Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations

Abstract

(t) + Au(t) + f(u(t)) = 0, (1.1)where H is a real Hilbert space, A is a nonnegative self-adjoint linear operator on Hwith dense domain, and f is a nonlinearity tangent to 0 at the origin.When f ≡ 0, then for rather general classes of strongly positive operators A it isknown that all solutions decay to 0 (as t → +∞) exponentially in the energy norm.Therefore, by perturbation theory it is reasonable to expect that also all solutions of(1.1) which decay to 0 have an exponential decay rate. The situation is different whenA has a non-trivial kernel. In this case solutions tend to 0 if f fulfils suitable signconditions, but we do not expect all solutions to have an exponential decay rate. Letus consider for example the hyperbolic equationu