Optimal energy decay rates for abstract second order evolution equations with non-autonomous damping

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