Beam equation, non cylindrical domain, asymptotic behaviour, strong solutions ..... Using Gronwall's inequality and taking into account (1.16) we get. (2.9). L. (m).
ASYMPTOTIC BEHAVIOUR FOR THE NONLINEAR BEAM EQUATION IN A TIME-DEPENDENT DOMAIN
Jorge Ferreira
˜oz Rivera Jaime E. Mun
Rachid Benabidallah
Abstract. In this paper we prove existence of strong solutions as well as the exponential decay of the energy to the mixed problem for the nonlinear beam equation Z |ux |2 dx)uxx + νut = 0
utt + uxxxx − M (
in Qt ,
It
where Qt is a non-cylindrical domain of R2 with lateral boundary Σt . By ν we are denoting a positive constant. Here M (λ) is a real function such that M (λ) ≥ −m0 , for all λ ≥ 0, where m0 > 0.
1. Introduction. In this work we will study the existence of strong solution as well as the exponential decay of the energy to the nonlinear beam equation of Kirchhoff type given by, (1.1)
(1.2)
(1.3)
utt + uxxxx − M (
u(x, 0) = u0 (x),
Z
It
|ux |2 dx)uxx + νut = 0
ut (x, 0) = u1 (x) in
u(x, t) = 0,
ux (x, t) = 0
in Qt ,
α(0) < x < β(0),
on Σt .
By Qt we are denoting a non-cylindrical domain of R2 defined by
(1.4)
Qt = {(x, t) ∈ R2 |α(t) < x < β(t),
0 < t < T}
Key words and phrases. Beam equation, non cylindrical domain, asymptotic behaviour, strong solutions. Typeset by AMS-TEX 1
2
J. FERREIRA R. BENABIDALLAH J. E. M. RIVERA
where α(·), β(·) are C 3 -functions such that α(t) < β(t)
for all 0 ≤ t ≤ T .
The lateral boundary Σt of Qt is given by [ Σt = (α(t) × {t}) ∪ (β(t) × {t}). 0 0 Z t 2 M γ −1 kvy(m0 ) (t)k 2 − M γ −1 kvy (t)k2 2 dt (2.21) L L 0
0
≤ ckv (m ) − vkL2 (0,t;H 1 )
where c is a positive constant independent of m0 and t, so that (2.22)
2
0
2
0
(m ) M (γ −1 kvy(m ) (t)kL2 )(vyy , wj ) −→ M (γ −1 kvy (t)kL2 )(vyy , wj ).
Therefore we have that v satisfies v ∈ L∞ (0, ∞; H02(0, 1) ∩ H 3 (0, 1)), (2.23) vt ∈ L∞ (0, ∞; H02 (0, 1)), vtt ∈ L∞ (0, ∞; L2(0, 1)), and Z 1 j j −2 −1 (2.24) (vtt , w ) + ν(vt , w ) + γ M γ |vy |2 dy (vyy , w j )+ 0
+γ
−4
j (vyy , wyy )
+ (a1 vyy , w j )+
+ (a2 vty , w j ) + (a3 vy , w j ) = 0, (2.25)
v(y, 0) = v0 ,
vt (y, 0) = v1 .
For any w j ∈ Vm . Letting m → ∞ we conclude that v satisfies equation (1.8) in the sense of L∞ (0, ∞; L2(0, 1)) therefore we have that (2.26)
v ∈ L∞ (0, ∞; H02(0, 1) ∩ H 4 (0, 1))
The Uniqueness follows by using standard arguments Thus we have the following result
u t
ASYMPTOTIC BEHAVIOUR TO THE NONLINEAR BEAM EQUATION
9
THEOREM 2.1. Let us take v0 ∈ H02 (0, 1) ∩ H 4 (0, 1), v1 ∈ H02 (0, 1) and let us suppose that assumptions (1.14)-(1.18) holds. Then there exists a unique solution v of the problem (1.8)-(1.11) in the class (2.26) satisfying the equation (1.8) in the sense of L∞ (0, ∞; L2(0, 1))
To show the existence in non cylindrical domains, we return to our original ˆ problem by using the change variable given in (1.5) by (y, t) = h(x, t), (x, t) ∈ Q. Let v the solution obtained from Theorem 2.1 and u defined by (1.7), then u belong to the class u ∈ L∞ (0, ∞; H02 (It ) ∩ H 4 (It )), ut ∈ L∞ (0, ∞; H 2 (It )), (2.27) utt ∈ L∞ (0, ∞; L2 (It )),
where It =]α(t), β(t)[ for any t ≥ 0. Denoting by
u(x, t) = v(y, t) = (v ◦ h)(x, t),
then from (1.8) it is easy to see that u satisfies the equation (1.1) in the sense L∞ (0, ∞; L2(It )). The uniqueness of u follows from the uniqueness of v. Therefore, we have the following result. THEOREM 2.2. Let us take u0 ∈ H02 (I0 ) ∩ H 4 (I0 ), u1 ∈ H02 (I0 ) and let us suppose that assumptions (1.14)-(1.18) holds. Then there exists a unique solution u of the initial boundary value problem (1.1)-(1.3) satisfying (2.27) and the equation (1.1) in the sense of L∞ (0, ∞; L2(It )). 3. Asymptotic behaviour In this section we will show the exponential decay of the solution given by the theorem 2.2. To do this we assume that M satisfies the condition ˆ (r), M (r) ≥ −m0 ∀r ≥ 0 (3.1) M (r)r ≥ M with m0 such that
(3.2)
0 ≤ m0
