We discuss admissibility and exact observability estimates of boundary observation and interior point observation of a one-dimensional wave equation on a time dependent domain for sufficiently regular boundary functions. We also discuss moving observers inside the noncylindrical domain and simultaneous observability results.Since we suppose f, g ∈ D((0, 1)), h satisfies the periodicity condition h (α) (−1)=h (α) (1) for all derivative orders α ≥ 0. As a consequence, the series of F , g and h above may be differentiated term by term. We let u(x, t) := n∈Z A n e 2πin ϕ(t+x)) − e 2πin ϕ(t−x)) † In particular, (bn) is a Riesz basis in L 2 ([−1, 1]).

We consider the observability problem for non-autonomous evolution systems (i.e., the operators governing the system depend on time). We introduce an averaged Hautus condition and prove that for skew-adjoint operators it characterizes exact observability. Next, we extend this to more general class of operators under a growth condition on the associated evolution family. We give an application to the Schrödinger equation with time dependent potential and the damped wave equation with a time dependent damping coefficient.

We investigate wave breaking criteria for the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system. We establish a new blowup criterion for the general case γ + c 0 α 2 ≥ 0 involving local-in-space conditions on the initial data.

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