New general decay results for a von Karman plate equation with memory-type boundary conditions

In this paper we consider a von Karman plate equation with memory-type boundary conditions. By assuming the relaxation function \begin{document}$ k_i $\end{document} \begin{document}$ (i = 1, 2) $\end{document} with minimal conditions on the \begin{document}$ L^1(0, \infty) $\end{document} , we establish an optimal explicit and general energy decay result. In particular, the energy result holds for \begin{document}$ H(s) = s^p $\end{document} with the full admissible range \begin{document}$ [1, 2) $\end{document} instead of \begin{document}$ [1, 3/2) $\end{document} . This result is new and substantially improves earlier results in the literature.

In the past decades, the mathematical analysis of Kirchhoff plates, which contain global existence, uniqueness and stability, with different boundary feedbacks was considered by many authors, see, for instance, Chueshov and Lasiecka [5,6], Favini et al. [7], Komornik [12], Lagnese [13,14], Lasiecka [15], Muñoz Rivera [18], Jorge Silva et al. [8,9]. For system (1)- (7), if α = 0, Santos and Junior [29] established the energy decays exponentially (polynomially) if the kernel g decays exponentially (polynomially). But they considered the following assumptions where a, b > 0 are constants and k i are the resolvent kernel of − g i gi(0) . Mustafa and Abusharkh [23] by assuming u 0 = 0 on one part of boundary and considering a more general assumption extend the decay result in Santos and Junior [29], where H(t) > 0 is strictly increasing and strictly convex near the origin with H(0) = H (0) = 0, was introduced by Alabau-Boussouira and Cannarsa [1]. They also considered a special case H(t) = t p and obtained the decay result for p ∈ [0, 3/2). When α = 0, Park and Park [27] proved the global existence and regularity of solution and established exponential stability. Park [24], using the same assumption on the kernel in Mustafa and Abusharkh [23], established a general decay result of energy and also extended the result to the case u 0 = 0 on one part of boundary. For more results on memory-type of von Karman equation, we refer to [4,10,11,19,20,26,25,28,30].
In this paper, we consider the von Karman plate equation with memory-type boundary conditions (1)-(7), by assuming the relaxation function k i (i = 1, 2) with minimal conditions on the L 1 (0, ∞), i.e., k i (t) ≥ η(t)H i (−k i (t)), where H 1 and H 2 are two linear or strictly increasing and strictly convex functions of class C 2 (R + ). We establish an optimal explicit and general energy decay result. In particular, the energy result holds for H(s) = s p with the full admissible range [1,2) instead of [1, 3/2). Hence our results extend and improve the stability results in [23] and in [24,27]. The arguments in this paper mainly rely on Lyapunov functional method and some properties of convex function developed by Alabau-Boussouira and Cannarsa [1] and Lasiecka and Tataru [16].
The paper is organized as follows. In Section 2, we give some assumptions used in this paper. In Section 3, we state our main results. The proof of stability result will be given in Section 4.

2.
Preliminaries. In this paper, we use the standard Lebesgue and Sobolev spaces with their usual scalar products and norms. We denote the norm of Banach space X by · X . For convenience, we write · L 2 (Ω) and · L 2 (Γ1) by · and · Γ1 , respectively. We define the spaces W andW by The bilinear symmetric form a(ω, φ) is defined by Since Γ 0 = 0, we have that there exist two positive constants c 1 and c 2 such that i.e., Ω a(u, u)dx is equivalent to the H 2 (Ω) norm on W , see [5]. Then combining Sobolev imbedding theorem, we get for u ∈ W , where c p > 0 andc p > 0 are embedding constants. For von Karman bracket and Airy stress function, we have the following two lemmas, see [3,5,7].
The following lemma plays a crucial role in proving our result, which can be found in [5]. This is very deep result of [7].
The following lemma is very useful in the proof of our main result, one can find in [27].
Here, we introduce our assumptions. (A1) There exists a fixed point x 0 ∈ R 2 and some constant δ > 0 such that for As in [17,21,22], we make the following assumptions on the kernels k 1 and k 2 .
(A2) The functions k i (i=1,2): R + → R + are nonincreasing and twice differentiable functions satisfying for any t ≥ 0, (A3) There exist two C 1 functions H i : R + → R + which are linear or are strictly increasing and strictly convex functions of class where η i (t) are C 1 nonincreasing continuous functions.
) is nonincreasing and nonnegative, we arrive at lim t→+∞ (−k i (t)) = 0. Hence we can get for some There exist some positive constants a i and b i such that for any t ∈ [0, t 1 ], Then for any t ∈ [0, t 1 ], which gives us for any t ∈ [0, t 1 ], where d is a positive constant.
3. Main results. For completeness, we give the well-posedness result in the following theorem proved in [27].
The total energy of the system is defined by where The stability result is stated in the following theorem.
where the positive constant µ 1 , µ 2 > 0 and , then the total energy E(t) satisfies that for any t > 0, where c 1 , c 3 and c 2 ≤ 1 are positive constants, and p = max{p 1 , p 2 }.

Remark 2.
Our result is obtained under a much larger class of kernels that guarantee the optimal decay rates of the energy. In particular, the energy result holds for H(s) = s p with the full admissible range [1,2) instead of [1, 3/2). It must to be pointed out that, if the viscoelastic term is as internal feedback, Lasiecka and Wang [17] provides the proof for optimal decay rates of second order systems in the full admissible range [1,2).
In the sequel we give some examples to illustrate explicit formulas for the decay rates of the energy.
, we know that the functions H 1 and H 2 satisfies (A3) on (0, r] for any 0 < r < 1. Therefore It follows from (20) 1 that Since c 2 ≤ 1, this is slower rate than [−k i (t)]. In addition, We infer from (20) 2 that for large t This is the same rate as [−k i (t)].

Proof of main result.
In this section, we will give the proof of Theorem 3.2, which is divided into the following two subsections.

Technical lemmas.
Lemma 4.1. The total energy functional E(t) is nonincreasing and satisfies for any t ≥ 0, Proof. See, for example, [24] or [23]. Under the assumptions of Theorem 3.2, for any ε > 0, we have the following estimate: where for i = 1, 2, and 0 < δ i < 1.
The following lemma is crucial to get the optimal energy decay.
satisfies for any t > 0, Proof. A direct computation gives us It follows from Young's and Hölder's inequalities that For i = 1, 2, noting the fact we obtain (28).

Proof of Theorem 3.2.
Proof. Here we define the functional L(t) by where N > 0 is a constant will be chosen later. It is easy to verify that one can take N large (if needed) such that Recalling k i = δ i k i + h i , we infer from (21) and (22) that for any ε > 0, Taking ε > 0 small enough such that 1 − ε 2 c > 0 and 1 − ε δ c > 0.
For i = 1, 2, since −k i > 0 and k i > 0, we obtain that for each s ∈ [0, ∞), By using Lebesgue dominated convergence theorem, we have Thus there exist 0 < γ i < 1 such that if δ i < γ i then At last take N large enough so that and choose δ i > 0 satisfying which gives us α 1 2 N − cC δ1 > 0 and α 2 2 N − cC δ2 > 0.
In [6], the following fact was proved which implies that Ω a(u, u)dx is bounded. Then it allows us to take α so small such that Noting that lim t→+∞ k i (t) = 0 and using (9) and the trace theorem, we can get that there exist a constant β 1 > 0 larger such that for large t 1 > 0, It follows from (17) and (21) that for any t ≥ t 1 , Then by (31), we conclude that there exists a constant m > 0 such that Define the functional F (t) := L(t) + 2cE(t) ∼ E(t), we get from (32) that We will consider the following two cases.
Define E(t) by Combining (28) and (31), we get E(t) ≥ 0 and for any t ≥ t 1 , Therefore there exists some constant β 2 > 0, This gives . Denote [u(t)−u(t−s)] 2 dΓds. Obviously, Without loss of generality assuming t 1 so large that I i (t 1 ) > 0, we obtain Since pi(p−1) pi−1 ≥ p, taking ε < 1 2 q, we obtain Define where q = q q−1 . Then we deduce that there exists some constant q 0 > 0 such that F (t) ≤ −q 0 η(t)F p (t), from which we get there exists a positive constant c 3 such that E(t) ≤ c 3 1 + t t1 η(s)ds Combining (I) and (II) and using the boundedness of η(t) and E(t), we can get (20).