Initial Boundary Value Problem and Asymptotic Stabilization of the Two-Component Camassa-Holm Equation

and Applied Analysis 3 Let T be a positive number. In the following we take ΩT 0, T × 0, 1 . Let vl and vr be in C0 0, T , R and m0 ∈ L∞ 0, 1 , ρ0 ∈ W1,∞ 0, 1 . We set Γl {t ∈ 0, T | vl t > 0}, Γr {t ∈ 0, T | vr t < 0}. 1.3 In the following, we will always suppose that the sets Pl {t ∈ 0, T | vl t 0}, Pr {t ∈ 0, T | vr t 0} 1.4 have a finite number of connected components. Finally, let ml, ρl ∈ L∞ Γl × W1,∞ Γl and mr, ρr ∈ L∞ Γr ×W1,∞ Γr . The given functions vl, vr ,ml, ρl, andmr , ρr will be the boundary values for the equation; m0, ρ0 are the initial data. Let now A t, x be the auxiliary function which lifts the boundary values vl and vr and is defined by 1 − ∂xx A t, x 0, ∀ t, x ∈ ΩT , A t, 0 vl t , A t, 1 vr t , ∀t ∈ 0, T . 1.5 Setting u θ A, we can further rewrite the system 1.1 as m t, x 1 − ∂xx θ t, x , θ t, 0 θ t, 1 0, 1.6 mt θ A mx −2m∂x θ A − ρρx − ρx, ρt θ A ρx − ( ρ 1 ) ∂x θ A , m 0, · m0, m ·, 0 |Γl ml, m ·, 1 |Γr mr, ρ 0, · ρ0, ρ ·, 0 |Γl ρl, ρ ·, 1 |Γr ρr. 1.7 Let y (m ρ ) , y0 (m0 ρ0 ) , b t, x ( −2∂x θ A 0 0 −∂x θ A ) , f t, x ( −ρρx−ρx −∂x θ A ) , and the system 1.7 can be written as ∂ty θ A ∂xy b t, x y f t, x , y 0, · y0, y ·, 0 |Γl yl, y ·, 1 |Γr yr. 1.8 We first define what we mean by a weak solution to 1.8 . Our test functions will be in the space: adm ΩT { ψ ∈ C1 ΩT × C1 ΩT | ∀x ∈ 0, 1 , ψ t, x 0; ∀t ∈ 0, T /Γl, ψ t, 0 0; ∀t ∈ 0, T /Γr , ψ t, 1 0 } . 1.9 4 Abstract and Applied Analysis Definition 1.1. Given y0 (m0 ρ0 ) ∈ L∞ ΩT × W1,∞ ΩT , when θ ∈ L∞ 0, T ; Lip 0, 1 , a function pair y (m ρ ) ∈ L∞ ΩT ×W1,∞ ΩT is a weak solution to 1.8 if y satisfies ∫∫ ΩT [ y∂tψ y θ A ∂xψ − yb t, x ψ − f t, x ψ ] dtdx