Frictional versus viscoelastic damping in a semilinear wave equation

摘要

In this article we show exponential and polynomial decay rates for the partially viscoelastic nonlinear wave equation subject to a nonlinear and localized frictional damping. The equation that models this problem is given by (0.1) u(tt) - kappa0Deltau + integral(0)(t) div[alpha(x)g(t - s)delu(s)]ds + f(u) + b(x)h(u(t)) = 0 in Omega x R+, where a, b are nonnegative functions, alpha is an element of C-1((&UOmega;) over bar), b is an element of L-infinity(Omega), satisfying the assumption (0.2) alpha(x) + b(x) greater than or equal to delta > 0 For Allx is an element of Omega, and f and h are power-like functions. We observe that the assumption (0.2) gives us a wide assortment of possibilities from which to choose the functions alpha(x) and b(x), and the most interesting case occurs when one has simultaneous and complementary damping mechanisms. Taking this point of view into account, a distinctive feature of our paper is exactly to consider different and localized damping mechanisms acting in the domain but not necessarily "strategically localized dissipations" as considered in the prior literature. [References: 16]