the free Dirichlet wave is observable, and a smooth $2π$-periodic potential for which there exists a nonzero smooth Dirichlet wave that vanishes on the observed cylinder for all time. The wave is the unique mild solution in the space $X_Ω=L^2(Ω)\times H^{-1}(Ω)$ on every finite interval, so no finite observation horizon yields an observability inequality for this fixed potential. The construction starts from a real-valued local counterexample to unique continuation across a noncharacteristic hyperplane; this local statement is of independent interest, since earlier constructions for the wave operator use complex-valued waves and potentials. The global counterexample is obtained by joining flat oscillatory solutions along the edges of a polygonal observation set, repairing their sum at the vertices without changing its zero set, and then using a spatial transition to a common time-independent pair near the boundary. For comparison, on the unit disk we show that an observation region containing a central disk and a neighborhood of an embedded cross-cut yields observability for every weak wave at sufficiently large observation time, with constants uniform over bounded measurable potentials; the proof uses a cut, logarithmic polar coordinates, and a Carleman weight whose favorable flux is confined to the observed artificial boundary.