ASYMPTOTIC STABILITY OF THE WAVE EQUATION ON COMPACT SURFACES AND LOCALLY DISTRIBUTED DAMPING-A SHARP RESULT

This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by u(tt) - Delta(M)u + a(x) g(u(t)) = 0 on M x ]0, infinity[, where M subset of R-3 is a smooth oriented embedded compact surface without boundary. Denoting by g the Riemannian metric induced on M by R-3, we prove that for each epsilon > 0, there exist an open subset V subset of M and a smooth function f : M -> R such that meas(V) >= meas(M) - epsilon, Hessf approximate to g on V and inf(x is an element of V)vertical bar del f(x)vertical bar > 0. In addition, we prove that if a(x) >= a(0) > 0 on an open subset M* subset of M which contains M\V and if g is a monotonic increasing function such that k vertical bar s vertical bar <= vertical bar g(s)vertical bar <= K vertical bar s vertical bar for all vertical bar s vertical bar <= 1, then uniform and optimal decay rates of the energy hold.