Remark on the decay for damped string and beam equations

Abstract

where A is an operator in a Hilbert space H, M is a real function. Our results improve those obtained by de Brito in a recent paper [l]. We briefly recall the notation and the assumptions in [l] referring to this paper for other information on physical meaning, existence and regularity theory, applications and bibliography. Let A be a self-adjoint positive operator in a (real) Hilbert space H with dense domain V. We assume that 5 > 0 is the least eigenvalue of A so a(u) = (Au, u) 3 51~1~ for u E V. Let M E Cl@+) be a real nondecreasing function such that M(s) 2 p + ks, k z 0, p-real. We have, with the notation