## Existence and boundary stabilization of a nonlinear hyperbolic equation with

## time-dependent coefficients ^{∗}

### M. M. Cavalcanti, V. N. Domingos Cavalcanti & J. A. Soriano

Abstract

In this article, we study the hyperbolic problem
K(x, t)utt−P_{n}

j=1 a(x, t)uxj

+F(x, t, u,∇u) = 0
u= 0 on Γ_{1}, ^{∂u}_{∂ν} +β(x)ut= 0 on Γ_{0}

u(0) =u^{0}, ut(0) =u^{1} in Ω,

where Ω is a bounded region inR^{n}whose boundary is partitioned into two
disjoint sets Γ_{0},Γ_{1}. We prove existence, uniqueness, and uniform stabil-
ity of strong and weak solutions when the coefficients and the boundary
conditions provide a damping effect.

### 1 Introduction

Let Ω be a bounded domain of R^{n} withC^{2} boundary Γ. Assume that Γ has
a partition Γ_{0},Γ_{1}, such that each set has positive measure, and Γ_{0}∩Γ_{1} is
empty. See the definition of these two sets in (2.1) below, and note that this
condition excludes domains with connected boundary. Our objective is to study
the problem

K(x, t)^{∂}_{∂t}^{2}^{u}2 +A(t)u+F(x, t, u,∇u) = 0 in Q= Ω×]0,∞[ (1.1)
u= 0 on Σ_{1}= Γ_{1}×]0,∞[

∂ν∂uA +β(x)^{∂u}_{∂t} = 0 on Σ_{0}= Γ_{0}×]0,∞[
u(0) =u^{0}, ^{∂u}_{∂t}(0) =u^{1} in Ω,
where A(t) =−P_{n}

j=1 ∂

∂xj

a(x, t)_{∂x}^{∂}

j

.

∗1991 Mathematics Subject Classifications: 35B40, 35L80.

Key words and phrases: Boundary stabilization, asymptotic behaviour.

c1998 Southwest Texas State University and University of North Texas.

Submitted July 6, 1997. Published March 10, 1998.

1

Stability of solutions for this problem with K(x, t) = 1, A(t) = −∆ and F = 0 has been studied by many authors; see for example J. P. Quinn & D. L.

Russell [10], G. Chen [2,3,4], J. Lagnese [6,7], and V. Komornik & E. Zuazua [5] who also studied the nonlinear problem with F = F(x, t, u). To the best of our knowledge, this is the first publication on boundary stabilization with time-dependent coefficients and the nonlinear termF =F(x, t, u,∇u).

Stability of problems with the nonlinear termF(x, t, u,∇u) require a care- ful treatment, because we do not have any information about the influence of integralR

ΩF(x, t, u,∇u)u^{0}dx on the energy
e(t) =1

2 Z

Ω(K(x, t)|u^{0}(x, t)|^{2}+a(x, t)|∇u(x, t)|^{2})dx , (1.2)
or about the sign of the derivativee^{0}(t).

When the coefficients depend on time, there are some technical difficulties that we need to overcome. First, semigroup arguments are not suitable for find- ing solutions to (1.1); therefore, we make use of a Galerkin approximation. For strong solutions, this approximation requires a change of variables to transform (1.1) into an equivalent problem with initial value equals zero. Secondly, the presence of∇uin the nonlinear part brings up serious difficulties when passing to the limit.

The goal of this work is to investigate conditions on the coefficients that lead
to exponential decay of an energy determined by the solution. To this end, we
use the perturbed-energy method developed by V. Komornik & E. Zuazua in
[5]. By establishing adequate hypotheses onK(x, t),a(x, t) and F(x, t, u,∇u),
the above method allow us to solve (1.1) whenβ(x) = (x−x^{0})·ν(x) withx^{0} a
point inR^{n} andν(x) the exterior unit normal.

Our paper is divided in 4 sections. In §2, we establish notation and state our results. In§3, we prove solvability of (1.1) using the Galerkin method. In

§4, we prove exponential decay of solutions.

### 2 Notation and statement of results

For the rest of this article, letx^{0} be a fixed point inR^{n}. Then put
m=m(x) =x−x^{0},

and partition the boundary Γ into two sets:

Γ_{0}={x∈Γ : m(x)·ν(x)≥0}, Γ_{1}={x∈Γ : m(x)·ν(x)<0}. (2.1)
Consider the Hilbert space

V ={v∈H^{1}(Ω) : v= 0 on Γ_{1}},
and define the following:

(u, v) =R

Ωu(x)v(x)dx, |u|^{2}=R

Ω|u(x)|^{2}dx,
(u, v)_{Γ}_{0} =R

Γ0u(x)v(x)dΓ, |u|^{2}_{Γ}_{0}=R

Γ0|u(x)|^{2}dx,
kuk∞= ess sup_{t≥0}ku(t)k_{L}^{∞}_{(Ω)}, u^{0}=u_{t}=^{∂u}_{∂t}, u_{x}_{i} =_{∂x}^{∂u}

i

and

R(x^{0}) = max

x∈Ωkx−x^{0}k (2.2)

Now, we state the general hypotheses.

(A.1) Assumptions on F(x, t, u,∇u). SupposeF : Ω×[0,∞[×R^{n+1}→Ris
an element of the spaceC^{1}(Ω×[0,∞[×R^{n+1}) and satisfies

|F(x, t, ξ, ζ)| ≤C_{0}(1 +|ξ|^{γ+1}+|ζ|) (2.3)
where C_{0} is a positive constant, andζ= (ζ_{1}, ..., ζ_{n}).

Letγ be a constant such that γ >0 for n = 1,2, and 0< γ ≤2/(n−2)
for n ≥ 3. Assume that there is a non-negative function C(t) in the space
L^{∞}(0,∞)∩L^{1}(0,∞), such that

F(x, t, ξ, ζ)η≥ |ξ|^{γ}ξη−C(t)(1 +|ηkζ|), ∀η∈R, (2.4)
F(x, t, ξ, ζ) (m·ζ)≥ |ξ|^{γ}ξ(m·ζ)−C(t)(1 +|ζkm·ζ|). (2.5)
Assume that there exist positive constantsC_{0}, . . . , C_{n}, such that

|Ft(x, t, ξ, ζ)| ≤C_{0} 1 +|ξ|^{γ+1}+|ζ|

, (2.6)

|Fξ(x, t, ξ, ζ)| ≤C_{0}(1 +|ξ|^{γ}), (2.7)

|F_{ζ}_{i}(x, t, ξ, ζ)| ≤C_{i} fori= 1,2, . . . , n . (2.8)
We also assume that there exist positive constantsD_{1}, D_{2}, such that for allη,
ˆ

η in Rand for allζ, ˆζ inR^{n},

(F(x, t, ξ, ζ)−F(x, t,ξ,ˆζ))(η−ˆ η)ˆ ≥ −D_{1}(|ξ|^{γ}+|ξ|ˆ^{γ})|ξ−ξkη−ˆ η|−Dˆ _{2}|η−ˆηkζ−ζ|ˆ .
(2.9)
The following is an example of a functionF that satisfies the above condi-
tions.

F(x, t, u,∇u) =|u|^{γ}u+ϕ(t)
Xn
i=1

sin ∂u

∂x_{i}

,

where ϕis a function sufficiently regular.

(A.2) Assumptions on the initial data.

u^{0}, u^{1}∈V ∩H^{2}(Ω) and ∂u^{0}

∂ν_{A}+β(x)u^{1}= 0 on Γ_{0}.
(A.3) Assumptions on the coefficients.

K∈W^{1,∞}(0,∞;C^{1}(Ω)), a∈W^{1,∞}(0,∞;C^{1}(Ω))∩W^{2,∞}(0,∞;L^{∞}(Ω))
a_{t}, K_{t}∈L^{1}(0,∞;L^{∞}(Ω)), β∈W^{1,∞}(Γ_{0}).

Also assume that there exist positive constantsa_{0},k_{0}, such that

K≥k_{0}, a≥a_{0}, inQ, and β(x)≥0 a.e. on Γ_{0}. (2.10)
For short notation, define

a(t, u, v) =P_{n}

j=1

R

Ωa(x, t)_{∂x}^{∂u}

j

∂x∂vjdx,
a^{0}(t, u, v) =P_{n}

j=1

R

Ωa_{t}(x, t)_{∂x}^{∂u}

j

∂x∂vj dx,
a^{00}(t, u, v) =P_{n}

j=1

R

Ωa_{tt}(x, t)_{∂x}^{∂u}

j

∂x∂vj dx .

We observe that from the above assumptions ona, there exist positive constants
a_{1},a_{2}, and a_{3} such that,

a_{0}|∇u|^{2}≤a(t, u, u)≤a_{1}|∇u|^{2} ∀u∈V and t≥0, (2.11)

|a^{0}(t, u, v)| ≤a_{2}|∇uk∇v| ∀u∈V and t≥0, (2.12)

|a^{00}(t, u, v)| ≤a_{3}|∇uk∇v| ∀u∈V and t≥0. (2.13)
Now, we are in a position to state our results.

Theorem 2.1 Under Assumptions (A1, A2, A3), Problem (1.1) possesses a unique strong solution,u:]0,∞[×Ω→R, such that

u∈L^{∞}(0,∞;V ∩H^{2}(Ω)), u^{0}∈L^{∞}(0,∞;V), andu^{00}∈L^{∞}(0,∞;L^{2}(Ω)).

Now, we present a result on stability of strong solutions, which will be ex- tended to weak solutions. Let

H(t) = k∇a(t)k_{L}^{∞}_{(Ω)}+k∇K(t)k_{L}^{∞}_{(Ω)}

+kat(t)k_{L}^{∞}_{(Ω)}+kKt(t)k_{L}^{∞}_{(Ω)}+C(t).

Theorem 2.2 Assume that there are positive constantsα, r, , θ_{0}, such that for
allt sufficiently large,

Z _{t}

0 exp(θ_{0}s)H(s)ds≤αt^{r}. (2.14)
Then the energy (1.2) determined by the strong solutionudecays exponentially.

This is, for some positive constants δ, , θ_{1},

E(t) =e(t) + 1 γ+ 2

Z

Ω|u(x, t)|^{2}dx≤δexp(−θ_{1}t). (2.15)
Notice that (2.14) requires the integral to have polynomial growth. There-
fore, each term in H(t) behaves as a function of the form Q(t) exp(−βt) with
Q(t) a polynomial andβ > θ_{0}.

An example of a function that satisfies (2.14) isH(t) =texp(−βt). In fact,
Z _{t}

0

exp(θ_{0}s)sexp(−βs)ds

= − t

β−θ_{0}exp(−(β−θ_{0})t)− 1

(β−θ_{0})^{2}exp(−(β−θ_{0})t) + 1
(β−θ_{0})^{2}

≤ αt+δ ,

for some positive constantsαandδ.

Theorem 2.3 Suppose that {u^{0}, u^{1}} is in V ×L^{2}(Ω), and that assumptions
(A1), (A3) hold. Then (1.1) has a unique weak solution, u: Ω×]0,∞[→R, in
the space

C([0,∞);V)∩C^{1}([0,∞);L^{2}(Ω)).
Furthermore, Theorem 2.2 holds for the weak solution u.

Remark Notice that astincreases, (1.1) converges to an equation of constant
coefficients, andF =|u|^{γ}u. Hence, (1.1) can be seen as a disturbance of a much
better known problem, which was studied in [5]. Also note that both equations
have solutions with the same exponential decay, (2.15).

### 3 Existence of strong and weak solutions

In this section, we prove the existence and uniqueness of strong and weak so- lutions to (1.1). First we consider strong solutions, and then using a density argument we extend the same result to weak solutions.

A variational formulation of Problem (1.1) leads to the equation Z

ΩKu^{00}w dx+
Z

Ωa(x, t)∇u∇w dx+ Z

ΩF(x, t, u,∇u)w dx+ Z

Γ0

βu^{0}w dΓ = 0,

for allwin the spaceV.

Strong solutions to (1.1) with the boundary conditionR

Γ0βu^{0}w dΓ can not be
obtained by the method of “special basis”; therefore, bases formed with eigen-
functions can not be used for (1.1). Differentiating the above expression with
respect to tdoes not help, because of the technical difficulties when estimating
u^{00}(0). To avoid these difficulties, we transform (1.1) into an equivalent problem
with initial value equal to zero. In fact, the change of variables

v(x, t) =u(x, t)−φ(x, t) (3.1)
φ(x, t) =u^{0}(x) +tu^{1}(x), t∈[0, T] (3.2)
leads to

K(x, t)v^{00}+A(t)v+F(x, t, φ+v,∇φ+∇v) =f in Q= Ω×(0, T),(3.3)
v= 0 on Σ_{1}= Γ_{1}×(0, T),

∂ν∂vA+β(x)v^{0} =g on Σ_{0}= Γ_{0}×(0, T),

v(x,0) =v^{0}(x,0) = 0, (3.4)

wheref(x, t) =−A(t)u^{0}(x)−tA(t)u^{1}(x), (x, t)∈Ω×[0, T], andg(x, t) =−t_{∂ν}^{∂u}_{A}^{1}.
Note that ifv is a solution of (3.3) on [0,T], thenu=v+φis a solution of
(1.1) in the same interval. From estimates obtained below, we are able to prove
that

|A(t)v(t)|^{2}+|∇v^{0}(t)|^{2}≤C, ∀t∈[0, T]. (3.5)
Thus, from (3.1) and (3.2) we obtain the same inequality (3.5) for the solution
u. Then using standard methods, we extenduto the interval (0,∞). Hence,
it is sufficient to prove that (3.3) has a local solution, which shall be done by
using the Galerkin method.

Let (ω_{ν})_{ν∈N} be a set of functions inV ∩H^{2}(Ω), that form and orthonormal
basis forL^{2}(Ω). LetV_{m}be the space generated byω_{1}, ω_{2}, . . . , ω_{m}and let

v_{m}(t) =
Xm
i=1

g_{jm}(t)ω_{j} (3.6)

be the solution to the Cauchy problem

(K(t)v^{00}_{m}(t), w) +a(t, v_{m}(t), w) + (βv_{m}^{0} (t), w)_{Γ}_{0}
+

Z

ΩF(x, t, v_{m}+φ,∇vm+∇φ)w dx

= (f(t), w) + (g(t), w)_{Γ}_{0}, ∀w∈V_{m}, (3.7)
v_{m}(0) =v_{m}^{0} (0) = 0.

Observe that all the terms in the above expression are well defined. In particular,R

ΩF(x, t, v_{m}+φ,∇vm+∇φ)w dxexists because of (2.3).

By standard methods in differential equations, we can prove the existence of
a solution to (3.7) on some interval [0, t_{m}). Then this solution can be extended
to the close interval by the use of the first estimate below.

### A priori estimates

First Estimate: Takingw= 2v^{0}_{m}(t) in (3.7), we have
d

dt{|p

K(t)v_{m}^{0} (t)|^{2}+a(t, v_{m}(t), v_{m}(t))}+ 2(β, v_{m}^{02}(t))_{Γ}_{0}
+2

Z

ΩF(x, t, v_{m}+φ,∇vm+∇φ)v_{m}^{0} dx

= (K_{t}(t), v^{02}_{m}(t)) +a^{0}(t, v_{m}(t), v_{m}(t)) + 2(f(t), v^{0}_{m}(t))
+2d

dt(g(t), v_{m}(t))_{Γ}_{0}−2(g^{0}(t), v_{m}(t))_{Γ}_{0}.
Integrating the above expression over [0,t], we obtain

|p

K(t)v^{0}_{m}(t)|^{2}+a(t, v_{m}(t), v_{m}(t)) + 2
Z _{t}

0 (β, v_{m}^{0} ^{2}(s))_{Γ}_{0}ds (3.8)

+2
Z _{t}

0

Z

ΩF(x, s, v_{m}+φ,∇v_{m}+∇φ)v^{0}_{m}dx ds

=
Z _{t}

0 (K_{s}(s), v_{m}^{02}(s))ds+
Z _{t}

0 a^{0}(s, v_{m}(s), v_{m}(s))ds
+2

Z _{t}

0 (f(s), v_{m}^{0} (s))ds+ 2(g(t), v_{m}(t))_{Γ}_{0}−2
Z _{t}

0 (g^{0}(s), v_{m}(s))_{Γ}_{0}ds .
Estimate for I_{1}:= 2R_{t}

0

R

ΩF(x, s, v_{m}+φ,∇v_{m}+∇φ)v^{0}_{m}dx ds. We have
I_{1} = 2

Z _{t}

0

Z

ΩF(x, s, v_{m}+φ,∇vm+∇φ)(v_{m}^{0} +φ^{0})dx ds

−2
Z _{t}

0

Z

ΩF(x, s, v_{m}+φ,∇v_{m}+∇φ)φ^{0}dx ds .
From (2.3) and (2.4) it follows that

I_{1} ≥ 2

γ+ 2kv_{m}(t) +φ(t)k^{γ+2}_{L}γ+2(Ω)− 2

γ+ 2kφ(0)k^{γ+2}_{L}γ+2(Ω) (3.9)

−2C
Z _{t}

0

Z

Ω

(1 +|v_{m}^{0} +φ^{0}||∇v_{m}+∇φ|)dx ds

−2C
Z _{t}

0

Z

Ω(1 +|v_{m}+φ|^{γ+1}+|∇v_{m}+∇φ|)|φ^{0}|dx ds .

Substituting (3.9) in (3.8), observing that (2.10), (2.11), (2.12) hold, and noting
that v_{m}(0) =v^{0}_{m}(0) = 0, it follows that

k_{0}|v_{m}^{0} (t)|^{2}+a_{0}|∇v_{m}(t)|^{2}+ 2

γ+ 2kv_{m}(t) +φ(t)k^{γ+2}_{L}γ+2(Ω)+ 2
Z _{t}

0

(β, v_{m}^{0} (s)^{2})_{Γ}_{0}ds

≤ 2

γ+ 2kφ(0)k^{γ+2}_{L}γ+2(Ω)+
Z _{t}

0 (K_{s}(s), v_{m}^{02}(s))ds+a_{2}
Z _{t}

0 |∇v_{m}(s)|^{2}ds
+2

Z _{t}

0 (f(s), v^{0}_{m}(s))ds+ 2(g(t), v_{m}(t))_{Γ}_{0}−2
Z _{t}

0 (g^{0}(s), v_{m}(s))_{Γ}_{0}ds
+2C

Z _{t}

0

Z

Ω(1 +|v_{m}^{0} +φ^{0}||∇vm+∇φ|)dx ds
+2C

Z _{t}

0

Z

Ω(1 +|v_{m}+φ|^{γ+1}+|∇v_{m}+∇φ|)|φ^{0}|dx ds .
Using Young, H¨older and the Schwarz inequalities we obtain

k_{0}|v_{m}^{0} (t)|^{2}+a_{0}

2 |∇v_{m}(t)|^{2}+ 2

γ+ 2kv_{m}(t) +φ(t)k^{γ+2}_{L}γ+2(Ω)+ 2
Z _{t}

0 (β, v^{02}_{m}(s))_{Γ}_{0}ds

≤ L_{0}+L_{1}
Z _{t}

0

|v_{m}^{0} (s)|^{2}+|∇vm(s)|^{2}+kvm(s) +φ(s)k^{γ+2}_{L}γ+2

ds .

From this inequality and the Gronwall’s inequality, we obtain the first estimate,

|v^{0}_{m}(t)|^{2}+|∇v_{m}(t)|^{2}+kv_{m}(t) +φ(t)k^{γ+2}_{L}γ+2(Ω)+
Z _{t}

0 (β, v_{m}^{0} (s)^{2})_{Γ}_{0}ds≤L , (3.10)
whereLis a positive constant independent ofmandt∈[0, T].

Second Estimate: First, we prove thatv_{m}^{00}(0) is bounded in theL^{2}(Ω) norm.

Indeed, consideringt= 0 in (3.7) we obtain

(K(0)v_{m}^{00}(0), w) +a(0, v_{m}(0), w) + (βv_{m}^{0} (0), w)_{Γ}_{0}+R

ΩF(x,0, u^{0},∇u^{0})w dx

= (−A(0)u^{0}, w) ∀w∈V_{m}.

From this inequality and the fact thatv_{m}(0) =v_{m}^{0} (0) = 0, we get
(K(0)v_{m}^{00}(0), w) =−

Z

ΩF(x,0, u^{0},∇u^{0})w dx−(A(0)u^{0}, w), ∀w∈V_{m}.
Withw=v^{00}_{m}(0) in the above equation, we obtain

(K(0), v_{m}^{002}(0)) =−
Z

ΩF(x,0, u^{0},∇u^{0})v^{00}_{m}(0)dx−(A(0)u^{0}, v^{00}_{m}(0))
From this equation, (2.3), and (2.10), we conclude that

k_{0}|v_{m}^{00}(0)|^{2} ≤ C
Z

Ω

(1 +|u^{0}|^{γ+1}+|∇u^{0}|)|v_{m}^{00}(0)|dx+|A(0)u^{0}kv^{00}_{m}(0)|

≤ C(Ω)[1 +|∇u^{0}|^{γ+1}+|∇u^{0}|+|A(0)u^{0}|]|v^{00}_{m}(0|.
That is

|v^{00}_{m}(0)| ≤C(Ω, k_{0})[1 +|∇u^{0}|^{γ+1}+|∇u^{0}|+|A(0)u^{0}|], ∀m∈N.
Therefore,

v_{m}^{00}(0) is bounded inL^{2}(Ω). (3.11)
Taking the derivative of (3.7) with respect tot, it follows that

(K_{t}(t)v_{m}^{00}(t), w) + (K(t)v_{m}^{000}(t), w) +a^{0}(t, v_{m}(t), w) +a(t, v_{m}^{0} (t), w)
(βv_{m}^{00}(t), w)_{Γ}_{0}+

Z

Ω

F_{t}(x, t, v_{m}+φ,∇v_{m}+∇φ)w dx
+

Z

Ω

F_{v}_{m}_{+φ}(x, t, v_{m}+φ,∇v_{m}+∇φ)(v^{0}_{m}+φ^{0})w dx
+

Xn i=1

Z

ΩF_{v}_{m}_{x}_{i}_{+φx}_{i}(x, t, v_{m}+φ,∇vm+∇φ)(v^{0}_{mx}_{i}+φ^{0}_{x}_{i})w dx

= (f^{0}(t), w) + (g^{0}(t), w)_{Γ}_{0}.

Substituting wby 2v^{00}_{m}(t) in the above expression it results that
d

dt{|p

K(t)v^{00}_{m}(t)|^{2}+a(t, v_{m}^{0} (t), v_{m}^{0} (t)) + 2a^{0}(t, v_{m}(t), v^{0}_{m}(t))}+ 2(β, v_{m}^{002}(t))_{Γ}_{0}

= −(K_{t}(t), v_{m}^{002}(t)) + 2a^{0}(t, v_{m}^{0} (t), v_{m}^{0} (t)) + 2a^{00}(t, v_{m}(t), v_{m}^{0} (t))
+a^{0}(t, v^{0}_{m}(t), v^{0}_{m}(t))−2

Z

Ω

F_{t}(x, t, v_{m}+φ,∇v_{m}+∇φ)v_{m}^{00} dx

−2 Z

Ω

F_{v}_{m}_{+φ}(x, t, v_{m}+φ,∇v_{m}+∇φ)(v_{m}^{0} +φ^{0})v^{00}_{m}dx

−2 Xn i=1

Z

ΩF_{v}_{m}_{x}_{i}_{+φx}_{i}(x, t, v_{m}+φ,∇vm+∇φ)(v_{mx}^{0} _{i}+φ^{0}_{x}_{i})v^{00}_{m}dx
+2(f^{0}(t), v^{00}_{m}(t)) + 2d

dt(g^{0}(t), v_{m}^{0} (t))_{Γ}_{0}.

Integrating both sides of this equation over [0,t] and observing that v^{0}_{m}(0) = 0,
we obtain

|p

K(t)v^{00}_{m}(t)|^{2}+a(t, v_{m}^{0} (t), v_{m}^{0} (t)) + 2
Z _{t}

0 (β, v_{m}^{002}(s))_{Γ}_{0}ds (3.12)

= |p

K(0)v_{m}^{00}(0)|^{2}−2a^{0}(t, v_{m}(t), v^{0}_{m}(t))−
Z _{t}

0 (K_{s}(s), v^{002}_{m}(s))ds
+3

Z _{t}

0 a^{0}(s, v_{m}^{0} (s), v^{0}_{m}(s))ds+ 2
Z _{t}

0 a^{00}(s, v_{m}(s), v_{m}^{0} (s))ds

−2
Z _{t}

0

Z

ΩF_{s}(x, s, v_{m}+φ,∇v_{m}+∇φ)v_{m}^{00} dx ds

−2
Z _{t}

0

Z

ΩF_{v}_{m}_{+φ}(x, s, v_{m}+φ,∇v_{m}+∇φ)(v_{m}^{0} +φ^{0})v_{m}^{00} dx ds

−2X^{n}

i=1

Z _{t}

0

Z

ΩF_{v}_{m}_{x}_{i}_{+φx}_{i}(x, s, v_{m}+φ,∇v_{m}+∇φ)(v^{0}_{mx}_{i}+φ^{0}_{x}_{i})v^{00}_{m}dx ds
+2

Z _{t}

0 (f^{0}(s), v_{m}^{00}(s))ds+ 2(g^{0}(t), v^{0}_{m}(t))_{Γ}_{0}.

From (2.6), (2.7), (2.8), (2.10), (2.11), (2.12), (2.13), (3.10), (3.11), (3.12) and using Young, H¨older and Schwarz inequalities, and the Sobolev injection, we have

k_{0}|v_{m}^{00}(t)|^{2}+a_{0}

2 |∇v_{m}^{0} (t)|^{2}+ 2
Z _{t}

0 (β, v_{m}^{002}(s))ds

≤ L_{2}+L_{3}
Z _{t}

0 (|v^{00}_{m}(s)|^{2}+|∇v^{0}_{m}(s)|^{2})ds .

Then using the Gronwall’s inequality, we obtain the second estimate,

|v_{m}^{00}(t)|^{2}+|∇v^{0}_{m}(t)|^{2}+
Z _{t}

0 (β, v_{m}^{002}(s))ds≤L ,

whereLis a positive constant independent ofm∈Nandt∈[0, T].

The above estimates, allows us passing to the limit in the linear terms. Next we analyze the nonlinear term.

### Analysis of the nonlinear term F

From (2.3) there is positive constantM such that Z

Ω|F(x, t, v_{m}+φ,∇v+∇φ)|^{2}dx

≤ M

1 +kv_{m}(t) +φ(t)k^{2(γ+1)}_{L}2(γ+1)+|∇v_{m}(t) +∇φ(t)|^{2}
.
Therefore, from the first estimate it follows that

{F(x, t, v_{m}+φ,∇v_{m}+∇φ)}_{m∈N} is bounded inL^{2}(0, T;L^{2}(Ω)). (3.13)
Consequently, there exists a subsequence of{vm}m∈N (which we still denote by
the same symbol) and a functionχ inL^{2}(0, T;L^{2}(Ω)) such that

F(x, t, v_{m}+φ,∇v_{m}+∇φ)* χ weak inL^{2}(0, T;L^{2}(Ω)). (3.14)
From the above estimates after passing to the limit, we conclude that

Kv^{00}+A(t)v+χ=f in L^{2}(0, T;L^{2}(Ω)). (3.15)
We observe that

v∈L^{∞}(0, T;V), v^{0}∈L^{∞}(0, T;V), v^{00}∈L^{∞}(0, T;L^{2}(Ω)).
Moreover,

∂v

∂ν_{A} +βv^{0}=g inL^{∞}(0, T H^{1/2}(Γ_{0})).

On the other hand, integrating the approximate problem (3.7) over [0, T] and
considering thatw=v_{m}, we obtain

Z _{T}

0 (K(t)v_{m}^{00}(t), v_{m}(t))dt+
Z _{T}

0 a(t, v_{m}(t), v_{m}(t))dt (3.16)
+

Z _{T}

0 (βv_{m}^{0} (t), v_{m}(s))_{Γ}_{0} ds+
Z _{T}

0

Z

ΩF(x, t, v_{m}+φ,∇v_{m}+∇φ)v_{m}dx, dt

=
Z _{T}

0

(f(t), v_{m}(t))dt+
Z _{T}

0

(g(t), v_{m}(t))_{Γ}_{0}dt .

To simplify notation, subsequences will be denoted by the same symbol as the corresponding original sequences.

Notice that from the first and second estimates, and the Aubin-Lions The-
orem there exists a subsequence of{v_{m}}_{m∈N}, such that

v_{m}→v strong in L^{2}(0, T;L^{2}(Ω)), (3.17)
v_{m}^{0} →v^{0} strong inL^{2}(0, T;L^{2}(Ω)). (3.18)

Now, the first estimate yields p

βv_{m}^{0} (s)^{2}

H^{1/2}(Γ0)≤C_{0}|∇v^{0}_{m}(s)|^{2}≤L; s∈[0, T], (3.19)
and from the second estimate we get

p

βv_{m}^{00}(s)^{2}

Γ0≤L; s∈[0, T]. (3.20)
Combining (3.19) and (3.20), noting that the injectionH^{1/2}(Γ_{0}),→L^{2}(Γ_{0})
is compact, and considering Aubin-Lions Theorem it follows that

pβv^{0}_{m}→p

βv^{0} in L^{2}(0, T;L^{2}(Γ_{0})). (3.21)
In a similar way

pβv_{m}→p

βv in L^{2}(0, T;L^{2}(Γ_{0})).

Moreover, because of the second estimate

v_{m}^{00} * v^{00} weak in L^{2}(0, T;L^{2}(Ω)).

Then, considering the strong convergences given in (3.17), (3.18) and (3.21) and the corresponding weak converges, we are able to pass to the limit in (3.16).

m→∞lim
Z _{T}

0 a(t, v_{m}(t), v_{m}(t))dt (3.22)

= −

Z _{T}

0 (K(t)v^{00}(t), v(t))dt−
Z _{T}

0 (βv^{0}(t), v(t))_{Γ}_{0} dt

−
Z _{T}

0

Z

Ωχ(t)v(t)dx dt+
Z _{T}

0 (f(t), v(t))dt+
Z _{T}

0 (g(t), v)_{Γ}_{0}dt .
Substituting (3.15) in (3.22), applying Green formula and noting that

∂v

∂ν_{A} =−βv^{0}+g a.e. on Γ_{0}
we deduce that

m→∞lim
Z _{T}

0 a(t, v_{m}(t), v_{m}(s))dt=
Z _{T}

0 a(t, v(t), v(t))dt and that

m→∞lim
Z _{T}

0 (∇vm(t),∇vm(t))dt=
Z _{T}

0 (∇v(t),∇v(t)) dt . (3.23)

Finally, taking into account that
Z _{T}

0 |∇v_{m}(t)− ∇v(t)|^{2} dt

=
Z _{T}

0 |∇vm(t)|^{2} dt−2
Z _{T}

0 (∇vm(t),∇v(t))dt+
Z _{T}

0 |∇v(t)|^{2}dt ,
from (3.23) and the first estimate we deduce that

m→∞lim
Z _{T}

0 |∇vm(t)− ∇v(t)|^{2} dt= 0.
Therefore,

∇v_{m}→ ∇v in L^{2}(0, T;L^{2}(Ω)),
and consequently

∇v_{m}→ ∇v a.e. in Q_{T} = Ω×(0, T).
From (3.17) and the above convergence, we obtain

F(x, t, v_{m}+φ,∇v_{m}+∇φ)→F(x, t, v+φ,∇v+∇φ) a. e. inQ_{T}.
Applying Lemma 1.3 in [8, Chant. 1], it follows from the above convergence,
(3.13) and (3.14) that

F(x, t, v_{m}+φ,∇vm+∇φ)* F(x, t, v+φ,∇v+∇φ) weak inL^{2}(0, T;L^{2}(Ω)).
Note that the function v : Ω → R is a weak solution to the Dirichlet-
Neumann problem

A(t)v=f^{∗} in Ω,
v= 0 in Γ_{1}, _{∂ν}^{∂v}

A =g^{∗} in Γ_{0},

where f^{∗} = f−Kv^{00}−F(x, t, v+φ,∇v+∇φ), f^{∗} ∈ L^{2}(Ω), g^{∗} =−βv^{0}+g,
g^{∗}∈H^{1/2}(Γ_{0}), andt is a fixed value in [0,T].

The theory of elliptic problems states that the solution v belongs to the
spaceL^{∞}(0, T;H^{2}(Ω)); therefore,v∈L^{∞}(0, T;V ∩H^{2}(Ω)).

### Uniqueness of the solution

Letu and ˆube two solutions of (1.1), and put z =u−ˆu. From (2.9), (2.10), (2.11) and (2.12), it follows that

k_{0}|z^{0}(t)|^{2}+a_{0}|∇z(t)|^{2}+ 2
Z _{t}

0 β, z^{02}(s)

Γ0 ds

≤ 2D_{1}
Z _{t}

0

Z

Ω(|u|^{γ}+|ˆu|^{γ})|z| |z^{0}|dx dt+ 2D_{2}
Z _{t}

0

Z

Ω|z^{0}| |∇z|dx ds
+kK1k∞

Z _{t}

0 |z^{0}(s)|^{2}ds+a_{2}
Z _{t}

0 |∇z(s)|^{2} ds .

Since 0< γ ≤2/(n−2),for n≥3, we have the Sobolev immersionH^{1}(Ω),→
L^{2(γ+1)}(Ω). This immersion is also true for allγ >0 when n= 1,2. Therefore,

with γ

2(γ+ 1) + 1

2(γ+ 1) +1 2 = 1,

and using the generalized H¨older and the Poincar´e inequalities, we conclude that

|z^{0}(t)|^{2}+a_{0}|∇z(t)|^{2}+ 2
Z _{t}

0 β, z^{02}(s)

Γ0 ds≤C
Z _{t}

0

n|z^{0}(s)|^{2}+|∇z(s)|^{2}o
ds .

Applying Gronwall’s lemma in the last inequality we obtainz= 0 and therefore, u= ˆu. This concludes the proof of Theorem (2.1).

Existence of weak solutions. We have just proved the existence of strong
solutions to (1.1) whenu^{0} andu^{1} are smooth. Now by a density argument and
a procedure analogous to the one in the third estimate, we prove the existence
of a weak solution. The main step in this approach is obtaining a sequence that
satisfy the hypothesis of compatibility (A.2). For this purpose, we define the
following sequence. Given{u^{0}, u^{1}}in V ×L^{2}(Ω), consider

u^{1}_{µ}∈H_{0}^{1}(Ω)∩H^{2}(Ω) such that u^{1}_{µ}→u^{1} in L^{2}(Ω),
and

u^{0}_{µ}∈D(−∆) ={u∈V ∩H^{2}(Ω);∂u

∂ν = 0 on Γ_{0}}, such thatu^{0}_{µ}→u^{0} in V.

Uniqueness of a weak solution is guaranteed by the Visik-Ladyshenskaya method.

See for example Lions and Magenes [9, section 8].

### 4 Asymptotic behaviour

In this section we prove exponential decay for strong solutions of (1.1), and by a density argument we obtain the same results for weak solutions.

Let us consider the modified energy E(t) =e(t) + 1

γ+ 2 Z

Ω|u(x, t)|^{γ+2}dx ,
which by (2.4) satisfies

E^{0}(t) ≤ 1

2a^{0}(t, u, u) +1
2

Z

ΩK_{t}(x, t)|u^{0}|^{2}dx (4.1)

− Z

Γ0

(m·ν)|u^{0}|^{2}dΓ +C(t)
Z

Ω(1 +|u^{0}k∇u|)dx .

Letµandλbe positive constants such that R

Γ0(m·ν)v^{2}dΓ≤µR

Ω|∇v|^{2}dx ∀v∈ V (4.2)

|v|^{2}≤λ|∇v|^{2} ∀v∈V . (4.3)
For an arbitrary >0 define the perturbed energy

E_{}(t) =E(t) +ψ(t), (4.4)

where

ψ(t) = 2 Z

Ω

K(x, t)u^{0}(m· ∇u)dx+θ
Z

Ω

K(x, t)u^{0}u dx , (4.5)
θ∈]n−2, n[, andθ > _{γ+2}^{2n} . For short notation, put

k_{1}= min

2(θ−n+ 2),2(n−θ),(γ+ 2)(θ− 2n γ+ 2)

>0. (4.6)
Proposition 4.1 There existsδ_{0}>0 such that

|E_{}(t)−E(t)| ≤δ_{0}E(t),∀t≥0∀ >0.

Proof: From (2.2), (2.11), (4.3), and (4.5) we obtain

|ψ(t)| ≤ 2a^{−1/2}_{0} kKk^{1/2}_{∞} R(x^{0})|√

Ku^{0}(t)|a^{1/2}(t, u, u)
+a^{−1/2}_{0} λ^{1/2}θkKk^{1/2}_{∞} |√

Ku^{0}(t)|a^{1/2}(t, u, u)

≤ a^{−1/2}_{0} kKk^{1/2}_{∞} (2R(x^{0}) +λ^{1/2}θ)E(t).
Puttingδ_{0}=a^{−1/2}_{0} kKk^{1/2}∞ (2R(x^{0}) +λ^{1/2}θ), we deduce

|E_{}(t)−E(t)|=|ψ(t)| ≤δ_{0}E(t).
Which proves this proposition.

For a positive constantM, let

H(t) = M k∇a(t)k_{L}^{∞}_{(Ω)}+k∇K(t)k_{L}^{∞}_{(Ω)}
+kat(t)k_{L}^{∞}_{(Ω)}+kKt(t)k_{L}^{∞}_{(Ω)}+C(t)

.

Proposition 4.2 There exist positive constants δ_{1}, δ_{2}, _{1} such that

E^{0}_{}(t)≤ −δ_{1}E(t) +H(t)E(t) +δ_{2}C(t),
for allt≥0and for all ∈(0, _{1}].

Proof: Differentiating each term in (4.5) with respect tot and substituting
Ku^{00}=−A(t)u−F(x, t, u,∇u) in the expression obtained,

ψ^{0}(t)

= 2

Z

ΩK_{t}u^{0}(m· ∇u)dx−2
Z

ΩA(t)u(m· ∇u)dx

−2 Z

ΩF(x, t, u,∇u)(m· ∇u)dx+ 2 Z

ΩKu^{0}(m· ∇u^{0})dx+θ
Z

ΩK_{t}u^{0}u dx

−θ Z

ΩA(t)uu dx−θ Z

ΩF(x, t, u,∇u)u dx+θ Z

ΩK|u^{0}|^{2}dx .
From (2.5) and the above identity we have

ψ^{0}(t) ≤ 2
Z

Ω

K_{t}u^{0}(m· ∇u)dx−2
Z

Ω

A(t)u(m· ∇u)dx

−2 Z

Ω|u|^{γ}u(m· ∇u)dx+ 2C(t)
Z

Ω

(1 +|∇ukm· ∇u|)dx +2

Z

ΩKu^{0}(m· ∇u^{0})dx+θ
Z

ΩK_{t}u^{0}u dx−θ
Z

ΩA(t)uu dx (4.7)

−θ Z

ΩF(x, t, u,∇u)u dx+θ Z

ΩK|u^{0}|^{2}dx .

Now, we estimate one by one the terms on the right-hand side of the above inequality.

Estimate forI_{1}:=−2R

ΩA(t)u(m· ∇u)dx. Using Green and Gauss formula, we obtain

I_{1} = (n−2)
Z

Ωa(x, t)|∇u|^{2}dx+
Z

Ω(∇a·m)|∇u|^{2}dx

− Z

Γa(x, t)(m·ν)|∇u|^{2}dΓ + 2
Z

Γ

∂u

∂ν_{A}(m· ∇u)dΓ. (4.8)
Estimate for I_{2}:=−2R

Ω|u|^{γ}u(m· ∇u)dx. By the Gauss formula,
I_{2} = − 2

γ+ 2 Z

Ω∇(|u|^{γ+2})·m dx (4.9)

= 2n

γ+ 2 Z

Ω|u|^{γ+2}dx− 2
γ+ 2

Z

Γ(m·ν)|u|^{γ+2}dΓ.
From (2.1) and noting thatu|Γ1 = 0, we have

− 2 γ+ 2

Z

Γ(m·ν)|u|^{γ+2}dΓ≤0. (4.10)

Estimate for I_{3}:= 2R

ΩKu^{0}(m· ∇u^{0})dx. By Gauss Theorem we get
I_{3} =

Z

ΩK(x, t)m· ∇(|u^{0}|^{2})dx (4.11)

= −

Z

Ω

(∇K·m)|u^{0}|^{2}dx−n
Z

Ω

K(x, t)|u^{0}|^{2}dx+
Z

Γ0

(m·ν)K(x, t)|u^{0}|^{2}dΓ.

Estimate for I_{4}:=−θR

ΩA(t)uu dx. By Green’s formula and observing that

∂ν∂uA =−(m·ν)u^{0} on Γ_{0}, it follows that

I_{4}=−θ
Z

Ωa(x, t)|∇u|^{2}dx−θ
Z

Γ0

(m·ν)u^{0}u dΓ. (4.12)

Estimate for I_{5}:=−θR

ΩF(x, t, u,∇u)u dx. From (2.4) we deduce that
I_{5}≤ −θ

Z

Ω|u|^{γ+2}dx+θC(t)
Z

Ω(1 +|uk∇u|)dx . (4.13) Thus, substituting (4.8)–(4.13) in (4.7) we conclude that

ψ^{0}(t) ≤ (θ−n)
Z

ΩK(x, t)|u^{0}|^{2}dx+ (n−2−θ)
Z

Ωa(x, t)|∇u|^{2}dx (4.14)
+( 2n

γ+ 2 −θ) Z

Ω|u|^{γ+2}dx+
Z

Ω(∇a·m)|∇u|^{2}dx

− Z

Ω(∇K·m)|u^{0}|^{2}dx+ 2
Z

ΩK_{t}u^{0}(m· ∇u)dx+θ
Z

ΩK_{t}u^{0}u dx
+2C(t)

Z

Ω(1 +|∇ukm· ∇u|)dx+θC(t) Z

Ω(1 +|uk∇u|)dx

− Z

Γ(m·ν)a(x, t)|∇u|^{2}dΓ + 2
Z

Γ

∂u

∂ν_{A}(m· ∇u)dΓ
+

Z

Γ0

(m·ν)K(x, t)|u^{0}|^{2}dΓ−θ
Z

Γ0

(m·ν)u^{0}u dΓ.
On the other hand, _{∂x}^{∂u}

k = ^{∂u}_{∂ν}ν_{k} on Γ_{1}implies
m· ∇u= (m·ν)∂u

∂ν and |∇u|^{2}= (∂u

∂ν)^{2} on Γ_{1}.
Consequently,

− Z

Γ(m·ν)a(x, t)|∇u|^{2}dΓ (4.15)

= −

Z

Γ0

(m·ν)a(x, t)|∇u|^{2}dΓ−
Z

Γ1

(m·ν)a(x, t)(∂u

∂ν)^{2}dΓ

and 2

Z

Γ

∂u

∂ν_{A}(m·∇u)dΓ =−2
Z

Γ0

(m·ν)u^{0}(m·∇u)dΓ + 2
Z

Γ1

a(x, t)(m·ν)(∂u

∂ν)^{2}dΓ.
(4.16)
In the above equality, we used that _{∂ν}^{∂u}

A =−(m·ν)u^{0} on Γ_{0}. Replacing (4.15)
and (4.16) in (4.14), and using thatR

Γ1a(x, t)(m·ν)(^{∂u}_{∂ν})^{2}dΓ≤0, we obtain
ψ^{0}(t) ≤ (θ−n)

Z

ΩK(x, t)|u^{0}|^{2}dx+ (n−2−θ)
Z

Ωa(x, t)|∇u|^{2}dx (4.17)
+( 2n

γ+ 2 −θ) Z

Ω|u|^{γ+2}dx+
Z

Ω(∇a·m)|∇u|^{2}dx

− Z

Ω(∇K·m)|u^{0}|^{2}dx+ 2
Z

ΩK_{t}u^{0}(m· ∇u)dx+θ
Z

ΩK_{t}u^{0}u dx
+2C(t)

Z

Ω(1 +|∇ukm· ∇u|)dx+θ Z

ΩC(t)(1 +|uk∇u|)dx

− Z

Γ0

(m·ν)a(x, t)|∇u|^{2}dΓ−2
Z

Γ0

(m·ν)u^{0}(m· ∇u)dΓ
+

Z

Γ0

(m·ν)K(x, t)|u^{0}|^{2}dΓ−θ
Z

Γ0

(m·ν)u^{0}u dΓ.
However, since

−2 Z

Γ0

(m·ν)u^{0}(m· ∇u)dΓ

≤ a^{−1}_{0} R^{2}(x^{0})
Z

Γ0

(m·ν)|u^{0}|^{2}dΓ +
Z

Γ0

(m·ν)a(x, t)|∇u|^{2}dΓ,
from (4.17) it results that

ψ^{0}(t)

≤ (θ−n) Z

ΩK(x, t)|u^{0}|^{2}dx+ (n−2−θ)
Z

Ω|∇u|^{2}dx (4.18)

+( 2n γ+ 2−θ)

Z

Ω|u|^{γ+2}dx+
Z

Ω(∇a·m)|∇u|^{2}dx−
Z

Ω(∇K·m)|u^{0}|^{2}dx
+2

Z

ΩK_{t}u^{0}(m· ∇u)dx+θ
Z

ΩK_{t}u^{0}u dx+ 2C(t)
Z

Ω(1 +|∇ukm· ∇u|)dx +θ

Z

ΩC(t)(1 +|uk∇u|)dx+a^{−1}_{0} R^{2}(x^{0})
Z

Γ0

(m·ν)|u^{0}|^{2}dΓ
+

Z

Γ0

(m·ν)K(x, t)|u^{0}|^{2}dΓ−θ
Z

Γ0

(m·ν)u^{0}u dΓ.

Letk_{2}be a positive real number such that 0< k_{2}< k_{1}. Then from (4.2),

−θ Z

Γ0

(m·ν)u^{0}u dΓ≤ µθ^{2}
2a_{0}k_{2}

Z

Γ0

(m·ν)|u^{0}|^{2}dΓ +k_{2}E(t). (4.19)

Therefore, from (4.6), (4.18), and (4.19) it follows that
ψ^{0}(t)

≤ −(k1−k_{2})E(t) +
Z

Ω(∇a·m)|∇u|^{2}dx−
Z

Ω(∇K·m)|u^{0}|^{2}dx (4.20)
+2

Z

ΩK_{t}u^{0}(m· ∇u)dx+θ
Z

ΩK_{t}u^{0}u dx+ 2C(t)
Z

Ω(1 +|∇ukm· ∇u|)dx +θ

Z

Ω

C(t)(1 +|uk∇u|)dx+a^{−1}_{0} R^{2}(x^{0})
Z

Γ0

(m·ν)|u^{0}|^{2}dΓ
+

Z

Γ0

(m·ν)K(x, t)|u^{0}|^{2}dΓ + µθ^{2}
2a_{0}k_{2}

Z

Γ0

(m·ν)|u^{0}|^{2}dΓ.
From (4.20), we obtain

ψ^{0}(t) ≤ −(k_{1}−k_{2})E(t) + (M_{1}k∇a(t)k_{L}^{∞}_{(Ω)}+M_{2}k∇K(t)k_{L}^{∞}_{(Ω)}
+M_{3}kK_{t}(t)k_{L}^{∞}_{(Ω)}+M_{4}C(t))E(t) + (θ+ 2) meas(Ω)C(t)
+(a^{−1}_{0} R^{2}(x^{0}) + µθ^{2}

2a_{0}k_{2} +kKk_{∞})
Z

Γ0

m·ν|u^{0}|^{2}dΓ, (4.21)
where

M_{1}= 2a^{−1}_{0} R(x^{0}), M_{2}= 2k^{−1}_{0} R(x^{0}),
M_{3}= 2k^{−1/2}_{0} a^{−1/2}_{0} R(x^{0}) +θλ^{1/2}k^{−1/2}_{0} a^{−1/2}_{0} ,

M_{4}= 4a^{−1}_{0} R(x^{0}) + 2θλ^{1/2}a^{−1}_{0} .
Define

G(t) =M_{1}k∇a(t)k_{L}^{∞}_{(Ω)}+M_{2}k∇K(t)||_{L}^{∞}_{(Ω)}+M_{3}kK_{t}(t)k_{L}^{∞}_{(Ω)}+M_{4}C(t).
(4.22)
Then from (4.1), (4.4), (4.21), and (4.22), we obtain

E_{}^{0}(t) = E^{0}(t) +ψ^{0}(t) (4.23)

≤ 1

2a^{0}(t, u, u) +1
2
Z

ΩK_{t}|u^{0}|^{2}dx+C(t)
Z

Ω(1 +|u^{0}k∇u|)dx

−(k_{1}−k_{2})E(t) +G(t)E(t) +(θ+ 2) meas(Ω)C(t)

− Z

Γ0

(m·ν)

1−(a^{−1}_{0} R^{2}(x^{0}) + µθ^{2}

2a_{0}k_{2} +kKk∞)

|u^{0}|^{2}dΓ.

By setting δ_{1}=k_{1}−k_{2}, from (4.23) we conclude that

E_{}^{0}(t) ≤ −δ1E(t) +G(t)E(t) +(θ+ 2) meas(Ω)C(t) +J(t)E(t)
+ meas(Ω)C(t) +k^{−1/2}_{0} a^{−1/2}_{0} C(t)E(t) (4.24)

− Z

Γ0

(m·ν)

1−(a^{−1}_{0} R^{2}(x^{0}) + µθ^{2}

2a_{0}k_{2} +kKk_{∞})

|u^{0}|^{2}dΓ,

where

J(t) =a^{−1}_{0} ka_{t}(t)k_{L}^{∞}_{(Ω)}+k_{0}^{−1}kK_{t}(t)||_{L}^{∞}_{(Ω)}.
Let_{1}= min{(a^{−1}_{0} R^{2}(x^{0}) + _{2a}^{µθ}^{2}

0k2 +kKk_{∞})^{−1},1}. Then from (4.24), we obtain
that for all∈(0, _{1}],

E_{}^{0}(t) ≤ −δ_{1}E(t) + (G(t) +J(t) +k_{0}^{−1/2}a^{−1/2}_{0} C(t))E(t)
+(θ+ 3) meas(Ω)C(t)

≤ −δ1E(t) +H(t)E(t) +δ_{2}C(t),

where M = max{M1, M_{2}, M_{3}+k^{−1}_{0} , a^{−1}_{0} , M_{4}+k^{−1/2}_{0} a^{−1/2}_{0} } and δ_{2} = (θ+
3) meas(Ω). Which completes the proof of Proposition 4.2.

Proposition 4.3 There exists a positive constantδ_{3} such that

E(t)≤δ_{3} ∀t≥0.
Proof: We shall show that the constant is given by

δ_{3}= (E(0) + meas(Ω)kCk_{L}^{1}_{(0,∞)}) exp(

Z _{∞}

0 F(t)dt),
where F(t) =a^{−1}_{0} ka_{t}(t)k_{L}^{∞}_{(Ω)}+k^{−1}_{0} ||K_{t}(t)k_{L}^{∞}_{(Ω)}+k^{−1/2}_{0} a^{−1/2}_{0} C(t).

From (4.1) we have

E^{0}(t)≤ F(t)E(t) + meas(Ω)C(t).

Hence d

dt(E(t)exp

−
Z _{t}

0 F(s)ds

)≤meas (Ω)C(t) exp

−
Z _{t}

0 F(s)ds

.

Therefore, E(t)≤E(0) exp

Z _{∞}

0 F(s)ds

+ meas(Ω) exp
Z _{∞}

0 F(s)ds
Z _{t}

0 C(s)ds . Which completes the proof of this porposition.

Now, we prove exponential decay. In what follows, let
_{0}= min{_{1}, 1

2δ_{0}}
where δ_{0} is the constant obtained in Proposition 4.1.

For all∈(0, _{0}], we have
1

2E(t)≤E_{}(t)≤ 3

2E(t)≤2E(t), ∀t≥0. (4.25)

Consequently, from (4.25) and Proposition 4.2 we obtain
E_{}^{0}(t)≤ −

2δ_{1}E_{}(t) +H(t)E(t) +δ_{2}C(t). (4.26)
From Proposition 4.3 and (4.26), we get

E_{}^{0}(t)≤ −

2δ_{1}E_{}(t) +δ_{3}H(t) +δ_{2}C(t).
Therefore,

d

dt(E_{}(t) exp(

2δ_{1}t))≤exp(

2δ_{1}t)(δ_{3}H(t) +δ_{2}C(t)). (4.27)
Integrating (4.27) over [0,t] and using (4.25), we conclude

1

2E(t) ≤ 3

2E(0) exp(−

2δ_{1}t)
+

δ_{3}

Z _{t}

0 exp(

2δ_{1}s)H(s)ds+δ_{2}
Z _{t}

0 exp(

2δ_{1}s)C(s)ds

exp(−

2δ_{1}t)

≤ 3

2E(0) exp(−

2δ_{1}t)
+ max{δ_{2}, δ_{3}}

Z _{t}

0 exp(

2δ_{1}s)(H(s) +C(s))ds

exp(−

2δ_{1}t).
From the above inequality and (2.14), we obtain exponential decay, which com-
pletes the proof of Theorem 2.2.

Remark 1. Exponential decay for weak solutions can be proved using a den- sity argument.

Remark 2. Theorem 2.3 remains valid for Γ_{0}∩Γ1not empty whenA(t) =−∆

andn≤3. In which case, the Rellich identity given in (4.8) can be replaced by the Grisvard inequality,

I_{1}≤(n−2)
Z

Ω|∇u|^{2}dx−
Z

Γ(m·ν)|∇u|^{2}dΓ + 2
Z

Γ

∂u

∂ν(m· ∇u)dΓ. The proof of this inequality can be found in Komornik and Zuazua [5].

Acknowledgments. The authors would like to thank the referees for their constructive comments.

### References

[1] M. M. Cavalcanti - J. A. Soriano, On Solvability and Stability of the Wave Equation with Lower Order Terms and Boundary Damping, Revista Matem´aticas Aplicadas18(2), (1997), 61-78.