Abstract
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.
Highlights
We are concerned with an abstract second order non-autonomous evolution equation in a real Hilbert space H, with time dependent damping, as follows
Where A is a nonnegative self-adjoint linear operator on H, the nonlinear term f : D(A1/2) → H is assumed to be conservative with polynomial growth, and γu is the time dependent damping with
What is the optimal decay rate of the energy of the non-autonomous equation (1.1), i.e., how exactly the decay of the energy of (1.1) is affected by the damping coefficient γ(t) and the exponent associated with the nonlinear term f ? This problem is still unsolved
Summary
In view of the importance in mathematical theory and applications in physics, engineering, mechanics, biology and others, various nonautonomous equations including the non-autonomous evolution equations in abstract (infinite-dimensional) spaces, have been studied by many researchers and a lot of good results on this issue have been established (cf., e.g. [1, 3, 8,9,10,11,12,13,14,15,16,17, 24,25,26, 28, 30, 32, 34, 35] and references therein). Keywords and phrases: Non-autonomous, abstract second order evolution equation, time dependent damping, energy estimates, slow solutions, nonlinear source, Hilbert space. We assume A to just have a nontrivial kernel; for the case of γ(t) being constant, optimal decay rates have been obtained, and fast and slow solutions are classified subtly (see [18,19,20]) It was shown in [7] that any bounded solution of (1.1) converges weakly in H1 to a stationary point of the potential energy, when f is a general monotone operator. To the best of our knowledge, there has been little development so far on the study of such problems, with regard to non-autonomous equations, even for strongly positive operator A; see Section 1.3 of [30] about an optimality result for one dimensional wave equation damped by a time-dependent boundary feedback, whose proof relies on d’Alembert’s formula.
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