The paper is concerned with the Cauchy problem for second order hyperbolic evolution equations with nonlinear source in a Hilbert space, under the effect of nonlinear time-dependent damping. With the help of the method of weighted energy integral, we obtain explicit decay rate estimates for the solutions of the equation in terms of the damping coefficient and two nonlinear exponents. Specialized to the case of linear, time-independent damping, we recover the corresponding decay rates originally obtained in [3] via a different way. Moreover, examples are given to show how to apply our abstract results to concrete problems concerning damped wave equations, integro-differential damped equations, as well as damped plate equations.

The paper is concerned with the semilinear wave equations with time-dependent damping (t) = ∕(1 + t) (> 0), under the effect of nonlinear source f behaving like a polynomial, and subject to Neumann boundary conditions. Constructing appropriate auxiliary functions, we obtain an explicit uniform decay rate estimate for the solutions of the equation in terms of the exponent of f , when is large enough. On the other hand, via a new hyperbolic version of Dirichlet quotients, we show that the upper estimate is optimal in some case, which implies the existence of slow solutions.

We consider an abstract second order non-autonomous evolution equation in a Hilbert space H : u″ + Au + γ(t)u′ + f(u) = 0, where A is a self-adjoint and nonnegative operator on H, f is a conservative H-valued function with polynomial growth (not necessarily to be monotone), and γ(t)u′ is a time-dependent damping term. How exactly the decay of the energy is affected by the damping coefficient γ(t) and the exponent associated with the nonlinear term f? There seems to be little development on the study of such problems, with regard to non-autonomous equations, even for strongly positive operator A. By an idea of asymptotic rate-sharpening (among others), we obtain the optimal decay rate of the energy of the non-autonomous evolution equation in terms of γ(t) and f. As a byproduct, we show the optimality of the energy decay rates obtained previously in the literature when f is a monotone operator.

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