Local solution of carrier's equation in a noncylindrical domain
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We study Carrier’s equation in a noncylindrical domain. We use the penalty method combined with Faedo-Galerkin and compactness arguments. We obtain results of the existence and uniqueness of the local solution. 1. Introduction The wave equation of the type
∂2 u − P0 + P1 ∂t 2
β α
2 ∂u ∂2 u (t) dx =0 ∂x ∂x2
(1.1)
is a model for small vibrations of an elastic string fixed at α, β with P0 and P1 constants, and was investigated by Kirchhoff [4] and Carrier [1]. By approximations on the above equation, Carrier obtained the model
∂2 u − P0 + P1 ∂t 2
β α
2
u(t) dx
∂2 u = 0, ∂x2
(1.2)
which is known in the literature as Carrier’s equation. Medeiros et al. [7, 8] studied the problem of small vibrations of an elastic string with moving boundary. In this work, we study the following generalization of (1.2):
∂2 u − M x,t, ∂t 2
β(t) α(t)
u(t) dx
u=0 u(x,0) = u0 (x),
2
∂2 u =f ∂x2
on ,
ut (x,0) = u1 (x),
in Q,
in α(0), β(0) ,
where
= (i) Q ]α(t),β(t)[×{t } is a noncylindrical domain, 0 0 and a unique u satisfying
u ∈ L∞ 0,T0 ,H01 Ωt ∩ H 2 Ωt ,
u ∈ L∞ 0,T0 ,H 1 Ωt ,
(2.6)
u ∈ L 0,T0 ,L Ωt , 2
2
and u is the solution of (1.3). First of all we prove the equivalent result in a cylindrical domain. Theorem 2.2. Let α(t), β(t), and γ(t) be as in (H1)–(H4) and let M satisfy 1,∞ (Q)) and for each L > 0, ∂M/∂λ ∈ L∞ (Q × (0,L)), (i) M ∈ L∞ loc ([0, ∞),W (ii) M,∂M/∂λ, ∂M/∂t are continuous with respect to λ, a.e. (y,t) ∈ Q. There exists a real number m0 > 0 such that M(y,t,λ) ≥ m0 ,
∀(y,t) ∈ Q, λ ≥ 0.
(2.7)
Given v0 ∈ H01 (0,1) ∩ H 2 (0,1), v1 ∈ H01 (0,1), and g ∈ L2 ([0,T],H01 (0,1)) with g ∈ L2 ([0,T],L2 (0,1)), there exist T0 > 0 and a unique v satisfying
v ∈ L∞ 0,T0 ,H01 (0,1) ∩ H 2 (0,1) ,
∞
v ∈ L 0,T0 ,H01 (0,1) , v ∈ L∞ 0,T0 ,L2 (0,1) , and v is the solution of (2.5).
(2.8a) (2.8b) (2.8c)
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Local solution of Carrier’s equation in a noncylindrical domain
3. Proof of results Let (wν )ν∈N be the orthonormal complete set of L2 (0,1) given by the eigenvectors of the operator −d2 /dx2 , that is,
−
d2 wν = λν wν , dx2
wν = 0
on Γ.
(3.1)
We represent by Vm = [w1 ,...,wm ] the subspace of H01 (0,1) ∩ H 2 (0,1) generated by the m first vectors wν . We consider F : (0, ∞) → R a function satisfying F ∈ C 1 (0, ∞),
F (ξ) < 0,
F(ξ) = 1, ∀ξ ≥ 1. µ0 There exist µ0 , η0 , δ > 0 such that F(ξ) ≥ η0 , ∀ξ ∈ (0,δ]. ξ ∀ξ > 0,
(3.2) (3.3)
Let K > 0 such that 2 v1 < K.
(3.4)
For each ε > 0, we consider the following penalized problem: for each m ∈ N, let vεm (t) ∈ Vm satisfy
2 b2 (y,t) ∂2 vεm 1 vεm (t) − 2 M y,t,γ(t)vεm (t) − (t),w γ (t) 4 ∂y 2
2
K − vεm (t) ∂v ∂v + c(y,t) εm (t) + b(y,t) εm (t) + εF ∂y ∂y ε
vεm (t),w = g(t),w , (3.5)
for all w ∈ Vm , with the initial conditions vεm (0) = v0m −→ v0
in H01 (0,1) ∩ H 2 (0,1),
vεm (0) = v1m −→ v1
in H01 (0,1).
(3.6a) (3.6b)
As in [9] we prove that we can apply Carath´eodory’s theorem to (
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