THE BOUNDARY STABILIZATION OF A LINEARIZED SELF-EXCITED WAVE-EQUATION

We study the linearized self-excited wave equation u(u) - c(2) Delta u - P(x)u(t) = 0, where P(x) greater than or equal to 0, and P(x) is an element of L(infinity)(Omega), in a bounded domain Omega subset of R(n) with smooth boundary Gamma, where boundary damping is present. We observe that the energy is not monotonically non-increasing, owing to negative internal damping P which causes self-excitation. Hence, the system may become unstable. Having considered the partition {Gamma(+), Gamma(-)} of the boundary Gamma on which u = 0 on Gamma(-) and partial derivative u/partial derivative n + Ku(t) + Lu = 0 on Gamma(+), we find two different bounds for P such that the energy of the system decays exponentially as t tends to infinity (here, we assume ($) over bar Gamma(+) boolean AND Gamma(-) = phi for n > 3). Both bounds depend on Omega. Moreover, the second bound depends on the feedback functions K, L is an element of L(infinity)(Gamma(+)), or more precisely it depends on a positive function k(x) is an element of L(infinity)(Gamma(+)) which determines K and L on the partition Gamma(+).