Achieving Arbitrarily Large Decay in the Damped Wave Equation

is referred to as the decay rate associated with a. If a is to be introduced in order to absorb an initial disturbance then one naturally wishes to strike upon that a with the least possible (most negative) decay rate. The mathematical attraction here lies in the oftnoted fact that, with respect to damping, ‘more is not better.’ More precisely, for constant a, the decay rate is not a decreasing function of a. Rather, for small a, ω decreases until a reaches π, after which ω rapidly increases to 0. Our aim is to show that there exist nonconstant a that circumvent this phenomena of overdamping and hence that more indeed can be better. Cox and Zuazua ([3], Thm. 6.5) have shown that a 7→ ω(a) attains its finite minimum over {a ∈ BV (0, 1) : T (a) ≤ M} where T (a) denotes the total variation of a. We show here that the total variation constraint is not superfluous. More precisely, we establish