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)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. Mathematics Subject Classification. 35B35, 93D20, 34G20, 35L70, 35L90. Received July 23, 2020. Accepted April 22, 2021.


Introduction
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, 8-17, 24-26, 28, 30, 32, 34, 35] and references therein).
In this paper, we are concerned with an abstract second order non-autonomous evolution equation in a real Hilbert space H, with time dependent damping, as follows    u (t) + Au(t) + γ(t)u (t) + f (u(t)) = 0, ∀t ≥ 0, u(0) = u 0 , u (0) = u 1 , where A is a nonnegative self-adjoint linear operator on H, the nonlinear term f : D(A 1/2 ) → H is assumed to be conservative with polynomial growth, and γu is the time dependent damping with where R 1 and R 2 are two positive constants, and 0 ≤ α 2 ≤ α 1 < 1.
For the study of the asymptotic behaviors and decay rates for the damped autonomous evolution equations, there have been many developments. We refer the reader to, e.g., [2, 4, 6, 12, 13, 15, 17-21, 27, 31, 35] and references therein. Especially, when A is strongly positive, the asymptotic behaviors for the autonomous case of (1.1) have been extensively studied too, cf. [10,22,23,29,30,32,33]. In this paper, 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 H 1 to a stationary point of the potential energy, when f is a general monotone operator. Following the work, it was proved in [31] that any solution energy E(t) satisfies that for everyᾱ < α 1 , as t → +∞, (1.3) under the condition that γ(t) is decreasing, (with a constant c > 0), and γ (t) ≤ −α 1 γ(t) 1 + t , t ≥ t 0 ≥ 0 a.e. (1.5) We note that the two inequalities (1.4) and (1.5) together imply that γ(t) ≤ C(1 + t) −α1 , t ≥ 0 (with a constant C > 0). For the estimate (1.3) was later improved in [4] as However, 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. 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.
In this paper, we devote ourselves to studying the problem.
Let γ(t) be as in (1.6), and f (u) = |u| p u (p > 0). By Theorem 2.4 of [28], the energy has the upper estimate On the other hand, to get a lower bound we may exploit the hyperbolic version of the Dirichlet quotient introduced in [18]. Estimating the time-derivative ofĜ(t), a small perturbation of G with γ as a product factor (see (4.9), Sect. 4), with the aid of either (1.7) or (1.8) we might probably obtain a lower estimate for a nonempty open set of initial data, but under the restriction α < p/(p + 4) or α < 1/3.
Obviously, there is a gap between the exponents −(1 − α)(p + 2)/p and −(1 + α)(p + 2)/p, when α > 0. Actually, it is still unknown what is the best possible upper estimate. By an idea of asymptotic rate-sharpening, we obtain the optimal decay rate of the energy of this abstract second order non-autonomous evolution equation in terms of γ(t) and f . Moreover, as a byproduct, we show that, when f is a monotone operator, some energy decay rates given previously in the literature are optimal.
More explicitly, we obtain upper estimates for the solutions and their energies of (1.1) in a general setting, and single out slow solutions (decaying exactly at the rates corresponding to the upper estimates) with a nonempty open set of initial data. When γ(t) is a constant, the estimates recover those given in Theorems 2.2 and 2.3 of [18]. Moreover, specialized to the case of (1.6), the exponents of the upper and lower bounds coincide and read −(1 + α)(p + 2)/p for all α ∈ [0, 1). In addition, since − (1 + α)(p + 2)/p → −(1 + α) as p → +∞, (1.10) we see that the decay rate in (1.7) is the best possible for a general monotone f (as considered in [4,7,31]). The outline of this paper is the following. In Section 2 we state assumptions and present our main theorems. Section 3 is devoted to formulating several preliminary results. We employ these results in Section 4 to prove the main theorems. Finally, in Section 5 we give two examples of our theorems being applied to Neumann or Dirichlet problems for nonautonomous, semilinear wave equations.

Main theorems
We denote by v, w the inner product of two vectors v, w in H, and by v the H-norm of v, and we define The following is the basic assumptions on γ and f . Assumption (H1) (i) γ ∈ W 1,∞ loc (R + ) satisfies (1.2); (ii) f = ∇F : H 1 → H is a locally Lipschitz continuous gradient operator of some nonnegative functional F on H 1 , with F (0) = 0.
One knows is called a mild solution of problem (1.1), if it satisfies the integral equation Here S(·) : [0, +∞) → L(H) (the space of bounded linear operators on H) is a solution operator for the linear equation with S(0) = 0 and S (0) = I (the identity; the derivative being in the sense of strong topology).
The following result concerning the wellposedness of (1.1) is from Proposition 2.3 of [28].
Proposition 2.1. Assume (H1). Then for every (u 0 , u 1 ) ∈ H 1 × H, problem (1.1) admits a unique mild solution u, which depends continuously on the initial data. In particular, u is a strong solution if (u 0 , u 1 ) ∈ D(A) × H 1 . Moreover, defining the energy (for strong solutions).
For energy decay, more conditions are required.
Theorem 2.2. Let Assumptions (H1) and (H2) hold. Let (u 0 , u 1 ) ∈ H 1 × H, and let u(t) be the unique global mild solution of problem (1.1). Then for some positive functions M 1 and M 2 on R + that are bounded on bounded sets.
For the existence of slow solutions, we need additional conditions.

Assumption (H3)
(1) ker A = {0}, and there exists ζ > 0 such that (2) there exist positive number ξ and M such that The following is our second main result.
for some positive function c 0 depending on u 0 , A 1/2 u 0 and u 1 .

Preliminary results and proofs
Throughout this section,C,C, C 1 , C 2 , . . . , C 17 denote positive constants depending on the values of E(0) and being bounded on bounded sets of the values, and they may be different at different positions.
From (2.2), for strong solutions (and so for mild solution as well, by a density argument). Thus, from (H2)(2) we have for some positive functionC on R + that is bounded on bounded sets. Now, we establish an important lemma with respect to a perturbed energy functional E ε,η (t) (see below), which is useful in the proof of Theorem 2.2. The functional with η(t) = 1 was employed in [18] to establish the upper estimate of the energy when γ(t) = 1.
Lemma 3.1. Assume (H1) and (H2)(1). Let u(t) be a strong solution of (1.1), and let η ∈ W 1,∞ loc (R + ) be a positive function, ε ∈ (0, 1), and Then with somec ∈ (0, 1) and some positive functionC that are bounded on bounded sets. Here, We first estimate I 2 . By (3.1) and the Cauchy-Schwarz inequality we obtain with a constant C 1 > 0. Hence For I 4 , we use Young's inequality, as well as the Cauchy-Schwarz inequality and (3.1) to infer that Since 2r + 2 p+2 = r + 1, this means that Similarly, we have Moreover, in view of (H2)(1) and (2.1) we deduce that with some constant c 1 > 0, so that This gives the estimate of E ε,η (t) as required.
Next, we derive a preliminary upper estimate of energy, by an application of Lemma 3.1.
for some positive function M 1 on R + that is bounded on bounded sets.
Proof. It suffices to show the estimate for strong solutions, by a density argument.
Finally, we make an improvement over the upper estimate of energy in Proposition 3.2, which will play a key role for the case α 1 > 1/2, by following the way as in Section 2 of [31] (see also [7], Sect. 3.1) with effective adaptations for the present situation. Proof. First we write It is clear that k 0 < k * . Applying Proposition 3.2 enables us to obtain Therefore, Integrating by parts yields that We see . Also, Therefore, the maximum of the above formal function of q(T ) Accordingly, we infer, for T ≥ T 0 , with constantsC 0 , C 9 > 0. From (1.2) and (3.10) it follows that Also using (3.10) gives
Since E(t) is decreasing, we see Hence, where If k 1 ≥ k * , then we have obtained the required energy estimate. If not, then this is because (3.12)-(3.14), with k 0 replaced by k 1 , are satisfied, by using (3.15) (instead of (3.10)). Thus, proceeding like this and denoting by n the positive integer satisfying we obtain This ends the proof.

Proof of Theorem 2.2
It suffices to deal with the strong solution case. We divide the proof into three steps.
Step 1. Obtain a better estimate of energy for case α 1 ≤ 1/2. Take η(t) = 1 in Lemma 3.1. Then V 2 (t) = γ(t) is bounded. For V 1 (t) we exploit Proposition 3.2 to deduce that Thus, similarly as in the proof of Proposition 3.2, we can let ε small enough such that This gives that Step 2. Obtain a better estimate of energy for case α 1 > 1/2. Take in Lemma 3.1. Making use of Proposition 3.3 we infer that by (H2) (5). This yields that which implies that whenever ε is small enough. Moreover, using Proposition 3.3 again, we obtain due to (H2) (5) and (1.2). Furthermore, we have by (1.2). These estimates combined with (4.2) indicate that V 2 (t) is bounded, and Therefore, there exists a sufficiently small ε such that (4.3) is satisfied, and Consequently, Step 3. Achieve the desired estimate of energy. We take Hence Moreover, Accordingly, choosing ε small enough we deduce that Therefore,

Proof of Theorem 2.3
We follow the strategy in Section 3.4 of [18] (for the case of γ(t) = 1) to prove the theorem. It is worth noting that exploitations of the (already obtained) fine upper bound estimates will play an important role.
Step 1. Construct a small perturbation of G(t).
Step 4. Estimate K 4 and K 5 in the case α ≥ 1/2. In view of (2.1), (4.4) and (H3) (2), we obtain with positive constants D 6 , D 7 independent of initial data. This means that the estimate (4.14) for K 4 holds too in this case. By virtue of Theorem 2.2, we get where D 8 , D 9 are positive constants independent of initial data. In addition, by (H2) (5). Hence where D 10 , D 11 are positive constants independent of initial data. So the estimate (4.15) for K 5 holds too in this case.
Consequently, for any (u 0 , u 1 ) ∈ S (given by (4.13) and (4.16)) and any T > 0, we have provided u(t) is a strong solution, and , for all t ∈ [0, T ). (4.19) This implies that sup{T > 0 : (4.19) holds} = +∞, due to the continuity of G and the fact that (4.19) is indeed satisfied for some T > 0, as can be seen from the estimate Therefore, we obtain for strong solutions (and so for mild solution as well). Hence, This combined with (3.1) gives (2.4). Thus, we complete the proof.

Applications
In this section, Ω ⊂ R n is a bounded domain with smooth boundary ∂Ω, ν is the unit outward normal on ∂Ω, and γ(t) is a decreasing function on R + satisfying (1.2) with α 1 = α 2 = α.
Example 5.1. We consider the following nonautonomous wave equation with the Neumann boundary condition: where p ≥ 1.