A MEMORY TYPE BOUNDARY STABILIZATION OF A MIDLY DAMPED WAVE EQUATION

We consider the wave equation with a mild internal dissipa-tion. It is proved that any small dissipation inside the domain is sufficient to uniformly stabilize the solution of this equation by means of a nonlinear feedback of memory type acting on a part of the boundary. This is established without any restriction on the space dimension and without geometrical conditions on the domain or its boundary.


INTRODUCTION
In this paper we are concerned with the uniform stability of the solution to the following mixed problem: u tt (t, x) + αu t (t, x) = u(t, x) + g(t, x), t > 0, x ∈ Ω, ∂u ∂ν (t, x) + t 0 k(t − s, x)u s (s, x)ds = h(t, x), t > 0, x ∈ Γ 0 , u(t, x) = 0, t > 0, x ∈ Γ 1 , u(0, x) = u 0 (x), u t (x) = u 1 (x), x ∈ Ω, (1.1) where Ω is a bounded domain in R n (the n-dimensional Euclidean space, n ≥ 1) with a boundary Γ = ∂Ω of class C 2 ; (Γ 0 , Γ 1 ) is a partition of Γ such that int(Γ 1 ) = ∅; ν(x) denotes the outward normal vector to Γ at x ∈ Γ; ∂ ∂ν is the normal derivative on Γ; α is a positive number and g, h, u 0 , u 1 are given functions; is the Laplacian with respect to the spatial variable x and the subscript t denotes differentiation with respect to the variable t.
Problem (1.1) models, for instance, the evolution of sound in a compressible fluid with reflection of sound at the surface of the material.The boundary condition in (1.1) is general and covers a fairly large variety of different physical configurations.The physical meaning of this boundary condition as well as the following three particular cases is discussed in [4].See also references therein for questions of existence, uniqueness, regularity and asymptotic behavior.In [1] the exponential decay of the energy of problem (1.1) with the boundary condition (1.2) in the case ζ(x) ≡ C a positive constant, g ≡ h ≡ 0 and α < 0 on the n-dimensional open unit cube was established.More delicate is the same problem with boundary condition (1.2), g ≡ h ≡ 0 without internal damping i.e α = 0.This is discussed in Komornik and Zuazua [2] and Zuazua [6].
Inspired by the method developed in [2], we shall prove exponential decay for solutions of problem (1.1) (h ≡ 0) using an appropriately chosen energy functional.In fact, we shall uniformly stabilize the solution of the wave equation by a nonlinear feedback of memory type acting on a part of the boundary provided the equation contains a mild damping (however small it is) in the interior of the domain.
Let R + denote the set of nonnegative real numbers and where H 1 (Ω) is the usual Sobolev space.By a real function a(t, x) ∈ L 1 loc (R + ; L ∞ (Γ 0 )) of positive type we mean a function satisfying ) and for every T > 0. See [3] for more information on functions of positive type.
In [4], Propst and Prüss have reformulated problem (1.1) (with α = 0) as an integral equation of variational type and then have used results and methods developed in the second author's monograph [5] to derive, among others the following theorem: ) and u(t, x) satisfies (1.1) for all t ≥ 0 and almost all x.W m,p and C m are the usual Sobolev space and the space of continuously differentiable functions up to order m respectively.BV is the space of functions of bounded variation.

Exponential decay
In this section we assume the existence of a regular strong solution to problem (1.1) in the sense of the preceding theorem with h ≡ 0.
Note that the Poincaré inequality holds in Combined with the trace inequality the preceding inequality (2.7) yields We suppose that our boundary material is characterized by the function Let us introduce the energy functional ( For the sake of brevity, we will write E ε (t) for E ε (u; t) and ϕ(t) for ϕ(u; t).Using the Poincaré inequality (2.7) we have It then follows that

.15)
We are now ready to prove our main theorem.
In this case we do not impose to the function h (and g) to vanish identically, we are restricted however to the space dimension condition n ≤ 3 when cl(Γ 0 ) ∩ cl(Γ 1 ) = ∅ because of the limited validity of Grisvard's inequality (see [2] and [6]).
It is clear from the proof that α may depend on the spatial variable x.