A NONLINEAR TRANSMISSION PROBLEM WITH TIME DEPENDENT COEFFICIENTS
˜ EUGENIO CABANILLAS LAPA, JAIME E. MUNOZ RIVERA

Abstract. In this article, we consider a nonlinear transmission problem for the wave equation with time dependent coefficients and linear internal damping. We prove the existence of a global solution and its exponential decay. The result is achieved by using the multiplier technique and suitable unique continuation theorem for the wave equation.

1. Introduction In this work, we consider the transmission problem ρ1 utt − buxx + f1 (u) = 0 u(0, t) = v(L, t), u(L0 , t) = v(L0 , t),
0

in ]0, L0 [×R+ , in ]L0 , L[×R , t > 0,
+

(1.1) (1.2) (1.3) (1.4) (1.5) (1.6)

ρ2 vtt − (a(x, t)vx )x + αvt + f2 (v) = 0 t > 0,
1

bux (L0 , t) = a(L0 , t)vx (L0 , t), ut (x, 0) = u (x), vt (x, 0) = v (x),
1

u(x, 0) = u (x), v(x, 0) = v (x),
0

x ]0, L0 [, x ]L0 , L[,

where ρ1 , ρ2 areconstants; α, b are positive constants, f, g are nonlinear functions and a(x, t) is a positive function. Controllability and Stability for transmission problem has been studied by many authors (see for example Lions [7], Lagnese [5], Liu and Williams [8], Mu˜oz Rivera and Portillo Oquendo [9], Andrade, Fatori and n Mu˜oz Rivera [1]). n The goal of this work is to study the existence and uniqueness of global solutions of (1.1)-(1.6) and the asymptotic behavior of the energy. All the authors mentioned above established their results with constant coefficients. To the best of our knowledge this is a first publication on transmission problem with time dependent coefficients and the nonlinear terms. In general,the dependence on spatial and time variables causes difficulties,semigroups arguments are not suitable for finding solutions to (1.1)-(1.6); therefore,we make use of a Galerkin’s process. Note that the time-dependent coefficient also appear in the second boundary condition, thus there are some technical difficulties that we need to overcome. To prove the exponential decay, the main difficulty is that the dissipation only
2000Mathematics Subject Classification. 35B40, 35L70, 45K05. Key words and phrases. Transmission problem; time dependent coefficients; stability. c 2007 Texas State University - San Marcos. Submitted May 2, 2007. Published October 9, 2007.
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works in [L0 , L] and we need estimates over the whole domain [0, L]; we overcome this problem introducing suitable multiplicadors and a compactness/uniqueness argument. 2. Notation and statement of results We denote (w, z) =
I

w(x)z(x)dx,

|z|2 =
I

|z(x)|2 dx

where I =]0, L0 [ or ]L0 , L[ for u’s and v’s respectively. Now, we state the general hypotheses. (A1) The functions fi C 1 (R), i = 1, 2, satisfy fi (s)s ≥ 0 for all s R and |fi (s)| ≤ c(1 + |s|)ρ−j ,
s (j)

s R, j = 0, 1

for some c > 0 and ρ ≥ 1. We assume that f1 (s) ≥ f2 (s) and set Fi (s) =
0

fi (ξ)dξ .

(A2) We assume that the coefficient a satisfies a W 1,∞ (0, ∞; C 1 ([L0 , L])) ∩ W 2,∞ (0, ∞; L∞ (L0 , L)) at L1 (0, ∞; L∞ (L0 , L)) a(x, t) ≥ a0 > 0, We define the Hilbert space V = . Also we define the first-order energy functionals associated to each equation, E1 (t, u) = E2 (t, v) = 1 ρ1 |ut |2 + b|ux |2 + 2 2
L0

(x, t) ]L0 , L[×]0, ∞[ .

F1 (u)dx
0

L 1 2 ρ2 |vt |2 + (a, vx ) + 2 F2 (v)dx 2 L0 E(t) = E1 (t, u, v) = E1 (t, u) + E2 (t, v).

We conclude this section with the following lemma which will play essential role when establishing the asymptotic behavior of solutions. Lemma 2.1 ([2, Lemma 9.1]). Let E : R+ → R+ be a non-increasing function and 0 0 assume that there exist two constants p > 0 and c > 0 such that
+∞

E (p+1)/2 (t)dt ≤ cE(s),
s

0 ≤ s < +∞.

Then for all t ≥ 0, E(t) ≤ cE(0)(1 + t)−2(p−1) cE(0)e1−wt if p > 1, if p = 1,

where c and w are positive constants.


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A NONLINEAR TRANSMISSION PROBLEM

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3. Existence and uniqueness of solutions First, we define weak solutions of problem (1.1)-(1.6). Definition 3.1. We say that the pair is a weak solution of (1.1)-(1.6) when L∞ (0, T ; V ) ∩ W 1,∞ (0, T ; L2 (0, L0 ) × L2 (L0 , L)) and satisfies
L0 L0 L

− ρ1
0 L

u1 (x)ϕ(x, 0)dx + ρ1
0 Tu0 (x)ϕt (x, 0)dx − ρ2
L0 L0

v 1 (x)ψ(x, 0)dx

+ ρ2
L0 T

v 0 (x)ψt (x, 0)dx + ρ1
0 L 0

(uϕtt + bux ϕx + f1 (u)ϕ) dx dt

+ ρ2
0 L0

(vψtt + a(x, t)vx ψx + αvt ψ + f2 (v)ψ) dx dt = 0

for any C 2 (0, T ; V ) such that ϕ(T ) = ϕt (T ) = 0 = ψ(T ) = ψt (T ) To show the existence of strong solutions we need a regularity result for the elliptic system associated to the problem (1.1)–(1.6) whose proof can be obtained, with little modifications, in the book by Ladyzhenskaya and Ural’tseva [3, theorem 16.2]. Lemma 3.2. For any given functions F L2 (0, L0 ), G L2 (L0 , L), there exists only one solution to the system −buxx = F in ]0, L0 [, in ]L0 , L[, −(a(x, t)vx )x = G u(L0 ) = v(L0 ),

u(0) = v(L) = 0, bux (L0 ) = a(L0 , t)vx (L0 ), with t a fixed value in [0, T ], with u in H 2 (0, L0 ) and v in H 2 (L0 , L). The existence result to the system (1.1)–(1.6) is summarized in the following theorem. Theorem 3.3. Suppose that V , L2 (0, L0 ) × L2 (L0 , L) and that assumptions (A1)–(A3) hold. Then there exists a unique weak solution of (1.1)–(1.6)satisfying C(0, T ; V ) ∩ C 1 (0, T ; L2 (0, L0 ) × L2 (L0 , L)). In addition, if H 2 (0, L0 ) × H 2 (L0 , L), V , verifying the compatibility condition 0 bu0 (L0 ) = a(L0 , 0) vx (L0 ) . (3.1) x Then
2



W k,∞ (0, T ; H 2−k (0, L0 ) × H 2−k (L0 , L))
k=0

Proof. The main idea is to use the Galerkin Method. Let , i = 1, 2, . . . be a basis of V . Let us consider the Galerkin approximation
m

=
i=1

him (t)


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where um and v m satisfy
m ρ1 (um , ϕi ) + b(um , ϕi ) + (f1 (um ), ϕi ) + ρ2 (vtt , ψ i ) tt x x m i m + (a(x, t)vx , ψx ) + α(vt , ψ i ) + (f2 (v m ), ψ i ) = 0

(3.2)

where i = 1, 2, . . . With initial data →
m t

in V,

→

1

1

in L (0, L) × L2 (L0 , L).

2

(3.3

Standard results about ordinary differential equations guarantee that there exists only one solution of this system on some interval [0, Tm [. The priori estimate that follow imply that in fact Tm = +∞.subsection*Existence of weak solutions Multiplying (3.2) by him (t) integrating by parts and summing over i, we get d |at (t L∞ m E(t, um , v m ) + α|vt |2 ≤ E(t, um , v m ). dt a0 (3.4)

From this inequality, the Gronwall’s inequality and taking into account the definition of the initial data of we conclude that E(t, um , v m ) ≤ C, thus we deduce that is bounded in L∞ (0, T ; V )
m is bounded in L∞ (0, T ; L2 (0, L0 ) × L2 (L0 , L)) t

t [0, T ], m N

(3.5)

which implies that → weak in L∞ (0, T ; V )
m → weak in L∞ (0, T ; L2 (0, L0 ) × L2 (L0 , L)). t

In particular, by application of the Lions-Aubin’s Lemma [6, Theorem 5.1], we have → strongly in L2 (0, T ; L2 (0, L0 ) × L2 (L0 , L)) and consequently um → u a.e in ]0, L0 [ and f1 (um ) → f1 (u) a.e in ]0, L0 [ v m → v a.e in ]L0 , L[ and f2 (v m ) → f2 (v) a.e in ]L0 , L[. Also, from the growth condition in (A1) we have f1 (um ) is bounded in L∞ (0, T ; L2 (0, L0 )) f2 (v m ) is bounded in L∞ (0, T ; L2 (L0 , L)); therefore, in L2 (0, T ; L2 (0, L0 ) × L2 (L0 , L)).

The rest of the proof of the existence of a weak solution is matter of routine.


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Regularity of solutions. To get the regularity, we take a basis B = {, i N} such that , are in the span of , }.
m Therefore, = and = . Let us differt entiate the approximate equation and multiply by him (t). Using a similar argument as before we obtain

d m m m E2 (t, um , v m ) + α|vtt |2 = −(f1 (um )um , um ) − (f2 (v m )vt , vtt ) t tt dt 1 m m m − (at vx , vxtt ) + (at , (vxt )2 ) 2 where E2 (t, u, v) = Note that
m m m m m m m − (at vx , vxtt ) = −(at vx , vxt )t + (att vx , vxt ) + at , (vxt )2 ,

(3.6)

b ρ2 1 ρ1 |utt |2 + |uxt |2 + |vtt |2 + (a, vxt )2 . 2 2 2 2

(3.7

E2 (0, um , v m ) is bounded, because of our choice of the basis. From the assumption (A1) and from the Sobolev imbedding we have
L0 L0

f1 (um )um um dx ≤ C t tt
0 0

(1 + |um |)2 dx x

(p−1)/2

|um ||um |,xt tt

(3.8)

and similarly
L m m f2 (v m )vt vtt dx ≤ C L0 L0 L m (1 + |vx |)2 dx (p−1)/2 m m |vxt ||vtt |

(3.9)

Substituting (3.7), the inequalities (3.8)–(3.9), using the estimative (3.5) in (3.6) and applying Gronwall’s inequality we conclude that E2 (t, um , v m ) ≤ C which imply
m → t m → tt

(3.10)

weak in L∞ (0, T ; H 1 (0, L0 ) × H 1 (L0 , L)) weak in L∞ (0, T ; L2 (0, L0 ) × L2 (L0 , L)).

Therefore, satisfies (1.1)–(1.4) and we have −buxx = −ρ1 utt − f1 (u) L2 (0, L0 ), −(a(x, t)vx )x = −ρ2 vtt − f2 (v) − αvt L2 (L0 , L), u(L0 , t) = v(L0 , t), bux (L0 , t) = a(L0 , t)vx (L0 , t). u(0, t) = 0 = v(L, t) Then using Lemma 3.2 we have the required regularity for .


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4. Exponential Decay In this section we prove that the solution of the system (1.1)–(1.6) decay exponentially as time approaches infinity. In the remainder of this paper we denote by c a positive constant which takes different values in different places. We shall suppose that ρ1 ≤ ρ2 and a(x, t) ≤ b,at (x, t) ≤ 0, (x, t) ]L0 , L[×]0, ∞[

ax (x, t) ≤ 0 . Theorem 4.1. Take in V and in L2 (0, L0 ) × L2 (L0 , L) with u0 (L0 ) = 0. x Then there exists positive constants γ and c such that E(t) ≤ cE(0)e−γt , t ≥ 0. (4.2) (4.1)

We shall prove this theorem for strong solutions; our conclusion follow by standard density arguments. The dissipative property of (1.1)–(1.6) is given by the following lemma. Lemma 4.2. The first-order energy satisfies d 2 E1 (t, u, v) = −α|vt |2 + (at , vx ). dt (4.3)

Proof. Multiplying equation (1.1) by ut , equation (1.2) by vt and performing an integration by parts we get the result.
∞ Let ψ C0 (0, L) be such that ψ = 1 in ]L0 − δ, L0 + δ[ for some δ > 0, small constant. Let us introduce the following functional L0 L

I(t) =
0

ρ1 ut qux dx +
L0

ρ2 vt ψqvx dx

where q(x) = x. Lemma 4.3. There exists c1 > 0 such that for all ε > 0, L0 a(L0 , t) 2 d 2 I(t) ≤ − dt 2 b 1 L0 − L0 (F1 (u(L0 , t)) − F2 (v(L0 , t))) − (ρ1 ut + bu2 + 2F (u))dx x 2 0 − + 1 4
L0 +δ 2 avx dx + c1 L0 L 2 L0 +δ L0 2 2(vt + avx )dx + L0 L 2 vt dx + 0 L0

u2 dx

v dx + εE(t, u, v) .

L0


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Proof. Multiplying (1.1) by qux , equation (1.2) by ψqvx , integrating by parts and using the corresponding boundary conditions we obtain d L0 (ρ1 ut , qux ) = [ρ1 u2 (L0 , t) + bu2 (L0 , t)] − L0 F1 (u(L0 , t)) t x dt 2 (4.4) L0 1 ρ1 u2 + bu2 + 2F1 (u)dx − t x 2 0 d L0 2 2 ρ2 vt (L0 , t) + a(L0 , t)vx (L0 , t) (ρ2 vt , ψqvx ) ≤ − dt 2 1 L0 +δ 2 1 L0 +δ 2 xax ψvx dx − avx dx (4.5) + L0 F2 (v(L0 , t)) + 2 L0 4 L0
L L 2 2 (vt + avx )dx + L0 +δ L0 2 (vt + F2 (v))dx]

+ c1 [

Summing up (4.4) and (4.5), and taking the assumption on ax into account, we get d L0 2 2 I(t) ≤ − [(ρ2 − ρ1 )vt (L0 , t) + a(L0 , t)vx (L0 , t) − bu2 (L0 , t)] x dt 2 − L0 [F1 (u(L0 , t)) − F2 (v(L0 , t))] − 1 2
L0

(ρ1 u2 + bu2 + 2F1 (u))dx − t x
0 L L 2 2 (vt + avx )dx + L0 +δ L0

1 4

L0 +δ 2 avx dx L0 L0

(4.6)

+ c1

2 (vt + F2 (v))dx + 0

F (u)dx

According to (A1), we have fi (0) = 0 and |fi (s)| ≤ c(|s| + |s|ρ ) this implies |Fi (s)| ≤ c(|s|2 + |s|ρ+1 ) ≤ c(|s|2 + |s|2ρ ). Fromthe interpolation inequality α 1−α 1 = + , α [0, 1] |y|p ≤ |y|α |y|1−α , 2 q p 2 q (4.8) (4.7)

and the immersion H 1 (a„Š) → L2(2p−1) (a„Š), a„Š =]0, L0 [, ]L0 , L[, we obtain for all t≥0 ε 2(2ρ−1) |u(t)|2ρ ≤ cε [E(0)]2(ρ−1) |u(t)|2 + , for all ε > 0. |ux (t)|2 2 2ρ [E(0)]2(ρ−1) Considering that |ux (t)|2 ≤ cE(0, u, v) ≡ c1 E(0) 2 it follows that |u(t)|2ρ ≤ cε [E(0)]2(ρ−1) |u(t)|2 + εE(t, u, v). 2 2ρ Replacing the inequalities (4.7)–(4.9) in (4.6) our conclusion follows.
δ δ Let ϕ C ∞ (R) a nonnegative function such that ϕ = 0 in Iδ/2 =]L0 − 2 , L0 + 2 [ and ϕ = 1 in RIδ and consider the functional L

(4.9)

J(t) =
L0

ρ2 vt ϕv dx.

We have the following lemma.


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Lemma 4.4. Given ε > 0, there exists a positive constant cε such that d 1 J(t) ≤ − dt 2
L 2 avx dx + ε L0 +δ L0 L0 +δ 2 avx dx + cε L0 L 2 (v 2 + vt )dx

Proof. Multiplying equation (1.2) by ϕv and integrating by parts we get d J(t) = −(avx , ϕvx ) − (avx , ϕx v) − α(vt , ϕv) − (ϕ, f2 (v)v) + (vt , ϕvt ). dt Applying Young’s Inequality and hypothesis (A1) we concludes ourassertion. Let us consider the functional K(t) = I(t) + (2c1 + 1)J(t) and we take ε = ε1 in lemma 4.4, where ε1 is the solution of the equation (2c1 + 1)ε1 = 1 . 8

Taking in to consideration (A1) in lemma 4.3, we obtain 1 d K(t) ≤ −E1 (t, u) − dt 8
L L 2 (avx + 2F2 (v))dx + εE(t, u, v) L0 L0 2 (vt L0 2

(4.10) u dx).
2

+ c2 (

+ v )dx +
0

Now in order to estimate the last two terms of (4.10) we need the following result. Lemma 4.5. Let be a solution in theorem 3.3. Then there exists T0 > 0 such that if T ≥ T0 we have
T T T t

(|v|2 +|u|2 )ds ≤ ε
S S

(b|ux |2 +|ut |2 )ds+
S

|a1/2 vx |2 ds +cε
S

|vt |2 ds (4.11)

for any ε > 0 and cε is a constant depending on T and ε, by independent of , for any initial data , satisfying E(0, u, v) ≤ R, where R > 0 is fixed and 0 < S < T < +∞. Proof. We use a contradiction argument. If (4.11) were false, there would exist a sequence of solutions such that
T t T ν |vt |2 ds + c0 S S

(|v ν |2 + |uν |2 )ds ≥ ν
S

(b|uν |2 + |ut |2 + |a1/2 vx |2 )ds x

and E(0, uν , v ν ) ≤ R for all ν. Let
T

λ2 =ν
S

(|v ν |2 + |uν |2 )ds, z ν (x, t) = v ν (x, t) , λν 0 ≤ t ≤ T.

uν (x, t) , w (x, t) = λν
ν

Then we have
T T ν |zt |2 ds + c0 S S ν ν ν b|wx |2 + |wt |2 + |a1/2 zx |2 ds ≤ 1

ν


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and consequently
T ν |zt |2 ds → 0 S T ν ν ν (b|wx |2 + |wt |2 + |a1/2 zx |2 )ds ≤ c. S

as ν → ∞,

(4.12) (4.13)

Also we have
T

|z ν |2 + |wν |2 ds = 1 .
S

(4.14)

Since S is chosen in the interval [0, T [, we obtain from (4.12)–(4.13) that, there exists a subsequence which we denote in the same way, such that wν → w
ν wt → wt

in L2 (0, T ; H 1 (0, L0 )), in L2 (0, T ; L2 (0, L0 )), in L2 (0, T ; H 1 (L0 , L)), in L2 (0, T ; L2 (L0 , L)).

zν → z
ν zt → 0

From which wν → w zν → z This implies
T

in L2 (0, T ; L2 (0, L0 )), in L2 (0, T ; L2 (L0 , L)).

|z|2 + |w|2 ds =
0

(4.15

Besides, from the uniqueness of the limit we conclude that zt (x, 0) = 0 and therefore z(x, t) = ϕ(x) . (4.16)

Note that satisfies 1 f1 (λν wν ) = 0 in ]0, L0 [×]0, T [, λν 1 ν ν ν ρ2 ztt − (a(x, t)zx )x + f2 (λν z ν) + αzt = 0 in ]L0 , L[×]0, T [, λν wν (0, t) = 0 = z ν (L, t),
ν ν ρ1 wtt − bwxx +

wν (L0 , t) = z ν (L0 , t),
ν bwx (L0 , t) ν,0

(4.17)

=

ν a(L0 , t)zx (L0 , t), ν wt (x, 0) =

wν (x, 0) = z ν (x, 0) =

u

(x) , λν

1 ν,0 v (x), λν

1 ν,1 u (x), λν 1 ν,1 ν zt (x, 0) = v (x). λν


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Now, we observe that ν≥1 is a bounded sequence,
T

λν =
S

(|v ν |2 + |uν |2 )ds
T ν (|vx |2 + |uν |2 )ds x S

1/2

1/2

≤c

≤ cE(0, u, v) ≤ cR, where R is a fixed value, because the initial data are in the ball B(θ, R). Hence, there exists a subsequence of ν≥1 (still denoted by (λν ) such that λν → λ ]0, +∞[. In this case passing to limit in (4.17), when ν → ∞ for , we get ρ1 wtt − bwxx + 1 f1 (λw) = 0 in ]0, L0 [×]0, T [, λ 1 (a(x, t)zx )x + f2 (λz) = 0 in ]L0 , L[×]0, T [, λ w(0, t) = 0 = z(L, t) w(L0 , t) = z(L0 , t) bwx (L0 , t) = a(L0 , t)zx (L0 , t), zt (x, 0) = 0 and for y = wt , ρ1 ytt − byxx + f (λw)y = 0 in ]0, L0 [×]0, T [, (4.19) y(0, t) = 0 = y(L0 , t), byx (L0 , t) = at (L0 , t)zx (L0 , t). Here, weobserve that wxt (L0 , t) at (L0 , t) = . wx (L0 , t) a(L0 , t) Then after an integration, wx (L0 , t) = k a(L0 , t) whee k is a constant. Using the hypotheses, we obtain 0 = lim wx (L0 , t) = ka(L0 , 0). +
t→0

(4.18)

in ]L0 , L[×]0, T [,

Consequently k = 0 and yx (L0 , t) = 0. Thus, the function y satisfies ρ1 ytt − byxx + f (λw)y = 0 y(0, t) = 0 = y(L0 , t) yx (L0 , t) = 0 in ]0, L0 [×]0, T [, on ]0, T [,

on ]0, T [.

Then, using the results in [4] (based on Ruiz arguments [10]) adapted to our case we conclude that y = 0, that is wt (x, t) = 0, for T suitable big. Returning to (4.18) we obtain the elliptic system 1 f1 (λw) = 0, λ 1 (a(x, t)zx )x + f2 (λz) = 0 . λ −bwxx +


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multiplying by u and v respectively and integrating, then summing up we arrive at
L0 L 2 wx dx + 0 L0 2 a(x, t)zx dx +

b

1 λ

L0

f1 (λw)wdx +
0

1 λ

L

f2 (λz)zdx = 0 .
L0

So we have w = 0 and z = 0, which contradicts (4.15). Suppose we are not in the above situation and there exists a subsequence satisfying λν → 0. Applying inequality (4.10) tothe solutions we have d ν K (t) ≤ −δ0 E(t, uν , v ν ) + c3 dt integrating from S to T , we obtain
T T L ν ((vt )2 + (v ν )2 )dx + L0 0 L0

(uν )2 dx ,

K ν (T ) + δ0
S

E(t, uν , v ν )dt ≤ K(S) + c3
S

ν (|vt |2 + |v ν |2 + |uν |2 ) dt.

Since K ν satisfies c0 E(t, uν , v ν ) ≤ K ν (T ) ≤ c1 E(t, uv , v ν ) and E is a decreasing function we have
T

E(T, uv , v ν ) + δ0
S

E(t, uν , v ν )dt
T ν (|vt |2 + |v ν |2 + |uν |2 )dt; S

c ≤ 1 T thus, we obtain

T

E(t, uν , v ν )dt + c3
S

E(T, uv , v ν ) + (δ0 −

c1 ) T

T

T

E(t, uν , v ν )dt ≤ c3
S S

ν |vt |2 + |v ν |2 + |uν |2 dt,

Dividing both sides of the above inequality by λ2 , using (4.12) and (4.14), taking T ν ν ν ν ν large enough we conclude that (|wt |2 + |zt |2 + |wx |2 + |zx |2 )(T ) is bounded. Now, ν ν multiplying equations (4.17) (4.17)2 by wt and zt respectively, performing an integration by parts we get
T T ν |zt |2 dt − S S ν (at , (zx )2 )dt.

E(t, wv , z ν ) ≤ E(T, wv , z ν ) + α

From (4.12) , (4.13) and Poincare Inequality we deduce that E(t, wv , z ν ) is bounded for all t [S, T ].Then in particular, on a subsequence we obtain wν → w
ν wt → wt

weak star in L∞ (0, T ; H 1 (0, L0 )), weak star in L∞ (0, T ; L2 (0, L0 )), weak star in L∞ (0, T ; H 1 (L0 , L)), weak star in L∞ (0, T ; L2 (L0 , L)), in L2 (0, T ; L2 (0, L0 )), in L2 (0, T ; L2 (L0 , L)).

zν → z
ν zt → zt

wν → w zν → z


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˜ E. CABANILLAS L., J. E. MUNOZ R.

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On the other hand,we note that 1 f1 (λν wν ) → f1 (0)w in L2 (0, T ; L2 (0, L0 )x]0, T [), (4.20) λν 1 f2 (λν z ν ) → f2 (0)z in L2 (0, T ; L2 (L0 , L)x]0, T [). (4.21) λν Indeed 1 f1 (λν wν )|2 2 ((0,L0 )x]0,T [) aˆ†ν = |f1 (0)wν − L λν 1 1 = |f1 (0)wν − f1 (uν )|2 dx dt + f1 (uν )|2 dx dt |f1 (0)wν − λν λν ν |≤ ν |> |u |u 1 ν 2 ν 2 f1 (u )| dx dt + 2|f1 (0)|2 |wν |2 dx dt ≤ |w f1 (0) − λν w ν ν |> ν |≤ |u |u 1 ν 2 +2 |f (u )| dx dt 2 1 |uν |> λν 1 1 2 ≤ Mε |wν |2 2 ((0,L0 )x]0,T [) + C ( 2 |uν |2 + 2 |uν |2ρ ) dx dt L λν λν ν |> |u 1 ν 2ρ 1 2 ≤ Mε |wν |2 2 (0,L0 x]0,T [) + C |u | (1 + 2ρ−2 ) dx dt L 2 ε ν |> λν |u
2 2ρ−2 ≤ Mε |wν |2 2 ((0,L0 )x]0,T [) + Cε λν |wν |2ρ ((0,L0 )x]0,T [) L L2ρ (s) where Mε = sup|s|≤ε |f1 (0) − f1s |, Mε →0 as ε → 0. ν From (4.13), is bounded in L∞ (0, T ; H 1 (0, L0 )) → L∞ (0, T ; L2ρ (0, L0 )), and consequently 2 lim sup aˆ†ν ≤ sup |wν |2 2 ((0,L0 )x]0,T [) .Mε L ν→∞ ν

Thus,taking ε → 0 we obtain (4.20). Applying a similar method as that used for we get (4.21). Now, the limit function satisfies ρ1 wtt − bwxx + f1 (0)w = 0 (a(x, t)zx )x + f2 (0)z = 0 in ]0, L0 [×]0, T [, in ]L0 , L[×]0, T [,

w(0, t) = 0 = z(L, t), w(L0 , t) = z(L0 , t), bwx (L0 , t) = a(L0 , t)zx (L0 , t), zt (x, t) = 0 in ]L0 , L[×]0, T [ Repeating the above procedure we get w = 0 and z = 0 which is a contradiction. The proof of lemma 4.5 is now complete. Proof of theorem 4.1. Let us introduce the functional L(t) = N E(t) + K(t) with N > 0. Using Young’s Inequality and taking N large enough we find that θ0 E(t) ≤ L(t) ≤ θ1 E(t) for some positive constants θ0 and θ1 . (4.22)


EJDE-2007/131

A NONLINEAR TRANSMISSION PROBLEM

13

Applying the inequalities (4.10) and (4.22), along with the ones in Lemma 4.5 and integrating from S to T where 0 ≤ S ≤ T < ∞ we obtain
T

E(t)dt ≤ c E(S).
S

In this situation, lemma2.1 implies E(t) ≤ c E(0)e−rt , this completes the proof. Remark. If in Equation (1.2) we consider a linear localized dissipation α = α(x) in C 2 (]L0 , L[), with α(x) = 1 in ]L0 , L0 + δ[ , α(x) = 0 in ]L0 + 2δ, L[, then our situation is very delicate and we need a new unique continuation theorem for the wave equation with variable coefficients. This is a work in preparation by the authors. References
[1] D. Andrade, L. H. Fatori, J. E. Mu˜oz Rivera; Nonlinear transmission Problem with a n Dissipative boundary condition of memory type. Electron. J. Diff. Eq, vol. 2006 (2006), No. 53, 1-16. [2] V. Komornik; Exact Controllability and Stabilization; The multiplier method. Masson. Paris [3] O. A. Ladyzhenskaya, N. N. Ural’tseva; Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. [4] I. Lasiecka, D. Tataru; Uniform boundary Stabilization of Semilinear wave Equations with Nonlinear Boundary Damping. Differential Integ. Eq. 6(3 1993) 507-533. [5] J. Lagnese; Boundary Controllability in Problem of Transmission for a class of second order Hyperbolic Systems, ESAIM: Control, Optim and Cal.Var. 2(1997), 343-357. [6] J. L. Lions; Quelques Methodes de RŽsolution dŽs R Žsolution dŽs problŽmes aux limites e e e e e Nonlineaires, Dunod, Gaulthier - Villars, Paris, 1969. [7] J. L. Lions; ControlabilitŽ Exacte, Perturbations et stabilization de Systems DistribuŽs (tome e e I), collection RMA, Masson, Paris 1988. [8] W. Liu, G. Williams; The exponential, The exponential stability of the problem of transmission of the wave equation, Balletin of the Austral. Math. Soc. 57 (1998), 305-327 [9] J. Mu˜oz Rivera and H. Portillo Oquendo; The transmission problem of Viscoelastic Waves, n Act. Appl. Math. 62 (2000) 1-21. [10] A. Ruiz; Unique Continuation for weak Solutions of the wave Equation plus a Potential, J. Math. Pures Appl. 71 (1992), 455-467. Eugenio Cabanillas Lapa Ž Ž Ž Instituto de Investigacion de Matematica, Facultad de Ciencias Matematicas, UniverŽ sidad Nacional Mayor de San Marcos, Lima, Peru E-mail address: cleugenio@yahoo.com ˜ Jaime E. Munoz Rivera Ž Laboratorio Nacional de Computacao Cientifica, Av. Getulio Vargas, 333, 25651-070ž Petropolis, Brazil E-mail address: rivera@lncc.br