Elastic Membrane Equation in Bounded and Unbounded Domains ∗ by

The small-amplitude motion of a thin elastic membrane is investigated in n-dimensional bounded and unbounded domains, with n 2 N. Existence and uniqueness of the solutions are established. Asymptotic behavior of the solutions is proved too.


Introduction
The one-dimensional equation of motion of a thin membrane fixed at both ends and undergoing cylindrical bending can be written as 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 and ν, ζ 1 is the nonlinear stiffness of the membrane, ζ 0 is an in-plane tensile load, and (x, t) belongs to Q = Ω × [0, ∞[ with Ω = (0, 1) .All quantities are physically non-dimensionalized, ν, σ, ζ 1 are fixed positive and ζ 0 is fixed non-negative.
Equation (1.1) is related to the flutter panel equation, i.e., when the internal aerodynamic pressure of plate motion ρ is assumed negligible.In equation (1.2) it means that the sum √ ρδu t + ρu x is negligible.
From the mathematical viewpoint, this hypothesis does not have a significant influence in the formulation (1.2) when the interest is to obtain existence and asymptotic behavior of the solutions, because the hight-order sum u xxxx + νu xxxxt has a dominant performance about √ ρδu t + ρu x .Equation (1.2) arises in a wind tunnel experiment for a panel at supersonic speeds.For a derivation of this model see, for instance, Dowell [12] and Holmes [15].
Existence of global solutions for the mixed problem associated with equation (1.2) was investigated by Hughes & Marsden [18], and with respect to asymptotic stability of solutions, Holmes & Marsden [16] supposing some restrictive hypotheses about the aerodynamic pressure ρ the authors concluded that the derivative of the solution is negative.
The investigation of existence of a solution for the Cauchy problem associated with equation (1.1) in n -dimensional bounded and unbounded domain will be made by the application of diagonalization theorem of Dixmier & Von Neumann.The use of the diagonalization theorem allows us to study the Cauchy problem associated with equation (1.1) independently of compactness, and in this way, the result obtained here leads to the conclusion that the inherent properties of such problem are valid in bounded, unbounded, and exteriors domains.
The use of the diagonalization theorem in the study of Cauchy problem associated with the Kirchhoff equation was initially utilized by Matos [22] to prove existence of a local solution.Subsequently, Clark [8] also utilizing the diagonalization theorem proved existence and uniqueness of a global classical solutions supposing that the initial datum are A-analytics such as Arosio-Spagnolo [1] on bounded domain.
This paper is divided in four sections, where the emphasis is to describe the properties in a mathematically rigorous fashion.In §2, the basic notations are laid out.Section §3 is devoted to investigate the existence and uniqueness of global solutions of the Cauchy problem associated with the equation (1.1).In §4, the asymptotic behavior to the energy of the solutions of the section 3 is established.Finally, in §4 is concerned with applications.

Notation and terminology
We shall use, throughout this paper, the following terminology.Let X be a Banach or Hilbert space, T is a positive real number or T = +∞ and 1 ≤ p ≤ ∞.L p (0, T ; X) denotes the Banach space of all measurable functions u :]0, T [→ X such that t → u(t) X belongs to L p (0, T ) and the norm in L p (0, T ; X) is defined by EJQTDE, 2002 No. 11, p. 1-21, 3 For m ∈ N, C m ([0, T ]; X) represents the space of m-times continuously differentiable functions v : [0, T ] → X, where X can be either R, a Banach space or a Hilbert space.
In the context of hilbertian integral a field of the Hilbert space is, by definition, a mapping λ → H(λ) where H(λ) is a Hilbert space for each λ ∈ R. A vector field on R is a mapping λ → u(λ) defined on R such that u(λ) ∈ H(λ).
We represent by F the real vector space of all vector fields on R and by µ a positive real measure on R.
A field of the Hilbert spaces λ → H(λ) is said to be µ-measurable when there exists a subspace M of F satisfying the following conditions The objects of M are called µ-measurable vector fields.In the following, λ → H(λ) represents a µ-measurable field of the Hilbert spaces and all the vector fields considered are µ-measurable.
The space H 0 = ⊕ H(λ)dµ(λ) will be defined by (2.1)With (2.1) the vector space H 0 becomes a Hilbert space which is called the hilbertian integral, or measurable hilbertian sum, of the field λ → H(λ).
Given a real number , we denote by H the Hilbert space of the vector fields u such that the field λ → λ u(λ) belongs to H 0 .In H we define the norms by Let us fix a separable Hilbert space H with scalar product (•, •) and norm |•, •|.We consider a self-adjoint operator A in H such that (Au, u) ≥ ϑ|u| 2 for all u ∈ D(A) and ϑ > 0.
( With the hypothesis (2.3) the operator A satisfies all the hypotheses of the diagonalization theorem, cf.Dixmier [11], Gelfand & Vilenkin [13], Huet [17] and Lions & Magenes [21].Thus, it follows that there exists a Hilbertian integral and an unitary operator U from H onto H 0 such that The domain D(A ) is equipped with the graph norm and µ is a positive Radon measure with support in ]λ 0 , ∞[ for 0 < λ 0 < ϑ where ϑ is the constant defined in (2.3).

Existence and uniqueness of solutions
Our goal in this section is to inquire the existence and uniqueness of solutions of the Cauchy problem associated with equation (1.1) in a bounded and unbounded domain.Thus, changing the Laplace operator − ∂ 2 ∂x 2 by a self-adjoint, positive and unbounded operator A in a real Hilbert space H, satisfying the hypothesis (2.3) we have the following Cauchy problem where A 1/2 represents the square root of A, A 2 is defined by for all w ∈ D(A).
The existence and uniqueness of solutions for the Cauchy problem (3.1) is guaranteed by Existence.Assuming the hypotheses on the operator A we have by diagonalization theorem that there exists an unitary operator U defined in (2.4) and (2.5) such that U : H → H 0 is an isomorphism.Thus, u is a solution of the Cauchy problem (3.1) if, and only if v = U (u) is a solution of the following system of ordinary differential equations Truncated problem -Local solution.Let p ∈ N be.We denote by H 0,p the subspace of H 0 of the fields v(λ) such that v(λ) = 0, µ -a.e. on the interval [p, +∞[.Under these conditions H 0,p equipped with the norm of H 0 is a Hilbert space.
For each vector field v ∈ H with ∈ R we denote by v p the truncated field associated with v, which is defined on the following way v p be.Hence the system (3.7) is equivalent to where and As M is a locally Lipschitz function then F is also locally Lipschitz function and by Cauchy theorem it follows that there exists a unique local solution V p of the Cauchy problem (3.8) in the class C 1 ([0, T p ] ; H 0,p × H 0,p ).
The interval [0, T p ] will be extended to the whole interval [0, ∞[ as a consequence of the first estimate below.
Estimate I. Taking the scalar product on H 0 of 2v p with both sides of (3.7) 1 and integrating from 0 to t ≤ T p yields v p (t) where (3.9) Estimate II.Taking the scalar product on H 0 of 2v p with both sides of (3.7) 1 , using Cauchy-Schwartz inequality in some terms and the identity where δ is a suitable positive constant to be chosen later.Hence, using the estimate (3.9), the continuous injection (2.6) and hypothesis (3.2) we obtain , where c 0 and c 1 represent positive constants dependent only on initial data.Hence, integrating from 0 to t ≤ T we have Choosing 0 < c 1 δ < 1 2 , using the initial conditions (3.7) 2,3 and the estimate (3.9) we have the estimates v p p∈N belongs to L ∞ (0, T ; H 1 ) , v p p∈N belongs to L 2 (0, T ; H 0 ) . (3.10) The estimates (3.9) and (3.10) are sufficient to take the limit in (3.7) 1 .
Limit of the truncated solutions.From estimates (3.9) and (3.10) it follows Hence, the sequences of real functions (f p ) p∈N and (g p ) p∈N defined by for all t ∈ [0, T ], are continuous on [0, T ].On the other hand, given t and s in [0, T ] we get As a consequence of (3.11), (3.12) and the Arzelá-Ascoli theorem there are subsequences of (f p ) p∈N and of (g p ) p∈N , which we will still continue to denote by (f p ) p∈N and (g p ) p∈N respectively, and functions f, g ∈ C 0 ([0, T ]; R) such that Hence, from estimates (3.9), (3.10) and hypothesis (3.2) we obtain that there exists a function v such that From the preceding convergence (3.14) and taking the limit in (3.7) 1 yields To do this we will prove the following result Proof.The function w p previously defined satisfies (3.17) Taking the scalar product on H 0 of w p with both sides of (3.17) 1 yields 1 2 Using the estimate (3.9), the convergence (3.13) and the continuity of the function M we have where c > 0 is a constant independent of p and 0 < ≤ 1 is a suitable constant.Observing that ν w p (t) 2 1 ≥ 0 for all t ∈ [0, T ] and g ∈ C 0 ([0, T ]; R), we get by substitution of (3.19) , where c is a general positive real constant independent of p. Hence, integrating form 0 to t ≤ T , using (3.17) Hence, from estimate (3.9) and inequality (2.7) we get Thus, from convergence (3.13) and (3.16) we obtain Analogously, we obtain As a consequence of the convergence the initial conditions (3.6) 2 hold.Therefore, the function v : [0, T ] → H 0 is a solution of the Cauchy problem (3.6) in the sense of definition 3.1.
Uniqueness.If v 1 and v 2 are two solutions of the Cauchy problem (3.6) then the function w = v 1 − v 2 satisfies the problem Taking the scalar product on H 0 of w with both sides of the equation (3.22) 1 , observing that ν |w (t)| 2 1 ≥ 0 for all t ∈ [0, T ], v and v belong to L ∞ (0, T ; H 1 ) and also using (2.7) and (3.2) yields Thus we obtain 1 2 Hence, integrating from 0 to t ≤ T yields . From this and Gronwall's inequality we have w = 0 for all t ∈ [0, T ].
Global solutions.Let us justify that the function limit v is a solution of the Cauchy problem (3.6) for all t in [0, ∞[.In fact, we proved that the function limit v is a unique solution of the Cauchy problem (3.6) in the sense of L 2 (0, T ; H 0 ).It means that for all θ ∈ D(0, T ) and φ eigenvector of A the following equation holds From estimate (3.9) we have that v belongs to L ∞ (0, ∞; H 1 ) and v belongs to L ∞ (0, ∞; H 0 ) ∩ L 2 (0, ∞; H 1 ) .Thus, we obtain that and consequently Therefore, from (3.10), (3.23) and uniqueness of solutions we have that the equation (3.6) 1 is verified in L 2 (0, ∞; H 0 ).This way we conclude that and from convergence (3.

Asymptotic behavior
The aim of this section is to prove that the total energy associated with the solutions of the Cauchy problem (3.1) has exponential decay when the time t goes to +∞.
For the sake of simplicity we will utilize for (3.1) 1 the representation +A 2 u(t) + νA 2 u (t) = 0 for all t ≥ 0. (4.1) The total energy of the system (4.1) is given, for all t ≥ 0, by Therefore, the energy E(t) is not increasing.To obtain the asymptotic behavior of E(t) we will use the method idealized by Haraux-Zuazua [14], see also, Komornik-Zuazua [20].Thus, we can prove the following result where ω = ω( ) > 0, Λ = 2E(0) + F (0).The function F (t) and the constants are defined by and c > 0 is the constant of the immersion of D(A) into H.
Proof.For each > 0 we consider the auxiliary function: in the second term of the right-hand side of the identity above and using the definition of the function F yields for all t ≥ 0. Hence, (4.3) and (4.5) 2 we obtain Using the definitions of the functions E, E and F it is easy to see that there exists a real positive constant c 0 = c 0 ( ) > 0 such that Thus, from inequalities (4.9) and (4.10) there exists a suitable real positive constant ω = ω( ) > 0 such that Therefore, (I) The theorem 3.1 is still valid if we suppose the operator A satisfying the property (Au, u) ≥ 0 for all u ∈ D(A) instead of ellipticity's propriety (2.3).In fact, in this case, we consider the operator A = A + I, where I is the identity operator on the Hilbert space H, and is a suitable constant such that 0 < ≤ 1.Under these conditions the operator A satisfies the hypotheses of the diagonalization theorem and the solution for the Cauchy problem (3.1) will be obtained as a limit of the family u of the solutions to the Cauchy problem +A 2 u (t) + νA 2 u (t) = 0 for all t ≥ 0, u (0) = u 0 and u (0) = u 1 . (5.1) The convergence of the family u of the solutions of ( where v (t) = U (u (t)) and c is a positive constant independent of and t.Consequently, we have the same convergence of (3.14) for u .
As an application of Theorem 3.1, Theorem 4.1 and the preceding commentary (I) we have the particular cases.

. 4 )
Equation (1.4) has also been extensively studied by several authors in both {1, 2, • • • , n} -dimensional cases and general mathematical models in a Hilbert space H .Both local and global solutions have been shown to exist EJQTDE, 2002 No. 11, p. 1-21, 2

Theorem 3 . 1
Let u 0 and u 1 belong to D(A).If the operator A is a selfadjoint, unbounded and positive on a real Hilbert space H satisfying the hypothesis (2.3) and the real function M satisfies the hypothesis (3.2), then there exists a unique function u : [0, ∞[→ H solution of the Cauchy problem (3.1) in the sense of the definition 3.1 in the class