Abstract
In this paper we study the behavior of the energy and the L-2 norm of solutions of the wave equation with localized linear damping in exterior domain. Let u be a solution of the wave system with initial data (u(0), u(1)). We assume that the damper is positive at infinity then under the Geometric Control Condition of Bardos et al [5] (1992), we prove that:
1. If (u(0), u(1)) belong to H-0(1)(Omega) x L-2 (Omega), then the total energy E-u(t) <= C-0 (1 + t)(-1) I-0 and parallel to u(t)parallel to(2)(L2) <= C0I0, where
I-0 = parallel to u(0)parallel to(2)(H1) + parallel to u(1)parallel to(2)(L2).
If the initial data (u(0), u(1)) belong to H-0(1)(Omega) x L-2 (Omega) and verifies
parallel to d(.)(u(1) + au(0))parallel to(2)(L2) < +infinity,
then the total energy E-u(t) <= C-2(1 + t)(-2) I-1 and parallel to u(t)parallel to(2)(L2) <= C-2 (1 + t)(-1) I-1, where
I1 = I-0 = parallel to u(0)parallel to(2)(H1) + parallel to u(1)parallel to(2)(L2) + parallel to d(.)(u(1) + au(0))parallel to(2)(L2)
and
d(x) = { vertical bar x vertical bar vertical bar x vertical bar ln (B vertical bar x vertical bar) d >= 3, d=2,
with B inf (x is an element of Omega) vertical bar x vertical bar >= 2.