Elastic Membrane Equation with Dynamic Boundary Conditions and Infinite Memory

where Ω is a bounded domain in R with smooth boundary ∂Ω such that ∂Ω = Γ0 ∪ Γ1, Γ̄0 ∩ Γ̄1 = ∅ and Γ0, Γ1 have positive measure λn−1 (Γi) , i = 0, 1, ν denotes the unit outer normal vector pointing towardthe exterior of Ω and M(t) = ξ0 +ξ1 ‖∇u (t)‖22 +σ (∇u (t) ,∇ut (t)), where u is the plate transverse displacement, x is the spatial coordinate in the direction of the fluid flow, and t is the time. The viscoelastic structural damping terms are denote by σ, ξ1 is the nonlinear stiffness of the membrane, ξ0 is an in-plane tensile load. All quantities are physically non-dimensionalized ξ0, ξ1, σ and α are fixed positive. Equation (1.1) is related to the flutter panel equation with memory term this equation arises in a wind tunnel experiment for a panel at supersonic speeds. For a derivation of this model see, for instance, Dowell [14] Holmes [24, 25], Bass [5] . For more results concerning Balakrishnan-Taylor equation, one can refer to Zaräı and Tatar [2, 3], For viscoelastic wave equation with Dirichlet boundary condition, the problems are truly overworked. Many existence and stability results have been established, Cavalcanti and Oquendo [10], Fabrizio and Polidoro [15], Messaoudi [30, 34]. For linear Cauchy viscoelastic problem, one can refer to Kafini and Mustafa [26]. With respect to viscoelastic wave equation with boundary stabilization, Cavalcanti [8 − 11] considered the following system


Introduction
The objective of this work is to study the following problem (1.1) where Ω is a bounded domain in R n with smooth boundary ∂Ω such that ∂Ω = Γ 0 ∪ Γ 1 ,Γ 0 ∩Γ 1 = ∅ and Γ 0 , Γ 1 have positive measure λ n−1 (Γ i ) , i = 0, 1, ν denotes the unit outer normal vector pointing towardthe exterior of Ω and M (t) = ξ 0 + ξ 1 ∇u (t) 2 2 + σ (∇u (t) , ∇u t (t)), where u is the plate transverse displacement, x is the spatial coordinate in the direction of the fluid flow, and t is the time. The viscoelastic structural damping terms are denote by σ, ξ 1 is the nonlinear stiffness of the membrane, ξ 0 is an in-plane tensile load. All quantities are physically non-dimensionalized ξ 0 , ξ 1 , σ and α are fixed positive. Equation (1.1) is related to the flutter panel equation with memory term this equation arises in a wind tunnel experiment for a panel at supersonic speeds. For a derivation of this model see, for instance, Dowell [14] Holmes [24,25], Bass [5] . For more results concerning Balakrishnan-Taylor equation, one can refer to Zaraï and Tatar [2,3], For viscoelastic wave equation with Dirichlet boundary condition, the problems are truly overworked. Many existence and stability results have been established, Cavalcanti and Oquendo [10], Fabrizio and Polidoro [15], Messaoudi [30,34]. For linear Cauchy viscoelastic problem, one can refer to Kafini and Mustafa [26]. With respect to viscoelastic wave equation with boundary stabilization, Cavalcanti [8 − 11] considered the following system the authors first proved the global existence of solutions, and obtained the energy decays exponentially if p = 1 and decays polynomially if p > 1.The results were generalized by Cavalcanti et al. [9]. They obtained the same results without imposing a growth condition on h and under a weaker assumption on g Messaoudi and Mustafa [26] extended these results and established an explicit and general decay rate result by exploiting some properties of convex functions. Recently, using the same method as in [26], Messaoudi et al. [33] considered the above wave system with infinite memory ∞ 0 g(s)∆u(t − s)ds and obtained a general decay result using multiplier method. Gerbi and Said-Houari [21] studied a viscoelastic wave equation with dynamic boundary conditions of the form Using the Faedo-Galerkin method and fixed point theorem, they proved the existence and uniqueness of a local in time solution, and proved the solution exists globally in time under some restrictions on the initial data. They also proved if α > 0, the solution is unbounded and grows as an exponential function, if α = 0, then the solution ceases to exist and blows up in finite time. Ferhat and Hakem [11] considered the same viscoelastic wave equation as in [14] but with the following dynamic boundary conditions They established a general decay result by introducing suitable energy and Lyapunov functionals and some properties of convex functions. Ferhat and Hakem [12] investigated the following system x ∈ Ω, and established an exponential decay result of energy by exploiting the frequency domain method which consists in combining a contradiction argument and a special analysis for the resolvent of the operator under the assumption −ζ 0 g(t) ≤ g ′ (t) ≤ ζ 0 g(t). For more results concerned with wave equation with boundary stabilization, one can refer to Doronin and Larkin [13], Muñoz Rivera and Andrade [35], Gerbi and Said-Houari [20 − 22], Liu and Yu [29], Since there are few works on wave equation with dynamic boundary conditions, source term and a nonlinear weak damping localized on a part of the boundary Elastic Membrane Equation

3
and past history, motivated by above scenario, we study in the present work the stability of solutions to problem (1.1) − (1.4). The main objective of the present work is to establish an explicit and general decay result using multiplier method and some properties of convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term. We end this section by establishing the usual history setting of problem (1.1) − (1.4). Following the same arguments of Dafermos [12], we introduce a new variable Assuming ξ 0 − ∞ 0 g(s)ds = l then we can get a new system, which is equivalent to problem (1.1) The rest of this paper is as follows. In Sect.2, we give some assumptions and our main results. In Sect.3, we establish the general decay result of the energy. In this paper we will use a lot of concepts and techniques contained in Feng [16].

Assumptions and Main Results
In this section, we present some materials and assumptions used in this paper. L q (Ω), (1 ≤ q ≤ ∞), and H 1 (Ω) denote Lebesgue integral and Sobolev spaces . q and . q,Γ1 are the norm in the space L q (Ω) and L q (Γ 1 ), respectively. For simplicity, we write . and . Γ1 instead of . 2 and . 2,Γ1 , respectively C is used to denote a generic positive constant. Denote then we have the embedding H 1 Γ0 (Ω) ֒→ L 2 (Γ 1 ) . We will usually use the following Green's formula To deal with the new variable η, we introduce a weighted L 2 space which is Hilbert space endowed with inner product and norm The phase spaceĤ is defined byĤ In the sequel, we shall give some assumptions. For the relaxation function g, we assume: (A1) g(t) : R + ֒→ R + is a nonincreasing C 1 function satisfying In addition, there exists an increasing strictly convex function G : is a nondecreasing C 0 function such that there exists a strictly increasing function h 0 ∈ C 1 (R + ) with h 0 (0) = 0 and positive constants c 1 , c 2 and ε such that In addition, we assume that (A4) There exists a positive constant m 0 such that The same arguments as in [5], [15] and [32], we can prove the global existence of solutions to problem (1.5) − (1.8) given in the following theorem.
The energy functional of problem (1.5) − (1.8) is defined by Then we can get the stability result of energy to problem (1.5) − (1.8) given in following theorem. where

General Decay
In this section, we shall study the general decay of energy to problem (1.5) − (1.8) to prove Theorem 2. For this purpose, we need the following technical lemmas. Lemma 3.1. Under the assumptions of T heorem2, the energy functional E(t) is non-increasing and satisfies that for any t ≥ 0, Proof. Multiplying (1.5) by u t , and using integration by parts, boundary conditions (1.6) − (1.7) and Green's formula, we can obtain that 2), we can get the desired estimate (3.1). Using (A2), we know that h(u t )u t ≥ 0. Then E ′ (t) ≤ 0. The proof is complete.

Lemma 3.3. Define the functional ψ (t) as
Under the assumptions of Theorem 2, then the functional ψ (t) satisfies for any δ 2 > 0, Proof. Differentiating ψ (t) with respect to t, we can obtain that which, using (1.7), yields Since E(t) is nonincreasing, then we can infer from (2.9) that which gives us Performing Hölder's and Young's inequalities, (2.5) and (2.8), we infer that for any δ 3 > 0 Combining (3.15) − (3.19) with (3.13), we have for any, δ 3 > 0 we can derive that The same arguments give us The proof is done.
Define the functional L (t) by where ε 1 and ε 2 are positive constants will be chosen later. It is easy to verify that for ε 1 > 0 and ε 2 > 0 small enough, Proof. It follows from (3.1), (3.5) and (3.11) that for any t ≥ 0, At this point we choose δ 2 > 0 satisfying ǫ For any fixed ,δ 2 > 0 we take ε 2 > 0 small enough so that (3.23) remains valid and further. For fixed δ 2 and ε 2 , we pick ε 1 > 0 so small that (3.23) remains valid and further Therefore, there exists a positive constant m such that for any t ≥ 0, which completes the proof.
The same arguments as in [17], we can get the following lemma.
. (3.26) Proof. of T heorem2. We distinguish the following two cases to prove T heorem2.
Following the arguments as in [20], we first suppose that max {r, h 0 (r)} < ε otherwise we choose r smaller.