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)

306

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 (