The decay of solutions of the exterior initial‐boundary value problem for the wave equation

Abstract

Suppose a disturbance which is initially confined to a finite region is propagated in free space according to the wave equation. It spreads outward and eventually reaches every point in space. However, at any fixed point the disturbance eventually dies out, decaying at a rate that depends on the number of space dimensions. If there are three space dimensions it follows from Huyghen's principle that there is eventually no disturbance at all. In two dimensions the rate of decay with time t can be exactly l/t.For physical reasons, it has been conjecturedl that the same situation holds if there is a reflecting body somewhere in space. The wave from the initial disturbance reaches the body and is reflected from it. At any point the disturbance can be measured in terms of "energy". Eventually one expects all the energy, and thus the disturbance too, to be carried outward. Of course, if a body has a very complicated indented shape, the rate of decay will be slowed by the reverberations within the indentations. But one would still expect the energy to leak out in time.For a spherical body it has been shown by C . H.

Concerning the Strauss Conjecture and Almost Global Existence for Nonlinear Dirichlet-Wave Equations in 4-Dimensions

“…This follows from the nontrapping assumption and certain resolvent estimates (see Burq [2], Theorem 3). Of course (2.13) is essentially equivalent to the classical local decay estimates in [16] and [18].…”

Steady state computations for wave propagation problems

“…Different conditions on the domain and the differential operator are derived in order to guarantee that the energy is not trapped in ß' and that there exists a decay law. See, e.g., [7], [8], [9].…”

Global existence for a quasilinear wave equation outside of star-shaped domains

“…The argument that we shall use is similar to that of Morawetz [18] (see also [17], p. 261-264) for a related energy-decay estimate in Minkowski space minus a star-shaped obstacle. In particular, we shall see that when one goes through the standard proof of energy estimates the (variable) boundary contributes a term with the "correct" sign if K is strictly star-shaped with respect to the origin.…”