A MEMORY TYPE BOUNDARY STABILIZATION OF A MIDLY DAMPED WAVE EQUATION

1) Mokhtar KIRANE and 2) Nasser-eddine TATAR

1) Université de Picarde Jules Verne,

Faculté de Mathématiques et d’Informatique, LAMFA UPRES A 6119

33, rue Saint Leu, 80039 Amiens Cedex 1, France e-mail: Mokhtar.Kirane@u-picardie.fr

2) Université de Annaba Institut de Mathématiques, B.P 12, 23000 Annaba, Algeria

ABSTRACT: We consider the wave equation with a mild internal dissipation. 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.

Mathematics Subject Classification: 35L20, 35B40 Key words and Phrases: wave equations, internal dissipation, stabilization

1 INTRODUCTION

In this paper we are concerned with the uniform stability of the solution to the following mixed problem:

\{ \begin{matrix}u_{tt}(t, x) + αu_{t}(t, x) = △u(t, x) + g(t, x),  t > 0, x ∈ Ω, \\ \frac{∂u}{∂ν}(t, x) + ∫_{0}^{t} 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, \end{matrix}

(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; \frac{∂}{∂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

\frac{∂p}{∂ν}(t, x) + ζ(x)p_{t}(t, x) = 0,  t > 0,  x ∈ Γ,  (1.2)

\frac{∂p}{∂ν}(t, x) + β(x)p_{t}(t, x) + α(x)p(t, x) = 0,  t > 0,  x ∈ Γ, (1.3)
\begin{matrix}m(x)δ_{tt}(t, x) + d(x)δ_{t}(t, x) + K(x)δ(t, x) = −p(t, x), \\ \frac{∂p}{∂ν}(t, x) + δ_{tt}(t, x) = 0,  t > 0,  x ∈ Γ, \end{matrix}  (1.4)

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

H_{Γ_{1}}^{1}(Ω) = {u ∈ H^{1}(Ω) : u |_{Γ_{1}} = 0}  (1.5)

where H^{1}(Ω) is the usual Sobolev space.

By a real function a(t, x) ∈ L_{loc}^{1}(R_{+}; L^{∞}(Γ_{0})) of positive type we mean a function satisfying

∫_{0}^{T} ∫_{Γ_{0}} v(t) ∫_{0}^{t} a(t − s)v(s)ds dσ dt ≥ 0  (1.6)

for all v ∈ C(R_{+}; H_{Γ_{1}}^{1}(Ω)) 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:

Theorem 1.1 Suppose that Γ_{0} and Γ_{1} are closed in Γ. Let u_{0} ∈ H^{2}(Ω) ∩ H_{Γ_{1}}^{1}(Ω), u_{1} ∈ H_{Γ_{1}}^{1}(Ω), g ∈ W_{loc}^{1,1}(R_{+}; L^{2}(Ω)), h ∈ W_{loc}^{2,1}(R_{+}; L^{2}(Γ_{0})) and h ∈ C(R_{+}; H^{1/2}(Γ_{0})), k ∈ BV_{loc}(R_{+}; C^{1}(Γ_{0})) of positive type, either u_{1} = 0 on Γ_{0} or k is locally absolutely continuous in t, uniformly with respect to x ∈ Γ_{0} and k′ (the derivative of k with respect to t) is in BV_{loc}(R_{+}; L^{∞}(Γ_{0})), then there is a unique solution u ∈ C(R_{+}; H^{2}(Ω)) ∩ C^{1}(R_{+}; H_{Γ_{1}}^{1}(Ω)) ∩ C^{2}(R_{+}; L^{2}(Ω)) 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.

2 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 H_{Γ_{1}}^{1}(Ω) i.e

∃β > 0, ∥v∥_{2}^{2} ≤ β ∥∇v∥_{2}^{2} , for all v ∈ H_{Γ_{1}}^{1}(Ω).  (2.7)

Combined with the trace inequality the preceding inequality (2.7) yields

∃γ > 0, ∫_{Γ_{0}} v^{2}dσ ≤ γ ∫_{Ω} |∇v|^{2} dσ, for all v ∈ H_{Γ_{1}}^{1}(Ω).  (2.8)

We suppose that our boundary material is characterized by the function

k(t, x) = p(x)e^{−t}, t ≥ 0,  x ∈ Γ_{0},  (2.9)

with 0 ≤ p(x) ∈ C^{1}(Γ_{0}) and ∥p(x)∥_{∞} = M. Let us introduce the energy functional

E(u; t) = \frac{1}{2} ∫_{Ω} (|u_{t}|^{2} + |∇u|^{2}) dx + \frac{1}{2} ∫_{Γ_{0}} p(x) (∫_{0}^{t} e^{−(t−s)}u_{s}(s)ds)^{2} dσ.

(2.10)

Differentiating the energy functional (2.10) using (1.1) we obtain

\frac{d}{dt}E(u; t) = −α ∫_{Ω} u_{t}^{2}dx − ∫_{Γ_{0}} p(x) (∫_{0}^{t} e^{−(t−s)}u_{s}(s)ds)^{2} dσ + ∫_{Ω} u_{t}gdx.

(2.11)

Remark 2.1 Note that if g ≡ 0, then it is readily seen that the energy is decreasing.

Next, for ε > 0 we will define

E_{ε}(u; t) = E(u; t) + εφ(u; t), t ≥ 0, (2.12)

where

φ(u; t) = ∫_{Ω} u_{t}udx.  (2.13)

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

|φ(t)| ≤ \frac{1}{2} ∫_{Ω} u_{t}^{2}dx + \frac{1}{2}β ∫_{Ω} |∇u|^{2} dx ≤ (1 + β)E(t).  (2.14)

It then follows that

|E_{ε}(t) − E(t)| ≤ ε(1 + β)E(t), t ≥ 0.  (2.15)

We are now ready to prove our main theorem. Theorem 2.2 Assume that h and k are as above. Let u_{0} ∈ H^{2}(Ω)∩H_{Γ_{1}}^{1}(Ω), u_{1} ∈ H_{Γ_{1}}^{1}(Ω) and g ∈ W_{loc}^{1,1}(R_{+}; L^{2}(Ω)). If

∫_{0}^{t} e^{εωs} (∫_{Ω} g^{2}dx) ds

grows no faster than a polynomial as t → ∞ for some ε satisfying

0 < ε < min \{\frac{2α}{5 + 4α^{2}β} , \frac{2}{1 + 2Mγ} , \frac{1}{1 + β} \}

where β and γ are the constants in (2.7) and (2.8) and 1/2 < ω < 1, then there exists a positive constant C such that

E(t) ≤ Ce^{−εωt}, t ≥ 0. EJQTDE, 1999 No. 6, p. 4 Proof: Differentiating the functional E_{ε}(t) we find

\begin{aligned}Eε^{′}(t) & = & E′(t) + εφ′(t) \\ & = & −α ∫_{Ω} u_{t}^{2}dx − ∫_{Γ_{0}} p(x) (∫_{0}^{t} e^{−(t−s)}u_{s}(s)ds)^{2} dσ \\ & & + ∫_{Ω} u_{t}gdx + ε ∫_{Ω} u_{tt}udx + ε ∫_{Ω} u_{t}^{2}dx. \end{aligned}  (2.16)

Using problem (1.1) we get

∫_{Ω} u_{tt}udx = −α ∫_{Ω} u_{t}udx + ∫_{Ω}(△u)udx + ∫_{Ω} ugdx
= −α ∫_{Ω} u_{t}udx − ∫_{Γ_{0}} p(x)u ∫_{0}^{t} e^{−(t−s)}u_{s}(s)dsdσ
− ∫_{Ω} |∇u|^{2} dx + ∫_{Ω} ugdx.

(2.17)

Making use of the Hölder inequality and the algebraic inequality

ab ≤ λa^{2} + \frac{1}{4λ}b^{2},  a, b ∈ R, λ > 0,  (2.18)

we have the following estimates

∫_{Ω} u_{t}udx ≤ c_{1} ∫_{Ω} u_{t}^{2}dx + \frac{1}{4c_{1}} ∫_{Ω} u^{2}dx  (2.19)
∫_{Ω} u_{t}gdx ≤ c_{2} ∫_{Ω} u_{t}^{2}dx + \frac{1}{4c_{2}} ∫_{Ω} g^{2}dx  (2.20)
∫_{Ω} ugdx ≤ c_{3} ∫_{Ω} u^{2}dx + \frac{1}{4c_{3}} ∫_{Ω} g^{2}dx  (2.21)

\begin{matrix}∫_{Γ_{0}} p(x)u ∫_{0}^{t} e^{−(t−s)}u_{s}(s)dsdσ ≤ \\ c_{4}M ∫_{Γ_{0}} u^{2}dx + \frac{1}{4c_{4}} ∫_{Γ_{0}} p(x) (∫_{0}^{t} e^{−(t−s)}u_{s}(s)ds)^{2} dσ \end{matrix}  (2.22)

Replacing the expression (2.17) into (2.16) and taking into account the estimates (2.19)-(2.22) we obtain

\begin{aligned}Eε^{′}(t) & ≤ & −2εE(t) − (α − 2ε) ∫_{Ω} u_{t}^{2}dx + (αc_{1}ε + c_{2}) ∫_{Ω} u_{t}^{2}dx \\ & & +β (\frac{αε}{4c_{1}} + c_{3}ε) ∫_{Ω} |∇u|^{2} dx + 2Mc_{4}γε ∫_{Ω} |∇u|^{2} dx \\ & & − (1 − ε − \frac{ε}{4c_{4}} ) ∫_{Γ_{0}} p(x) (∫_{0}^{t} e^{−(t−s)}u_{s}(s)ds)^{2} dσ \\ & & + (\frac{ε}{4c_{3}} + \frac{1}{4c_{2}} ) ∫_{Ω} g^{2}dx. \end{aligned}  (2.23)

Note that we have used the inequalities (2.7) and (2.8) in (2.23). Let us choose c_{1} = 2αβ, c_{2} = ε, c_{3} = 1/8β and c_{4} = 1/4Mγ, then (2.23) yields

\begin{aligned}Eε^{′}(t) & ≤ & −εE(t) − \{α − (\frac{5}{2} + 2α^{2}β)ε\} ∫_{Ω} u_{t}^{2}dx + (2βε + \frac{1}{4ε}) ∫_{Ω} g^{2}dx \\ & & − \{1 − (Mγ + \frac{1}{2})ε\} ∫_{Γ_{0}} p(x)^{2} (∫_{0}^{t} e^{−(t−s)}u_{t}(s)ds)^{2} dσ. \end{aligned}

(2.24)

Now we choose ε > 0 so that α − (\frac{5}{2} + 2α^{2}β)ε ≥ 0 and 1 − (Mγ + \frac{1}{2})ε ≥ 0, i.e.,

ε ≤ min \{\frac{2α}{5 + 4α^{2}β} , \frac{2}{1 + 2Mγ} \} .  (2.25)

Hence,

Eε^{′}(t) ≤ −εE(t) + K(ε, β) ∫_{Ω} g^{2}dx.  (2.26)

It follows from (2.15) that

(1 − (1 + β)ε)E(t) ≤ E_{ε}(t) ≤ (1 + (1 + β)ε)E(t), t ≥ 0.  (2.27)

If moreover ε > 0 satisfies ε < 1/(1 + β), let a be any real number such that 0 < a ≤ 1 − (1 + β)ε, then

aE(t) ≤ E_{ε}(t) ≤ (2 − a)E(t), t ≥ 0.  (2.28)

Using (2.28) in (2.16) we deduce

Eε^{′}(t) ≤ −\frac{ε}{2 − a}E_{ε}(t) + K(ε, β) ∫_{Ω} g^{2}dx.  (2.29)

Consequently,

E_{ε}(t) ≤ {E_{ε}(0) + K(ε, β) ∫_{0}^{t} e^{εωs} (∫_{Ω} g^{2}dx) ds} e^{−εωt},  t ≥ 0, (2.30)

where ω = 1/(2 − a). Once again in view of (2.28) we infer from (2.30) that

E(t) ≤ \{\frac{2 − a}{a}E(0) + \frac{K(ε, β)}{a} ∫_{0}^{t} e^{εωs} (∫_{Ω} g^{2}dx) ds\} e^{−εωt}, t ≥ 0. (2.31)

The proof is now complete.

Remark 2.2 It is clear from the proof that α may depend on the spatial variable x.

Remark 2.3 If the mild damping is in the boundary instead of the equation, i.e,

\{ \begin{matrix}u_{tt}(t, x) = △u(t, x) + g(t, x), t > 0, x ∈ Ω, \\ \frac{∂u}{∂ν}(t, x) + αu_{t}(t, x) + ∫_{0}^{t} k(t − s, x)u_{t}(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, \end{matrix}

then considering the energy functional (2.10) we proceed as in [2] with α(x) = m(x).ν(x) where m(x) = x − x^{0}, x^{0} ∈ R^{n}, and

\begin{matrix}Γ_{0} = \{x ∈ Γ : m(x).ν(x) > 0\}, \\ Γ_{1} = \{x ∈ Γ : m(x).ν(x) ≤ 0\}. \end{matrix}

The appropriate perturbed energy functional is

E_{ε}(u; t) = E(u; t) + ε ∫_{Ω} u_{t} {(n − 1)u + 2(m(x).∇u)} dx, t ≥ 0.

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]).

References

[1] A. B. Aliev and A. Kh. Khanmamedov, Energy estimates for solutions of the mixed problem for linear second-order hyperbolic equations, Math. Notes, Vol. 59, No 4 (1996), 345-149.

[2] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures et appl., 69 (1990), 33-54.

[3] J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations, Adv. Math., 22 (1976), 278-304.

[4] G. Propst and J. Prüss, On wave equations with boundary dissipation of memory type, J. Integral eq. Appl., Vol. 8, No 1 (1996), 99-123.

[5] J. Prüss, Evolutionary integral equations and applications, Birkhäuser Verlag, Basel, 1993.

[6] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim., Vol 28, No 2 (1990), 466-477.