DECAY RATES FOR SECOND ORDER EVOLUTION EQUATIONS IN HILBERT SPACES WITH NONLINEAR TIME-DEPENDENT DAMPING

. 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.

Energy decay of the solutions for second order semilinear or nonlinear damped hyperbolic equations has been extensively studied, when the linear part is governed by a strongly positive operator A; see, e.g., [1,11,12,13,15,17]. When A has a nontrivial kernel (like negative Neumann Laplacian), the situation is quite tricky and different. In [3], Ghisi, Gobbino, and Haraux proved, for the case of linear time-independent damping, namely, for the equation u (t) + Au(t) + u (t) + ∇F (u(t)) = 0, (1.2) that the solutions decay at least as fast as t −1/p , if ∇F (u) meets a sign condition and its norm is comparable to |u| p+1 , and this decay rate is optimal (see also [4,6] for more refined analysis).
In the work, we study the decay property for (1.1) subject to a nonlinear, timedependent damping, and obtain uniform decay rates of the solutions and energies. Specialized to the equation (1.2), this recovers the rates in [3,Theorem 2.2]. While a modified Lyapunov functional was used in the proof of [3, Theorem 2.2], we exploit a different approach, rather focusing on the original energy, showing that some of its power fulfils a weighted integral inequality, and so achieving the final decay estimates. On the other hand, how to show the existence of slow solutions (and so extend the main result of [3]) still remains to be explored.
For related information, we refer to [14], which handles the equation with the time-dependent linear damping γ(t)u (t), both γ and ∇F being monotone. Also, we refer to [7,8,10] for the case of F being analytical or satisfying a gradient inequality of Lojasiewicz type. The outline of this paper is the following. In Section 2 we state notations and assumptions, and present our decay rate theorem, as well as a result of wellposedness. In Section 3 we prove our results. Finally, in Section 4 we give examples 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.
2. Preliminaries and the main result. We assume that H is a real Hilbert space, denote by v, w the inner product of two vectors v, w in H, and by v the H-norm of v. Since A is nonnegative (that is Av, v ≥ 0 for every v ∈ D(A)), the power A 1/2 is well defined, and we define The following is the basic assumptions on the damping and source terms.
, v ≥ 0 for v ∈ H, and is locally Lipschitz continuous, i.e., for some positive function L on R + × R + , which is bounded on bounded sets.
for some positive function L 1 on R + × R + , which is bounded on bounded sets.
The following is the result of wellposedness. 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).
In view of Proposition 2.3, we know that E(t) is nonincreasing because of the properties of γ and g. Hence, Now we present our theorem on decay rates.
, v 1/(k+1) (2.6) for v ∈ H 1 , with constants p > 0, k ≥ 1, and some positive functions M 0 , M 1 on R + , which are bounded on bounded sets. Moreover, we assume that γ ∈ L 1 (R + ), and ∞ 0 γ(t)dt = +∞. (2.7) 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 function M on R + , which is bounded on bounded sets. In [9], asymptotic stability for second-order evolution equations (with the linear parts governed by self-adjoint and coercive operators) is established, when the feedbacks act intermittently. Theorem 2.4 here suits some special cases of intermittent feedbacks. For instance, consider a sequence of intervals in [1, +∞): (a n , b n ), n = 1, 2, 3, · · · , where b n + 2 ≤ a n+1 for each n, and ∞ n=1 n −2 (b n − a n ) = +∞. We define and so γ(t) satisfies the conditions in Theorem 2.4. Clearly, the associated feedback acts intermittently, yet the lengths of the damping-effect intervals growing to infinity.

Proof of Proposition 2.3.
Proof. Let H := H 1 × H, endowed with the inner product

Define two operators by
And for t ≥ 0, set the operator Clearly, problem (1.1) can be rewritten as and it is easy to check that Thus, −A 0 is the generator of a strongly continuous contraction semigroup on H, according to the Lumer-Phillips theorem (cf. Section 4.1 of [16]). Therefore, −A (as a bounded perturbation of −A 0 ) generates a strongly continuous semigroup {T (t)} on H, which is given by (cf., e.g., [2,5]). In addition, given T 0 > 0, we observe that for t, s ∈ [0, T 0 ], for some positive function L 2 on R + × R + that is bounded on bounded sets by the properties of γ, g and ∇F . Therefore, F is a locally Lipschitz continuous operator. Thus, according to Section 6.1 of [16], we know that equation (3.1) with the initial datum U(0) ∈ H has a unique mild solution U(·) = (u(·), v(·)) in a maximal interval [0, T * ), with either T * = +∞, or lim sup namely, U(·) is the solution of the integral equation Also, we know that the solution depends continuously on initial data; moreover, U(·) is a strong solution whenever U(0) ∈ D(A).
Letting E(t) be as in (2.2) and using (3.1), we get, for strong solutions, and hence, The latter holds too for mild solutions, by the continuous dependence of solutions on initial data. From the nonincreasing character of E(t) and our assumption that F (u) ≥ 0, we see that U(t) H can be controlled by E(t), which means that T * = +∞.
Assume that there exist q ≥ 0 and ω > 0 such that Then, for t ≥ 0, Throughout the proof, C, C 1 , C 2 , C 3 , C ε denote generic positive constants, which may depend on E(0) and may vary from line to line. Moreover, we only need to treat the strong solution case. The general case can be handled by a density argument, with the aid of the continuous dependence of solutions on initial data.
We divide the proof into three steps.
By (2.6), we have which gives that for any T > S ≥ 0, As for T S E q+ 1 p+2 γ g(u ) dt, we infer that by using Young's inequality and (3.7), for any T > S ≥ 0 and ε > 0.
Step 3. Obtain the final energy estimates.
Then, we can choose a small ε ensuring that (3.16) holds as well in this case.
We can rewrite (4.1) as an abstract problem of the type (1.1). In fact, let H = L 2 (Ω) be endowed with the usual inner product and norm, and define the operator A : It is clear that A is a self-adjoint nonnegative operator on H, and H 1 : Then, it is obvious that g is Lipschitz continuous on H, and g(v), v ≥ 0 for v ∈ H. So we just need to show (2.6). We observe from (4.2) that for v ∈ H, (here and in the sequel, C > 0 denote a generic constant), whenever |v(x)| ≤ 1. Moreover, by (4.3) we know that Noting that where L h is the Lipschitz constant of h, we see that (2.6) is satisfied.

JUN-REN LUO AND TI-JUN XIAO
For the nonlinear term f (u), we define Apparently F is nonnegative and differentiable at any point u ∈ H 1 , and From (4.5) we infer that for v, w ∈ H 1 , This shows local Lipschitz continuity of ∇F. Moreover, by (4.4) we have and so Thus, (2.4) and (2.5) are easily verified. It is obvious that γ(t) := (t + 1) −β satisfies condition (2.7). Therefore, by Proposition 2.3 and Theorem 2.4 we conclude that for (u 0 , u 1 ) ∈ H 1 (Ω) × L 2 (Γ), the problem (4.1) admits a unique mild solution such that the energy and the solution satisfy the following decay estimates that for t ≥ 0, with some positive function M on R + that is bounded on bounded sets.

Example 4.2.
Consider the initial-boundary value problem for an integrodifferential damped hyperbolic equation where p ≥ 1, λ 1 is the first eigenvalue of the negative Dirichlet-Laplacian on Ω, and γ, h are as in Example 4.1.
Our last example is related to a plate model, subject to guided boundary conditions.  Define g, F as in Example 4.1. We observe (as in (4.11)) that for v, w ∈ H 1 , Therefore, the conditions stated in Proposition 2.3 and Theorem 2.4 are satisfied.