Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 34, 1-27; http://www.math.u-szeged.hu/ejqtde/

Existence and Boundary Stabilization of the Semilinear Mindlin-Timoshenko System

F. D. Araruna^{∗}, J. E. S. Borges^{†}

Departamento de Matemática Universidade Federal da Paraíba 58051-900, João Pessoa - PB, Brasil

Abstract

We consider dynamics of the one-dimensional Mindlin-Timoshenko model for beams with a nonlinear external forces and a boundary damping mechanism. We investigate existence and uniqueness of strong and weak solution. We also study the boundary stabilization of the solution, i.e., we prove that the energy of every solution decays exponentially as t → ∞.

AMS Subject Classifications. 35L70, 35B40, 74K10 Key words. Mindlin-Timoshenko beam, continuous nonlinearity, boundary stability

1 Introduction

A widely accepted dynamical model describing the transverse vibrations of beams is the Mindlin-Timoshenko system of equations. This system is chosen because it is a more accurate model than the Euler-Bernoulli beam one and because it also takes into account transverse shear effects. The Mindlin-Timoshenko system is used, for example, to model aircraft wings. For a beam of length L > 0 this one-dimensional system reads as

| \begin{matrix}\frac{ρh^{3}}{12} u_{tt} − u_{xx} + k (u + v_{x}) + f (u) = 0 in Q, \\ ρhv_{tt} − k (u + v_{x})_{x} + g (v) = 0 in Q, \end{matrix}  (1.1)

where Q = (0, L) × (0, T) and T > 0 is a given time. In (1.1) subscripts mean partial derivatives. Here the function u = u (x, t) is the angle of deflection of a filament (it is measure of transverse shear effects) and v = v (x, t) is the transverse displacement of the beam at time t. The constant h > 0 represents the thickness of the beam that, for this model, is considered to be small and uniform, independent of x. The constant ρ is the mass density per unit volume of the beam and the parameter k is the so called modulus of elasticity in shear. It is given by the formula k = k̂Eh/2 (1 + μ) , where k̂ is a shear correction coefficient, E is the Young’s modulus and μ is the Poisson’s ratio, 0 < μ < 1/2. The functions f and g represent nonlinear external forces. For details concerning the Mindlin-Timoshenko hypotheses and governing equations see, for example, Lagnese [7] and Lagnese-Lions [8].

∗Partially supported by CAPES–Brazil and the Millennium Institute for the Global Advancement of Brazilian Mathematics IM–AGIMB, CNPq–Brazil (fagner@mat.ufpb.br).

†Partially supported by CAPES–Brazil (dudusampaioborges@hotmail.com).

We impose the following boundary conditions:

| \begin{matrix}u (0, ·) = v (0, ·) = 0 on (0, T) , \\ u_{x} (L, ·) + u_{t} (L, ·) = 0 on (0, T) , \\ u (L, ·) + v_{x} (L, ·) + v_{t} (L, ·) = 0 on (0, T) . \end{matrix}  (1.2)

The conditions (1.2) assure that the beam stays clamped in the end x = 0 and in the end x = L it is supported and suffering action of a dissipative force.

To complete the system, let us include the initial conditions:

u (·, 0) = u^{0}, u_{t} (·, 0) = u^{1}, v (·, 0) = v^{0}, v_{t} (·, 0) = v^{1} in (0, L) .  (1.3)

Several authors analyzed different aspects of the Mindlin-Timosheko system. In the linear case (f ≡ g ≡ 0) we can cite Lagnese-Lions [8], Medeiros [12], which studied the exact controllability property using the Hilbert Uniqueness Method (HUM) introduced by Lions (see [11]) and Lagnese [7] which analyzed the asymptotic behavior (as t → ∞) of the system. In Araruna-Zuazua [2] was made a spectral analysis of the system allowing to obtain a controllability using HUM combined with arguments of non-harmonic analysis. In the semilinear case, we can mention Parente et. al. [16], which treat about existence and uniqueness for the problem (1.1) − (1.3) , with the functions f and g being Lipschitz continuous, applying the same method used in Milla Miranda-Medeiros [15]. The existence of a compact global attractor, in the 2-dimensional case, was studied in Chueshov-Lasiecka [4] with the nonlinearities f and g being locally Lipschitz. All the mentioned papers are treated with different boundary conditions involving several situations that appear in the engineering.

In this work we state a result of existence of solutions for the system (1.1)−(1.3) , when the nonlinearities f and g satisfy the following conditions:

f, g are continuous function, such that f (s) s ≥ 0 and g(s)s ≥ 0, ∀s ∈ ℝ.  (1.4)

Furthermore, we analyze the asymptotic behavior (as t → ∞) of the solutions with the nonlinearities satisfy the additional growth condition:

∃δ_{1} > 0 such that f(s)s ≥ (2 + δ_{1})F (s), ∀s ∈ ℝ, where F (s) = ∫_{0}^{s} f (t) dt  (1.5)

and

∃δ_{2} > 0 such that g(s)s ≥ (2 + δ_{2})G(s), ∀s ∈ ℝ, where G (s) = ∫_{0}^{s} g (t) dt.  (1.6)

Precisely, we show the existence of positive constants C > 0 and κ > 0 such that the energy of the system (1.1) defined by

\begin{aligned}E(t) = & \frac{1}{2} [\frac{ρh^{3}}{12} ∫_{0}^{L} |u_{t} (x, t)|^{2} dx + ρh∫_{0}^{L} |v_{t} (x, t)|^{2} dx + k∫_{0}^{L} |(u + v_{x}) (x, t)|^{2} dx \\ & +∫_{0}^{L} |u_{x} (x, t)|^{2} dx + 2∫_{0}^{L}F (u(x, t))dx + 2∫_{0}^{L}G(v(x, t))dx] \end{aligned} (1.7)

verifies the estimate

E(t) ≤ CE (0) e^{−κt}, ∀t ≥ 0.  (1.8)

The uniqueness for the semilinear Mindlin-Timoshenko system (1.1) − (1.3) with the general nonlinearities considered here is a open problem.

To obtain existence of solution of the semilinear Mindlin-Timoshenko problem (1.1) − (1.3) , we found difficulties to show that the solution verifies the boundary conditions (1.2) and to overcome them, we use the same techniques applied in [1], that consists essentially in to combine results involving non-homogeneous boundary value problem with hidden regularity arguments. Boundary stability is also analyzed, that is, we show that the energy (1.7) associated to weak solution of the problem (1.1)−(1.3) tends to zero exponentially as t → ∞. In order, the exponential decay was obtained by constructing perturbed energy functional for which differential inequality leads to this rate decay. We apply this method motivated by work of Komornik-Zuazua [6], whose authors treated this issue for semilinear wave equation.

The paper is organized as follows. Section 2 contains some notations and essential results which we apply in this work. In Section 3 we prove existence and uniqueness of strong solution for (1.1) − (1.3) employing the Faedo-Galerkin’s method with a special basis like in [15] with f and g being Lipschitz continuous functions satisfying a sign condition. Section 4 is devoted to get existence of weak solution of (1.1)−(1.3) , with f and g satisfying (1.4). For this, we approached the functions f and g by Lipschitz functions, as in Strauss [17], and we obtain the weak solution as limit of sequence of strong solutions acquired in the Section 3. We still analyze the uniqueness only for some particular cases of f and g which permit the application of the energy method as in Lions [9]. Finally, in Section 5 we prove the exponential decay for the energy associated to weak solution of the problem (1.1) − (1.3) making use of the perturbed energy method as in [6].

2 Some Notations and Results

Let us represent by D(0, T) the space of the test functions defined in (0, T) and H^{1} (0, L) the usual Sobolev space. We define the Hilbert space

V = {v ∈ H^{1} (0, L) ; v (0) = 0}

equipped with the inner product and norm given by

((u, v)) = (u_{x}, v_{x}) , ∥u∥^{2} = |u_{x}|^{2} ,

where (·, ·) and |·| are, respectively, the inner product and norm in L^{2} (0, L) . By V′ we denote the dual of V.

Let us consider the operator −\frac{d^{2}}{dx^{2}} defined by tripled \{V, L^{2} (0, L) ; ((·, ·))\} with domain

D = {u ∈ V ∩ H^{2} (0, L) ; u_{x} (L) = 0} .

Let us represent by E the Banach space

E = {v ∈ L^{2} (Ω) ; v_{xx} ∈ L^{1} (Ω)}

with the norm

∥v∥_{E} = |v| + ∥v_{xx}∥_{L^{1}(Ω)} .

The trace application γ : E → ℝ^{4} defined by γv = (v (0) , v (L) , v_{x} (0) , v_{x} (L)) is linear and continuous, see Milla Miranda-Medeiros [14, Proposition 3.2].

In what follows, we will use C to denote a generic positive constant which may vary from line to line (unless otherwise stated).

We will now establish some results of elliptic regularity essential for the development of this work.

Proposition 2.1 Let us consider f ∈ L^{2} (0, L) and β ∈ ℝ. Then the solution u of the boundary value problem

| \begin{matrix}−u_{xx} = f in (0, L) , \\ u (0) = 0, \\ u_{x} (L) = β, \end{matrix}  (2.1)

belongs to V ∩ H^{2} (0, L) . Furthermore, there exists a constant C > 0 such that

∥u∥_{H^{2}(0,L)} ≤ C [|f| + |β|] .  (2.2)

Proof. We consider the function h : [0, L] → ℝ, defined by h (x) = βx. Thus

∥h∥_{H^{2}(0,L)} = C |β| ,  (2.3)

where C = \sqrt{(L^{3}/3) + L}.

Let w be the unique solution of the following boundary value problem:

| \begin{matrix}−w_{xx} = f in (0, L) , \\ w (0) = 0, \\ w_{x} (L) = 0. \end{matrix}

Since f ∈ L^{2}(0, L), we have by classical elliptic result (see for instance [3]) that w ∈ D and the existence of a constant C > 0 such that

∥w∥_{H^{2}(0,L)} ≤ C |f| .  (2.4)

In this way, u = w + h ∈ V ∩ H^{2} (0, L) solves (2.1) satisfying (2.2) .

We would like to prove existence and uniqueness of solution for the problem

| \begin{matrix}−u_{xx} = f in (0, L) , with f ∈ L^{1} (0, L) , \\ u (0) = 0, \\ u_{x} (L) = 0. \end{matrix}  (2.5)

Formally, we obtain from (2.5) that

∫_{0}^{L}u(−v_{xx})dx + u_{x} (0) v (0) + u (L) v_{x} (L) = ∫_{0}^{L}fvdx.  (2.6)

Taking in (2.6) v ∈ D, we obtain

∫_{0}^{L}u(−v_{xx})dx = ∫_{0}^{L} fvdx, ∀v ∈ D.  (2.7)

We adopt (2.7) as definition of solution of (2.5) in the sense of transposition (see [10]). To guarantee the existence and uniqueness of (2.5) we consider the follow result:

Proposition 2.2 If f ∈ L^{1} (0, L) , then there exists a unique function u ∈ E satisfying (2.7). The application T : L^{1} (0, L) → L^{2} (0, L) such that T f = u is linear, continuous and −u_{xx} = f.

Proof. Let g ∈ L^{2} (0, L) and v be a solution of the problem

| \begin{matrix}−v_{xx} = g in (0, L) , \\ v (0) = 0, \\ v_{x} (L) = 0. \end{matrix}  (2.8)

We have v ∈ D.

Let us consider the application S : L^{2} (0, L) → C^{0} ([0, L]) such that Sg = v, where v is the solution of (2.8). Then S is linear and continuous. Let S^{∗} be the transpose of S, that is,

S^{∗} : [C^{0} ([0, L])]′ → L^{2} (0, L) ; ⟨S^{∗}θ, ϕ⟩ = ⟨θ, Sϕ⟩ , ∀ϕ ∈ L^{2} (0, L) ,

where ⟨·, ·⟩ represents different pairs of duality. Let us prove that the function u = S^{∗}f satisfies (2.7). In fact, we have ⟨S^{∗}f, g⟩ = ⟨f, Sg⟩ , which means

∫_{0}^{L}u(−v_{xx})dx = ∫_{0}^{L} fvdx.

For the uniqueness, we consider u_{1}, u_{2} ∈ L^{2} (0, L) satisfying (2.7). Then

∫_{0}^{L} (u_{1} − u_{2}) (−v_{xx}) dx = 0, ∀v ∈ D.  (2.9)

Considering g ∈ L^{2} (0, L) and v be a solution of (2.8), we get

∫_{0}^{L} (u_{1} − u_{2}) gdx = 0, ∀g ∈ L^{2} (0, L) .

Therefore u_{1} = u_{2} and the uniqueness is proved. Since T = S^{∗} and S^{∗} is linear and continuous, it follows that T has the same properties.

For the non-homogeneous boundary value problem

| \begin{matrix}−u_{xx} = f in (0, L) , \\ u (0) = 0, \\ u_{x} (L) = β, \end{matrix}  (2.10)

we consider the following result:

Proposition 2.3 Let f ∈ L^{1} (0, L) and β ∈ ℝ. Then there exists a unique solution u ∈ E for the problem (2.10).

Proof. Let us consider the function ξ : [0, L] → ℝ, defined by ξ(x) = βx. Let w be the solution of the problem

| \begin{matrix}−w_{xx} = f in (0, L) , \\ w (0) = 0, \\ w_{x} (L) = 0. \end{matrix}

Since f ∈ L^{1} (0, L) , by Proposition 2.2, it follows that w ∈ E. Taking u = w + ξ, we have u ∈ E is a solution of (2.10).

For the uniqueness, let u_{1} and u_{2} two solutions of (2.10). Then v = u_{1} − u_{2} is solution of

| \begin{matrix}−v_{xx} = 0 in (0, L) , \\ v (0) = 0, \\ v_{x}(L) = 0. \end{matrix}

Hence, by Proposition 2.2, we have v = 0, which implies u_{1} = u_{2}.

Proposition 2.4 In V ∩ H^{2} (0, L) the norms H^{2} (0, L) and the norm

u ↦ (|−u_{xx}|^{2} + |u_{x} (L)|^{2})^{\frac{1}{2}} ,  (2.11)

are equivalents.

Proof. Let u ∈ V ∩ H^{2} (0, L) . Then, according to Proposition 2.1, we can guarantee that

∥u∥_{H^{2}(0,L)} ≤ C (|−u_{xx}|^{2} + |u_{x} (L)|^{2})^{\frac{1}{2}} .

On the other hand, since the embedding of H^{2} (0, L) in C^{1} ([0, L]) is continuous, we have

|u_{x} (L)| ≤ ∥u∥_{C^{1}([0,L])} ≤ C ∥u∥_{H^{2}(0,L)} .

We also have |−u_{xx}|^{2} ≤ C ∥u∥_{H^{2}(0,L)} . In this way we obtain the result.

We consider V ∩ H^{2} (0, L) equipped with the norm (2.11) .

Proposition 2.5 Let us suppose u^{0}, v^{0} ∈ V ∩ H^{2} (0, L) and u^{1}, v^{1} ∈ V such that u_{x}^{0} (L) + u^{1} (L) = 0 and u^{0} (L) + v_{x}^{0} (L) + v^{1} (L) = 0. Then, for each ϵ > 0, there exist w^{(1)}, z^{(1)}, w^{(2)} and z^{(2)} in V ∩ H^{2} (0, L) such that

∥w^{(1)} − u^{0}∥_{V ∩H^{2}(0,L)} < ϵ, ∥z^{(1)} − u^{1}∥_{V} < ϵ,
∥w^{(2)} − v^{0}∥_{V ∩H^{2}(0,L)} < ϵ and ∥z^{(2)} − v^{1}∥_{V} < ϵ,

with

w_{x}^{(1)} (L) + z^{(1)} (L) = 0 and u^{0} (L) + w_{x}^{(2)} (L) + z^{(2)} (L) = 0.

Proof. Since V ∩ H^{2} (0, L) is dense in V, for each ϵ > 0, there exist z^{(1)}, z^{(2)} ∈ V ∩ H^{2} (0, L) such that ∥z^{(1)} − u^{1}∥_{V} < ϵ and ∥z^{(2)} − v^{1}∥_{V} < ϵ.

Let us consider w^{(1)} to be a solution of the problem

| \begin{matrix}−w_{xx}^{(1)} = −u_{xx}^{0} in (0, L) , \\ w^{(1)} (0) = 0, \\ w_{x}^{(1)} (L) = −z^{(1)} (L) . \end{matrix}

According to Proposition 2.1, it follows that w^{(1)} ∈ V ∩ H^{2} (0, L) and

\begin{matrix}∥w^{(1)} − u^{0}∥_{V ∩H^{2}(0,L)}^{2} = |−w_{xx}^{(1)} + u_{xx}^{0}|^{2} + |w_{x}^{(1)} (L) − u_{x}^{0} (L)|^{2} = |−z^{(1)} (L) + u^{1} (L)|^{2} \\ ≤ C ∥z^{(1)} − u^{1}∥_{V}^{2} < Cϵ^{2}. \end{matrix}

Analogously, let us consider w^{(2)} to be a solution of the problem

| \begin{matrix}−w_{xx}^{(2)} = −v_{xx}^{0} in (0, L) , \\ w^{(2)} (0) = 0, \\ w_{x}^{(2)} (L) = −u^{0} (L) − z^{(2)} (L) . \end{matrix}

By Proposition 2.1, we have that w^{(2)} ∈ V ∩ H^{2} (0, L) and

∥w^{(2)} − v^{0}∥_{V ∩H^{2}(0,L)}^{2} = |−w_{xx}^{(2)} + v_{xx}^{0}|^{2} + |w_{x}^{(2)} (L) − v_{x}^{0} (L)|^{2}
= |−u^{0}(L) − z^{(2)} (L) − (−u^{0}(L) − v^{1} (L))|^{2} = |−z^{(2)} (L) + v^{1} (L)|^{2}
≤ C ∥z^{(2)} − v^{1}∥_{V}^{2} = Cϵ^{2},

concluding the result.

3 Strong Solution

Our goal in this section is to prove existence and uniqueness of solutions for the problem (1.1) − (1.3) , when u^{0}, v^{0}, u^{1} and v^{1} are smooth.

Let be f, g functions defined in ℝ and u^{0}, v^{0}, u^{1}, v^{1} functions defined in (0, L) satisfying

f, g : ℝ → ℝ are Lipschitz function with constant c_{f}, c_{g}, respectively,

and sf (s) ≥ 0, sg (s) ≥ 0, ∀s ∈ ℝ, (3.1)

(u^{0}, u^{1}) ∈ [V ∩ H^{2} (0, L)] × V,  (3.2)
(v^{0}, v^{1}) ∈ [V ∩ H^{2} (0, L)] × V,  (3.3)
u_{x}^{0} (L) + u^{1} (L) = 0,  (3.4)
u^{0} (L) + v_{x}^{0} (L) + v^{1} (L) = 0.  (3.5)

Theorem 3.1 Let f, g, u^{0}, v^{0}, u^{1} and v^{1} satisfying the hypotheses (3.1) − (3.5). Then there exist unique functions u, v : Q → ℝ, such that

u, v ∈ L^{∞} (0, T, V) ∩ L^{2} (0, T, H^{2} (0, L)) ,  (3.6)
u_{t}, v_{t} ∈ L^{∞} (0, T, V) ,  (3.7)
u_{tt}, v_{tt} ∈ L^{2}(Q),  (3.8)
\frac{ρh^{3}}{12} u_{tt} − u_{xx} + k (u + v_{x}) + f(u) = 0 in L^{2} (Q) ,  (3.9)
ρhv_{tt} − k (u + v_{x})_{x} + g(v) = 0 in L^{2} (Q) ,  (3.10)
u_{x} (L, ·) + u_{t} (L, ·) = 0 in (0, T) ,  (3.11)
u (L, ·) + v_{x} (L, ·) + v_{t} (L, ·) = 0 in (0, T) ,  (3.12)
u(0) = u^{0}, u_{t}(0) = u^{1}, v(0) = v^{0}, v_{t}(0) = v^{1} in (0, L) .  (3.13)

Proof. We employ the Faedo-Galerkin’s method with the special basis in V ∩ H^{2} (0, L) . Since the data u^{0}, v^{0}, u^{1} and v^{1} verify (3.2)−(3.5) , it follows by Proposition 2.5 the existence of four sequences (u^{0ν})_{ν∈ℕ}, (u^{1ν})_{ν∈ℕ}, (v^{0ν})_{ν∈ℕ} and (v^{1ν})_{ν∈ℕ} of vectors in V ∩ H^{2} (0, L) such that

u^{0ν} → u^{0} strongly in V ∩ H^{2} (0, L) ,  (3.14)
v^{0ν} → v^{0} strongly in V ∩ H^{2} (0, L) ,  (3.15)
u^{1ν} → u^{1} strongly in V,  (3.16)
v^{1ν} → v^{1} strongly in V,  (3.17)
u_{x}^{0ν}(L) + u^{1ν}(L) = 0, ∀ν ∈ ℕ,  (3.18)
u^{0ν}(L) + v_{x}^{0ν}(L) + v^{1ν}(L) = 0, ∀ ν ∈ ℕ.  (3.19)

We fix ν ∈ ℕ. If A = {u^{0ν}, v^{0ν}, u^{1ν}, v^{1ν}} is a linearly independent set, we take

w_{1}^{ν} = \frac{u^{0ν}}{∥u^{0ν}∥_{V ∩H^{2}(0,L)}} , w_{2}^{ν} = \frac{u^{1ν}}{∥u^{1ν}∥_{V ∩H^{2}(0,L)}} , w_{3}^{ν} = \frac{v^{1ν}}{∥v^{1ν}∥_{V ∩H^{2}(0,L)}} and w_{4}^{ν} = \frac{v^{0ν}}{∥v^{0ν}∥_{V ∩H^{2}(0,L)}}

as being the first four vectors of the basis. By Gram-Schmidt’s orthonormalization process, we construct, for each ν ∈ ℕ, a basis in V ∩H^{2} (Ω) represented by {w_{1}^{ν}, w_{2}^{ν}, w_{3}^{ν}, w_{4}^{ν}, …, w_{n}^{ν}, …}. Otherwise, if A is a linearly dependent set, we can extract a linearly independent subset of A and continue the above process. For each m ∈ ℕ, we consider V_{m}^{ν} = [w_{1}^{ν}, w_{2}^{ν}, w_{3}^{ν}, w_{4}^{ν}, …, w_{m}^{ν}] the subspace of V ∩ H^{2} (Ω) generated by the first m vectors of basis. Let us find an ”approximate solution” (u^{νm}, v^{νm}) ∈ V_{m}^{ν} × V_{m}^{ν} of the type

u^{νm} (x, t) = ∑_{j=1}^{m} μ^{jνm} (t) w_{j}^{ν}(x), v^{νm} (x, t) = ∑_{j=1}^{m} h^{jνm} (t) w_{j}^{ν}(x),

where μ^{jνm}(t) and h^{jνm}(t) are solutions of the initial value problem

| \begin{matrix}\frac{ρh^{3}}{12} (u_{tt}^{νm} (t) , ψ) − (u_{xx}^{νm} (t) , ψ) + k ((u^{νm} + v_{x}^{νm}) (t) , ψ) + (f(u^{νm} (t)), ψ) = 0, \\ ρh (v_{tt}^{νm} (t) , φ) − k (((u^{νm} + v_{x}^{νm}) (t))_{x}, φ) + (g(v^{νm} (t)), φ) = 0, \\ u^{νm}(0) = u^{0νm}, u_{t}^{νm}(0) = u^{1νm}, v^{νm}(0) = v^{0νm}, v_{t}^{νm}(0) = v^{1νm} in (0, L) , \end{matrix} (3.20)

for all ψ, φ ∈ V_{m}^{ν}, where

(u^{0νm}, u^{1νm}, v^{0νm}, v^{1νm}) → (u^{0}, u^{1}, v^{0}, v^{1}) strongly in [V ∩ H^{2} (0, L) × V ]^{2} .  (3.21)

The system (3.20) has solution on an interval [0, t_{νm}] , with t_{νm} < T. This solution can be extended to the whole interval [0, T] as a consequence of a priori estimates that shall be proved in the next step.

Adding the equations in (3.20) results

\begin{matrix}\frac{ρh^{3}}{12} (u_{tt}^{νm} (t) , ψ) + ρh (v_{tt}^{νm} (t) , φ) + k ((u^{νm} + v_{x}^{νm}) (t) , ψ + φ_{x}) + ((u^{νm} (t) , ψ)) \\ +u_{t}^{νm} (L, t) ψ (L, t) + v_{t}^{νm} (L, t) φ (L, t) + (f(u^{νm} (t)) , ψ) + (g(v^{νm} (t)) , φ) = 0, \end{matrix} (3.22)

for all ψ, φ ∈ V_{m}^{ν}. Estimates I. Making ψ = 2u_{t}^{νm} (t) , φ = 2v_{t}^{νm} (t) in (3.22) , integrating from 0 to t ≤ t_{νm} and using (3.21), we get

\begin{matrix}\frac{ρh^{3}}{12} |u_{t}^{νm}(t)|^{2} + ρh |v_{t}^{νm}(t)|^{2} + k |(u^{νm} + v_{x}^{νm}) (t)|^{2} + ∥u^{νm}(t)∥^{2} \\ +2∫_{0}^{t} |u_{t}^{νm}(L, t)|^{2} dt + 2∫_{0}^{t} |v_{t}^{νm}(L, t)|^{2} dt + 2∫_{0}^{L}F (u^{νm} (x, t))dx \\ +2∫_{0}^{L}G(v^{νm} (x, t))dx ≤ C + 2∫_{0}^{L}F (u^{0νm})dx + 2∫_{0}^{L}G(v^{0νm})dx, \end{matrix}  (3.23)

where F (t) = ∫_{0}^{t} f(s)ds, G(t) = ∫_{0}^{t} g(s)ds and the constant C > 0 is independent of m, ν and t. We must obtain estimates for the terms 2 ∫_{0}^{L} F (u^{0νm})dx and 2 ∫_{0}^{L} G(v^{0νm})dx. Since f(s)s ≥ 0 and g(s)s ≥ 0, it follows that F (t) ≥ 0 and G(t) ≥ 0, for all t ∈ [0, T] and f(0) = g(0) = 0. So, by (3.1), we have

∫_{0}^{L}F (u^{0νm})dx ≤ c_{f} |u^{0νm}|^{2} and ∫_{0}^{L}G(v^{0νm})dx ≤ c_{g} |v^{0νm}|^{2} .  (3.24)

From (3.21) and (3.24), the inequality (3.23) becomes

\begin{matrix}\frac{ρh^{3}}{12} |u_{t}^{νm}(t)|^{2} + ρh |v_{t}^{νm}(t)|^{2} + k |(u^{νm} + v_{x}^{νm})(t)|^{2} + ∥u^{νm}(t)∥^{2} + 2∫_{0}^{t} |u_{t}^{νm}(L, s)|^{2} ds \\ +2∫_{0}^{t} |v_{t}^{νm}(L, s)|^{2} ds + 2∫_{0}^{L}F (u^{νm} (x, t))dx + 2∫_{0}^{L}G(v^{νm} (x, t))dx ≤ C, \end{matrix}

(3.25)

where C > 0 is a constant which is independent of m, ν and t. In this way, we can prolong the solution to the whole interval [0, T] . Estimates II. Considering the temporal derivative of the approximate equation (3.22), setting ψ = u_{tt}^{νm} (t) and φ = v_{tt}^{νm} (t) in the resulting equation and integrating from 0 to t ≤ T we get

\frac{ρh^{3}}{12} |u_{tt}^{νm}(t)|^{2} + ρh |v_{tt}^{νm}(t)|^{2} + k |(u^{νm} + v_{x}^{νm})_{t} (t)|^{2} + ∥u_{t}^{νm}(t)∥^{2} + 2∫_{0}^{t} |u_{tt}^{νm} (L, s)|^{2} ds

+2∫_{0}^{t} |v_{tt}^{νm} (L, s)|^{2} dt ≤ \frac{ρh^{3}}{12} |u_{tt}^{νm} (0)|^{2} + ρh |v_{tt}^{νm} (0)|^{2} + k |u^{1νm} + v_{x}^{1νm}|^{2} + ∥u^{1νm}∥^{2}

+2∫_{0}^{t} |(f_{t}(u^{νm} (s) u_{t}^{νm} (s) , u_{tt}^{νm} (s))| ds + 2∫_{0}^{t} |(g_{t}(v^{νm} (s) v_{t}^{νm} (s) , v_{tt}^{νm} (s))| ds.

(3.26)

We need estimates for the terms involving u_{tt}^{νm} (0) , v_{tt}^{νm} (0) and for last two integrals in (3.26) . For this, we consider in (3.22) t = 0, ψ = u_{tt}^{νm}(0) and φ = v_{tt}^{νm}(0). So, using (3.1) and (3.21) we obtain

\frac{ρh^{3}}{12} |u_{tt}^{νm} (0)|^{2} + ρh |v_{tt}^{νm} (0)|^{2} ≤ C,  (3.27)

where C > 0 is a constant independent of m, ν and t. We also have by (3.1) that |f_{t} (s)| ≤ c_{f} and |g_{t} (s)| ≤ c_{g}, a. e. in ℝ. Then

\begin{matrix}2∫_{0}^{t} |(f_{t}(u^{νm} (s) u_{t}^{νm} (s) , u_{tt}^{νm} (s))| ds + 2∫_{0}^{t} |(g_{t}(v^{νm} (s) v_{t}^{νm} (s) , v_{tt}^{νm} (s))| ds \\ ≤ \frac{c_{f}}{2} ∫_{0}^{t} |u_{t}^{νm} (t)|^{2} dt + \frac{c_{f}}{2} ∫_{0}^{t} |u_{tt}^{νm} (t)|^{2} dt + \frac{c_{g}}{2} ∫_{0}^{t} |v_{t}^{νm} (t)|^{2} dt + \frac{c_{g}}{2} ∫_{0}^{t} |v_{tt}^{νm} (t)|^{2} dt. \end{matrix} (3.28)

Thus, using (3.21) and the estimates (3.27), (3.28) in (3.26) we get

\begin{matrix}|u_{tt}^{νm}(t)|^{2} + |v_{tt}^{νm}(t)|^{2} + |(u^{νm} + v_{x}^{νm})_{t} (t)|^{2} + ∥u_{t}^{νm}(t)∥^{2} + 2∫_{0}^{t} |u_{tt}^{νm} (L, t)|^{2} dt \\ +2∫_{0}^{t} |v_{tt}^{νm} (L, t)|^{2} dt ≤ C, \end{matrix}  (3.29)

where C = C (c_{f}, c_{g}) > 0 is a constant independent of t, ν and m.

According to (3.25) and (3.29), we have

(u^{νm}) is bounded in L^{∞}(0, T, V),  (3.30)
(u_{t}^{νm}) is bounded in L^{∞}(0, T, V),  (3.31)
(u_{tt}^{νm}) is bounded in L^{2}(Q),  (3.32)
(v^{νm}) is bounded in L^{∞}(0, T, V),  (3.33)
(v_{t}^{νm}) is bounded in L^{∞}(0, T, V),  (3.34)
(v_{tt}^{νm}) is bounded in L^{2}(Q),  (3.35)

From (3.30) − (3.35), we can obtain subsequences of (u^{νm}) and (v^{νm}), which will be also denoted by (u^{νm}) and (v^{νm}), such that

u^{νm} → u^{ν} weak ∗ in L^{∞}(0, T, V),  (3.36)
u_{t}^{νm} → u_{t}^{ν} weak ∗ in L^{∞}(0, T, V),  (3.37)
u_{tt}^{νm} → u_{tt}^{ν} weakly in L^{2}(Q),  (3.38)
v^{νm} → v^{ν} weak ∗ in L^{∞}(0, T, V),  (3.39)
v_{t}^{νm} → v_{t}^{ν} weak ∗ in L^{∞}(0, T, V),  (3.40)
v_{tt}^{νm} → v_{tt}^{ν} weakly in L^{2} (Q) .  (3.41)

According to (3.1), (3.30), (3.33) and the compact injection of H^{1}(Q) in L^{2}(Q), there exists a subsequence of (u^{νm}) and (v^{νm}), which will be also denoted by (u^{νm}) and (v^{νm}), such that

f(u^{νm}) → f(u^{ν}) strongly in L^{2} (Q) ,  (3.42)
g(v^{νm}) → g(v^{ν}) strongly in L^{2} (Q) .  (3.43)

We can see that the estimates (3.25) and (3.29) are also independent of ν. So, using the same arguments to obtain u^{ν} and v^{ν}, we can pass to the limit, as ν → ∞, to obtain functions u and v such that

u^{ν} → u weak ∗ in L^{∞}(0, T, V),  (3.44)
u_{t}^{ν} → u_{t} weak ∗ in L^{∞}(0, T, V),  (3.45)
u_{tt}^{ν} → u_{tt} weakly in L^{2}(Q),  (3.46)
v^{ν} → v weak ∗ in L^{∞}(0, T, V),  (3.47)
v_{t}^{ν} → v_{t} weak ∗ in L^{∞}(0, T, V),  (3.48)
v_{tt}^{ν} → v_{tt} weakly in L^{2}(Q),  (3.49)
f(u^{ν}) → f(u) strongly in L^{2} (Q) ,  (3.50)

g(v^{ν}) → g(v) strongly in L^{2} (Q) .  (3.51)

Making m → ∞ and ν → ∞ in the equations in (3.20) and using the convergences (3.36) − (3.51) we have

\begin{matrix}\frac{ρh^{3}}{12} ∫_{0}^{T} (u_{tt} (t) , ψ) θdt + ∫_{0}^{T} ((u (t) , ψ)) θdt + ∫_{0}^{T} u_{t}(L, t)ψ(L, t)θdt \\ +k∫_{0}^{T} ((u + v_{x}) (t) , ψ) θdt + ∫_{0}^{T} (f(u (t)), ψ) θdt = 0, ∀ψ ∈ V, ∀θ ∈ D(0, T) \end{matrix} (3.52)

and

\begin{matrix}ρh∫_{0}^{T} (v_{tt} (t) , φ) θdt + k∫_{0}^{T} ((u + v_{x}) (t) , φ_{x})θdt + k∫_{0}^{T} v_{t}(L, t)φ(L, t)θdt \\ +∫_{0}^{T} (g(v (t)), φ) θdt = 0, ∀φ ∈ V, ∀θ ∈ D(0, T). \end{matrix}  (3.53)

Taking φ, ψ ∈ D(0, L), it follows that

\frac{ρh^{3}}{12} u_{tt} − u_{xx} + k(u + v_{x}) + f(u) = 0 in L^{2}(Q)  (3.54)

and

ρhv_{tt} − k(u + v_{x})_{x} + g(v) = 0 in L^{2}(Q).  (3.55)

Multiplying (3.54) by ψθ, ψ ∈ V and θ ∈ D(0, T), integrating in Q and comparing with (3.52), we get

∫_{0}^{T} [u_{t}(L, t) + u_{x}(L, t)] ψ(L, t)θdt = 0, ∀θ ∈ D(0, T), ∀ψ ∈ V.

Consequently

u_{t}(L) + u_{x}(L) = 0 on (0, T) .  (3.56)

Now, multiplying (3.55) by φθ, φ ∈ V and θ ∈ D(0, T), integrating in Q and comparing with (3.53), we obtain

k∫_{0}^{T} (v_{t}(L, t) + u(L, t) + v_{x}(L, t))φ(L, t)θdt = 0, ∀θ ∈ D(0, T), ∀φ ∈ V,

which implies

v_{t}(L) + u(L) + v_{x}(L) = 0 on (0, T) .  (3.57)

To complete the proof of the theorem, we need to show that u, v ∈ L^{2}(0, T, H^{2}(0, L)). For this, we consider the following boundary value problem:

| \begin{matrix}−u_{xx} (t) = −\frac{ρh^{3}}{12} u_{tt} (t) − k(u + v_{x}) (t) − f (u (t)) in (0, L) , \\ u(0, t) = 0, \\ u_{x}(L, t) = −u_{t}(L, t) \end{matrix}  (3.58)

and

| \begin{matrix}−v_{xx} (t) = −\frac{ρh}{k}v_{tt} (t) + u_{x} (t) − \frac{1}{k}g (v (t)) in (0, L), \\ v(0, t) = 0, \\ v_{x}(L, t) = −(u(L, t) + v_{t}(L, t)). \end{matrix}  (3.59)

Since −\frac{ρh^{3}}{12} u_{tt} −k(u+v_{x})−f(u), \frac{ρh}{k}v_{tt} +u_{x} − \frac{1}{k}g(v) ∈ L^{2}(Q), it follows by Proposition 2.1 that u, v ∈ L^{2}(0, T, H^{2}(0, L)). Using a standard argument, we can verify the initial conditions. The uniqueness of solution is proved by energy method.

4 Weak Solution

The purpose of this section is to obtain existence of solutions for the problem (1.1) − (1.3), with less regularity on the initial data and now f, g being continuous functions and sf (s) ≥ 0, sg (s) ≥ 0, ∀s ∈ ℝ. Owing to few regularity of the initial data, the corresponding solutions shall be called weak.

Theorem 4.1 Let us consider

f, g : ℝ → ℝ are continuous functions such that f(s)s ≥ 0 and g(s)s ≥ 0, ∀s ∈ ℝ, (4.1)
(u^{0}, u^{1}, v^{0}, v^{1}) ∈ [V × L^{2}(0, L)]^{2} ,  (4.2)
F (u^{0}), G(v^{0}) ∈ L^{1}(0, L).  (4.3)

Then there exist at least two functions u, v : Q → ℝ such that

u, v ∈ L^{∞}(0, T, V),  (4.4)
u_{t}, v_{t} ∈ L^{∞}(0, T, L^{2}(0, L)),  (4.5)
\frac{ρh^{3}}{12} u_{tt} − u_{xx} + k (u + v_{x}) + f(u) = 0 in L^{1}(0, T, V′ + L^{1}(0, L)),  (4.6)
ρhv_{tt} − k (u + v_{x})_{x} + g(v) = 0 in L^{1}(0, T, V′ + L^{1}(0, L)),  (4.7)
u_{x} (L, ·) + u_{t} (L, ·) = 0 in L^{2}(0, T),  (4.8)
u (L, ·) + v_{x} (L, ·) + v_{t} (L, ·) = 0 in L^{2}(0, T),  (4.9)
u(0) = u^{0}, u_{t}(0) = u^{1}, v(0) = v^{0}, v_{t}(0) = v^{1} in (0, L) .  (4.10)

Proof. There exist two sequences of functions (f_{ν})_{ν∈ℕ} and (g_{ν})_{ν∈ℕ}, such that, for each ν ∈ ℕ, f_{ν}, g_{ν} : ℝ → ℝ are Lipschitz functions with constants c_{f_{ν}} and c_{g_{ν}}, respectively, satisfying sf_{ν}(s) ≥ 0 and sg_{ν}(s) ≥ 0, ∀s ∈ ℝ and (f_{ν})_{ν∈ℕ}, (g_{ν})_{ν∈ℕ} approximate f and g, respectively, uniformly on bounded sets of ℝ. The construction of these sequences can be seen in Strauss [17].

Since the initial data u^{0} and v^{0} are not necessarily bounded, we approximate u^{0} and v^{0} by bounded functions of V. We consider the functions ξ_{j} : ℝ → ℝ defined by

ξ_{j}(s) = \{ \begin{matrix}−j, & if s < −j, \\ s, & if |s| ≤ j, \\ j, & if s > j. \end{matrix}

Considering ξ_{j}(u^{0}) = u^{0j} and ξ_{j}(v^{0}) = v^{0j}, we have by Kinderlehrer-Stampacchia [5] that the sequences (u^{0j})_{j∈ℕ} and (v^{0j})_{j∈ℕ} in V are bounded in [0, L] and

u^{0j} → u^{0} strongly in V,  (4.11)
v^{0j} → v^{0} strongly in V.  (4.12)

Let us take the sequences (u^{0jp})_{p∈ℕ}, (v^{0jp})_{p∈ℕ} in V ∩ H^{2}(0, L) and (u^{1p})_{p∈ℕ}, (v^{1p})_{p∈ℕ} in V such that

u^{0jp} → u^{0j} strongly in V,  (4.13)
v^{0jp} → v^{0j} strongly in V,  (4.14)
u^{1p} → u^{1} strongly in L^{2}(0, L),  (4.15)
v^{1p} → v^{1} strongly in L^{2}(0, L),  (4.16)
u_{x}^{0jp}(L, ·) + u^{1p}(L, ·) = 0 in (0, T) ,  (4.17)
u_{x}^{0jp}(L, ·) + v_{x}^{0jp}(L, ·) + v_{t}^{0jp}(L, ·) = 0 in (0, T) .  (4.18)

We fix (j, p, ν) ∈ ℕ. For the initial data (u^{0jp}, u^{1p}, v^{0jp}, v^{1p}) ∈ {[V ∩ H^{2}(0, L)] × V }^{2}, there exist unique functions u_{jpν}, v_{jpν} : Q → ℝ in the conditions of the Theorem 3.1. By the same argument employed in the Estimates I (see (3.23)), we obtain

\frac{ρh^{3}}{12} |u_{t}^{jpν}(t)|^{2} + ρh |v_{t}^{jpν}(t)|^{2} + k |(u^{jpν} + v_{x}^{jpν})(t)|^{2} + ∥u^{jpν}(t)∥^{2} + 2∫_{0}^{t} |u_{t}^{jpν} (L, s)|^{2} ds

+2∫_{0}^{t} |v_{t}^{jpν} (L, s)|^{2} ds + 2∫_{0}^{L}F_{ν} (u^{jpν}(x, t)) dx + 2∫_{0}^{L}G_{ν} (v^{jpν}(x, t)) dx ≤ \frac{ρh^{3}}{12} |u^{1p}|^{2}

+ρh |v^{1p}|^{2} + k |u^{0jp} + v_{x}^{0pν}|^{2} + ∥u^{0jp}∥^{2} + 2∫_{0}^{L}F_{ν}(u^{0jp})dx + 2∫_{0}^{L}G_{ν}(v^{0jp})dx,

(4.19)

where F_{ν}(t) = ∫_{0}^{t} f_{ν}(s)ds and G_{ν}(t) = ∫_{0}^{t} g_{ν}(s)ds.

We need estimates for the terms ∫_{0}^{L}F_{ν} (u^{0jp}) dx and ∫_{0}^{L}G_{ν} (v^{0jp}) dx. Since u^{0j} and v^{0j} are bounded a. e. in [0, L], ∀j ∈ ℕ, it follows that

f_{ν}(u^{0j}) → f (u^{0j}) uniformly in (0, L) ,
g_{ν}(v^{0j}) → g (v^{0j}) uniformly in (0, L) .

So

∫_{0}^{L}F_{ν} (u^{0j}(x)) dx → ∫_{0}^{L}F (u^{0j}(x)) dx uniformly in ℝ,  (4.20)

∫_{0}^{L}G_{ν} (v^{0j}(x)) dx → ∫_{0}^{L}G (v^{0j}(x, t)) dx uniformly in ℝ.  (4.21)

From (4.11) and (4.12), there exist subsequences of (u^{0j})_{j∈ℕ} and (v^{0j})_{j∈ℕ}, which still be also denoted by (u^{0j})_{j∈ℕ} and (v^{0j})_{j∈ℕ}, such that

u^{0j} → u^{0} a. e. in (0, L) ,
v^{0j} → v^{0} a. e. in (0, L) .

By continuity of F and G, it follows that F (u^{0j}) → F (u^{0}) and G(v^{0j}) → G(v^{0}) a. e. in [0, L]. We also have F (u^{0j}) ≤ F (u^{0}) and G(v^{0j}) ≤ G(v^{0}). Thus, by (4.3) and the Lebesgue’s dominated convergence theorem, we get

F (u^{0j}) → F (u^{0}) strongly in L^{1}(0, L),  (4.22)
G(v^{0j}) → G(v^{0}) strongly in L^{1}(0, L).  (4.23)

Making the same arguments for F_{ν} and G_{ν}, it follows that

F_{ν}(u^{0jp}) → F_{ν}(u^{0j}) strongly in L^{1}(0, L),  (4.24)
G_{ν}(v^{0jp}) → G_{ν}(v^{0j}) strongly in L^{1}(0, L).  (4.25)

By (4.20) − (4.25), we obtain

∫_{0}^{L}F_{ν}(u^{0jp} (x))dx → ∫_{0}^{L}F (u^{0}(x)) dx in ℝ,  (4.26)

∫_{0}^{L}G_{ν}(v^{0jp}(x))dx → ∫_{0}^{L}G(v^{0}(x))dx in ℝ.  (4.27)

Then

∫_{0}^{L}F_{ν}(u^{0jp}(x)dx ≤ C and ∫_{0}^{L}G_{ν}(v^{0jp}(x)dx ≤ C,  (4.28)

where the constant C > 0 is independent of j, p and ν.

Using (4.11) − (4.16) and (4.28) in (4.19), we have

\begin{matrix}\frac{ρh^{3}}{12} |u_{t}^{jpν}(t)|^{2} + ρh |v_{t}^{jpν}(t)|^{2} + k |(u^{jpν} + v_{x}^{jpν})(t)|^{2} + ∥u^{jpν}(t)∥^{2} + 2∫_{0}^{t} |u_{t}^{jpν} (L, s)|^{2} ds \\ +2∫_{0}^{t} |v_{t}^{jpν} (L, s)|^{2} ds ≤ C, \end{matrix}

(4.29) where C > 0 is independent of j, p, ν and t.

From (4.29), we get

(u^{jpν}) is bounded in L^{∞}(0, T, V),  (4.30)
(v^{jpν}) is bounded in L^{∞}(0, T, V),  (4.31)
(u_{t}^{jpν}) is bounded in L^{2}(Q),  (4.32)
(v_{t}^{jpν}) is bounded in L^{2}(Q),  (4.33)
(u_{t}^{jpν}(L, ·)) is bounded in L^{2}(0, T),  (4.34)
(v_{t}^{jpν}(L, ·)) is bounded in L^{2}(0, T).  (4.35)

According to (3.11), (3.12), (4.30), (4.34) and (4.35), we have

u_{x}^{jpν} (L, ·) is bounded in L^{2}(0, T),  (4.36)
v_{x}^{jpν} (L, ·) is bounded in L^{2}(0, T).  (4.37)

As the estimates above are hold for all (j, p, ν) ∈ ℕ^{3} and, in particular for (ν, ν, ν) ∈ ℕ^{3}, we can take subsequences (u^{ννν})_{ν∈ℕ} and (v^{ννν})_{ν∈ℕ}, which we denote by (u^{ν})_{ν∈ℕ} and (v^{ν})_{ν∈ℕ}, such that

u^{ν} → u weak ∗ in L^{∞}(0, T, V),  (4.38)
v^{ν} → v weak ∗ in L^{∞}(0, T, V),  (4.39)
u_{t}^{ν} → u_{t} weakly in L^{2}(Q),  (4.40)
v_{t}^{ν} → v_{t} weakly in L^{2}(Q),  (4.41)
u_{x}^{ν}(L, ·) → χ weakly in L^{2}(0, T),  (4.42)
v_{x}^{ν}(L, ·) → Σ weakly in L^{2}(0, T),  (4.43)
u_{t}^{ν}(L, ·) → u_{t}(L, ·) weakly in L^{2}(0, T),  (4.44)
v_{t}^{ν}(L, ·) → v_{t}(L, ·) weakly in L^{2}(0, T),  (4.45)

We note that the Theorem 3.1 gives us

\frac{ρh^{3}}{12} u_{tt}^{ν} − u_{xx}^{ν} + k(u^{ν} + v_{x}^{ν}) + f_{ν}(u^{ν}) = 0 in L^{2}(Q),  (4.46)
ρhv_{tt}^{ν} − k (u^{ν} + v_{x}^{ν})_{x} + g_{ν}(v^{ν}) = 0 in L^{2}(Q)  (4.47)
u_{x}^{ν} (L, ·) + u_{t}^{ν} (L, ·) = 0 in (0, T) ,  (4.48)
u^{ν} (L, ·) + v_{x}^{ν} (L, ·) + v_{t}^{ν} (L, ·) = 0 in (0, T) .  (4.49)

From (4.38) − (4.41) and the compact embedding of H^{1}(Q) in L^{2}(Q), we can guarantee the existence of subsequences of (u^{ν}) and (v^{ν}), which we still denote with the index ν, such that

u^{ν} → u a. e. in Q,  (4.50)

v^{ν} → v a. e. in Q.  (4.51)

As f, g are continuous, it follows

f(u^{ν}) → f(u) a. e. in Q,
g(v^{ν}) → g(v) a. e. in Q.

We also have

f_{ν}(u^{ν}) → f(u^{ν}) a. e. in Q,
g_{ν}(v^{ν}) → g(v^{ν}) a. e. in Q,

because u^{ν}(x, t) and v^{ν}(x, t) are bounded in ℝ. Therefore

f_{ν}(u^{ν}) → f(u) a. e. in Q,  (4.52)
g_{ν}(v^{ν}) → g(v) a. e. in Q.  (4.53)

Making the inner product in L^{2} (Q) of (4.46) with u^{ν} (t), we obtain

\begin{matrix}∫_{0}^{T} (f_{ν}(u^{ν} (t)), u^{ν} (t)) = \frac{ρh^{3}}{12} ∫_{0}^{T} |u_{t}^{ν} (t)|^{2} dt − \frac{ρh^{3}}{12} (u_{t}^{ν}(T), u^{ν}(T)) \\ +\frac{ρh^{3}}{12} (u_{t}^{ν}(0), u^{ν}(0)) − ∫_{0}^{T} u_{t}^{ν}(L, T)u^{ν}(L, T)dt − ∫_{0}^{T} ((u^{ν} (t) , u^{ν} (t)))dt \\ −k∫_{0}^{T} ((u^{ν} + v_{x}^{ν}) (t) , u^{ν} (t))dt. \end{matrix}  (4.54)

Observing (4.13) , (4.15) , (4.30) − (4.32) and (4.34), we have by (4.54) that

∫_{0}^{T} (f_{ν}(u^{ν} (t)), u^{ν} (t)) ≤ C,  (4.55)

where C > 0 is independent of ν.

From (4.52), (4.55) and Strauss’ Theorem (see [17]), it follows

f_{ν}(u^{ν}) → f(u) strongly in L^{1}(Q).  (4.56)

Analogously, taking the inner product in L^{2} (Q) of (4.47) with v^{ν} (t) and after using (4.14) , (4.16) , (4.30), (4.31), (4.33) and (4.35) we get

∫_{0}^{T} (g_{ν}(v^{ν} (t)), v^{ν} (t))dt ≤ C,  (4.57)

where C > 0 is independent of ν.

From (4.53), (4.57) and Strauss’ Theorem (see [17]), it follows

g_{ν}(v^{ν}) → g(v) strongly in L^{1}(Q).  (4.58)

Applying the convergences (4.38) − (4.45), (4.56) and (4.58) in (4.46) − (4.49) , we conclude

\frac{ρh^{3}}{12} u_{tt} − u_{xx} + k(u + v_{x}) + f(u) = 0 in L^{1}(0, T, V′ + L^{1}(0, L)),  (4.59)
ρhv_{tt} − k(u + v_{x})_{x} + g(v) = 0 in L^{1}(0, T, V′ + L^{1}(0, L)),  (4.60)
χ + u_{t} (L, ·) = 0 in L^{2}(0, T),  (4.61)
u (L, ·) + Σ + v_{t} (L, ·) = 0 in L^{2}(0, T).  (4.62)

Let us prove that u_{x} (L, ·) = χ and v_{x} (L, ·) = Σ.

• u_{x}(L, ·) = χ

According to (4.59), we deduce

−u_{xx} = −\frac{ρh^{3}}{12} u_{tt} − k(u + v_{x}) − f(u).  (4.63)

Since u_{t}, (u+v_{x}) ∈ L^{2}(Q) and f(u) ∈ L^{1}(Q), by Propositions 2.1 and 2.3, there exist functions z, w ∈ L^{2}(0, T, V ∩ H^{2}(0, L)) and η ∈ L^{1}(0, T, E) such that −z_{xx} = u_{t}, −w_{xx} = u + v_{x} and −η_{xx} = f(u). Hence

−u_{xx} = \frac{ρh^{3}}{12} (z_{xx})_{t} + kw_{xx} + η_{xx}.  (4.64)

Multiplying (4.64) by θ ∈ D(0, T) and integrating from 0 to T, we obtain

− [∫_{0}^{T} uθdt − \frac{ρh^{3}}{12} ∫_{0}^{T} zθ′dt + k ∫_{0}^{T} wθdt]_{xx} = − [∫_{0}^{T} −ηθdt]_{xx} .

From the uniqueness given by Proposition 2.3, we get

∫_{0}^{T} (u + \frac{ρh^{3}}{12} z_{t} + kw + η) θdt = 0, ∀θ ∈ D(0, T),

that is,

u = −\frac{ρh^{3}}{12} z_{t} − kw − η.  (4.65)

Since z_{xt}(L, ·) = (z_{x}(L, ·))_{t} (see [14, Lemma 3.2]), we can apply the trace theorem in (4.65) to obtain

u_{x} (L, ·) = −\frac{ρh^{3}}{12} (z_{x} (L, ·))_{t} − kw_{x} (L, ·) − η_{x} (L, ·) ∈ H^{−1} (0, T) + L^{1} (0, T) .

For other side, by (4.46) we have

−u_{xx}^{ν} = −\frac{ρh^{3}}{12} u_{tt}^{ν} − k(u^{ν} + v_{x}^{ν}) − f_{ν}(u^{ν})

with u_{t}^{ν}, (u^{ν} + v_{x}^{ν}), f_{ν}(u^{ν}) ∈ L^{2}(Q). By Propositions 2.1, there exist functions z^{ν}, w^{ν}, η^{ν} ∈ L^{2}(0, T, V ∩ H^{2}(0, L)) such that −z_{xx}^{ν} = u_{t}^{ν}, −w_{xx}^{ν} = u^{ν} + v_{x}^{ν} and −η_{xx}^{ν} = f_{ν}(u^{ν}). Thus, as it was done before, we have

−u_{xx}^{ν} = \frac{ρh^{3}}{12} (z_{xx}^{ν})_{t} + kw_{xx}^{ν} + η_{xx}^{ν}

and

u^{ν} = −\frac{ρh^{3}}{12} z_{t}^{ν} − kw^{ν} − η^{ν}.  (4.66)

By (4.38) − (4.40) and (4.56), we get

z^{ν} → z weakly in L^{2}(0, T, V ∩ H^{2}(0, L)),  (4.67)
w^{ν} → w weakly in L^{2}(0, T, V ∩ H^{2}(0, L)),  (4.68)
η^{ν} → η strongly in L^{1}(0, T, E).  (4.69)

According to [13] and (4.67), we have

z_{t}^{ν} → z_{t} weakly in H^{−1}(0, T, V ∩ H^{2}(0, L)).  (4.70)

From (4.67) − (4.70) and by continuity of the trace, we obtain

η_{x}^{ν}(L, ·) → η_{x}(L, ·) strongly in L^{1}(0, T),  (4.71)
z_{xt}^{ν}(L, ·) → z_{xt}(L, ·) weakly in H^{−1}(0, T),  (4.72)
w_{x}^{ν}(L, ·) → w_{x}(L, ·) weakly in L^{2}(0, T).  (4.73)

Taking into account the convergences (4.71) − (4.73), it follows by (4.65) and (4.66) that

u_{x}^{ν}(L, ·) → u_{x}(L, ·) weakly in [H^{1}(0, T) ∩ L^{∞}(0, T)]′ .  (4.74)

In this way, comparing (4.42) and (4.74), we can conclude

χ = u_{x} (L, ·) in L^{2}(0, T).  (4.75)

• v_{x} (L, ·) = Σ.

Making the same procedure as before, from (4.60), it follows that

−v_{xx} = u_{x} − \frac{ρh}{k}v_{tt} − \frac{1}{k}g(v),  (4.76)

with u_{x}, v_{t} ∈ L^{2}(Q) and g(v) ∈ L^{1}(Q). By Propositions 2.1 and 2.3, there exist functions β, ϕ ∈ L^{2}(0, T, V ∩ H^{2}(0, L)) and ζ ∈ L^{1}(0, T, E) such that −β_{xx} = u_{x}, − ϕ_{xx} = \frac{ρh}{k}v_{t} and −ζ_{xx} = \frac{1}{k}g(v). So, we can find

−v_{xx} = −β_{xx} + \frac{ρh}{k}(ϕ_{xx})_{t} + \frac{1}{k}ζ_{xx}  (4.77)

and

v = β − \frac{ρh}{k}ϕ_{t} − \frac{1}{k}ζ.  (4.78)

Applying the trace theorem in (4.78), we obtain

v_{x} (L, ·) = β_{x} (L, ·) − \frac{ρh}{k} (ϕ_{x} (L, ·))_{t} − \frac{1}{k}ζ_{x} (L, ·) ∈ H^{−1} (0, T) + L^{1} (0, T) .

We know by (4.47) that

−v_{xx}^{ν} = u_{x}^{ν} − \frac{ρh}{k}v_{tt}^{ν} − \frac{1}{k}g_{ν}(v^{ν}).

Since u_{x}^{ν}, v_{t}^{ν}, g_{ν}(v^{ν}) ∈ L^{2}(Q), it follows the existence of functions β^{ν}, ϕ^{ν}, ζ^{ν} ∈ L^{2}(0, T, V ∩ H^{2}(0, L)) such that −β_{xx}^{ν} = u_{x}^{ν}, −ϕ_{xx}^{ν} = v_{t}^{ν} and −ζ_{xx}^{ν} = g_{ν}(v^{ν}). Thus, for analogy to the that we did before, we get

−kv_{xx}^{ν} = −kβ_{xx}^{ν} + ρh(ϕ_{xx}^{ν})_{t} + ζ_{xx}^{ν}

and

kv^{ν} = kβ^{ν} − ρhϕ_{t}^{ν} − ζ^{ν}  (4.79)

By (4.38), (4.41) and (4.58), we have

β^{ν} → β weakly in L^{2}(0, T, V ∩ H^{2}(0, L)),  (4.80)
ϕ^{ν} → ϕ weakly in L^{2}(0, T, V ∩ H^{2}(0, L)),  (4.81)
ζ^{ν} → ζ strongly in L^{1}(0, T, E),  (4.82)
ϕ_{t}^{ν} → ϕ_{t} weakly in H^{−1}(0, T, V ∩ H^{2}(0, L)).  (4.83)

According to convergences (4.80) − (4.83) and the continuity of trace, it follows that

β_{x}^{ν}(L, ·) → β(L, ·) weakly in L^{2}(0, T),  (4.84)
ζ_{x}^{ν}(L, ·) → ζ(L, ·) strongly in L^{1}(0, T),  (4.85)
ϕ_{x}^{ν}(L, ·) → ϕ(L, ·) weakly in H^{−1}(0, T).  (4.86)

Using the convergences (4.84) − (4.86) , we can conclude from (4.78) and (4.79) that

v_{x}^{ν}(L, ·) → v_{x}(L, ·) weakly in [H^{1}(0, T) ∩ L^{∞}(0, T)]′ ,  (4.87)

which comparing with (4.43), we deduce

Σ = v_{x}(L, ·) in L^{2}(0, T).

To verify the initial conditions (4.10) , we use the standard method.

Remark 4.1 The uniqueness of solution in the conditions of the Theorem 4.1 is a open question. But, for some particular cases of the nonlinearities, for example f (s) = |s|^{p−1} s and g (s) = |s|^{q−1} s with p, q ∈ [1, ∞) , we can use the energy method as in Lions [9, p. 15] to obtain the uniqueness of solution.

5 Asymptotic Behavior

The aim of this section is to study the asymptotic behavior of the energy E(t) associated to weak solution of the problem (1.1) − (1.3). As it was mentioned in the introduction, this energy is defined by

\begin{aligned}E(t) = & \frac{1}{2} [\frac{ρh^{3}}{12} |u_{t} (t)|^{2} + ρh |v_{t} (t)|^{2} + k |(u + v_{x}) (t)|^{2} + ∥u (t)∥^{2} \\ & +2∫_{0}^{L}F (u(x, t))dx + 2∫_{0}^{L}G(v(x, t))dx] . \end{aligned}  (5.1)

Let us consider the following additional hypotheses:

∃ δ_{1} > 0 such that f(s)s ≥ (2 + δ_{1})F (s), ∀s ∈ ℝ,  (5.2)

and

∃ δ_{2} > 0 such that g(s)s ≥ (2 + δ_{2})G(s), ∀s ∈ ℝ.  (5.3)

The functions f and g given in the Remark 4.1 satisfy the conditions (4.1) , (4.3) , (5.2) and (5.3) .

The mean result of this section is:

Theorem 5.1 Let L < min {2, 2/k} and f, g, u^{0}, v^{0}, u^{1}, v^{1} in the conditions of the Theorem 4.1 plus the hypotheses (5.2) and (5.3). Then there exists a positive constant κ > 0 such that the energy E(t) satisfy

E(t) ≤ 4E(0)e^{−κt}, ∀t ≥ 0.  (5.4)

Proof. Taking the inner product in L^{2}(0, L) of (4.46) and (4.47) with u_{t}^{ν} (t) and v_{t}^{ν} (t) , respectively, we obtain

E′_{ν}(t) = − |u^{ν}(L, ·)|^{2} − |v^{ν}(L, ·)|^{2} ,  (5.5)

where E_{ν}(t) is the energy associated to strong solution (u^{ν}, v^{ν}), obtained in Section 3, when f and g are replaced by f_{ν} and g_{ν}, respectively. Thus this energy is non-increasing.

It is important to emphasize that, for each ν ∈ ℕ, the functions f_{ν} and g_{ν} of the approximating sequences also satisfy the conditions (5.2) and (5.3), respectively (cf. [17]).

For an arbitrary ϵ > 0, let us define the perturbed energy

E_{ν_{ϵ}}(t) = E_{ν}(t) + ϵΨ(t),  (5.6)

with

Ψ(t) = α(\frac{ρh^{3}}{12} u_{t}^{ν}(t), xu_{x}^{ν}(t)) + α(ρhv_{t}^{ν}(t), xv_{x}^{ν}(t)) + β(\frac{ρh^{3}}{12} u_{t}^{ν}(t), u^{ν}(t)) + β(ρhv_{t}^{ν}(t), v^{ν}(t)),

(5.7)

where α > 0 and β > 0 are constants such that

α + 2β > max {αkL, αL, 4β} ,  (5.8)

∃ γ_{1} > 0 such that βf(s)s ≥ (α + γ_{1})F (s), ∀s ∈ ℝ,  (5.9)

and

∃ γ_{2} > 0 such that βg(s)s ≥ (α + γ_{2})G(s), ∀s ∈ ℝ.  (5.10)

The choice of β is possible because

\frac{(α + γ_{1})}{(2 + δ_{1})}f(s)s ≥ (α + γ_{1})F (s), \frac{(α + γ_{2})}{(2 + δ_{2})}g(s)s ≥ (α + γ_{2})G(s)

and

0 < \frac{(α + γ_{i})}{(2 + δi)} < \frac{α}{2}, for 0 < γ_{i} < \frac{αδ_{i}}{2} (i = 1, 2) .

After some calculations, we find

\begin{matrix}|Ψ(t)| ≤ (\frac{αL + βL}{2}) \frac{ρh^{3}}{12} |u_{t}^{ν} (t)|^{2} + (αL + βL)ρh |v_{t}^{ν} (t)|^{2} \\ + (\frac{αρhL + βρhL}{2k}) k |(u^{ν} + v_{x}^{ν}) (t)|^{2} + [\frac{ρh^{3}L(α + β) + 12ρhL^{3}(α + β)}{24}] ∥u^{ν} (t)∥^{2} , \end{matrix}

which implies

|Ψ(t)| ≤ C_{1}E_{ν}(t),  (5.11)

where C_{1} = [3 + (ρh/k) + (ρh^{3}/12) + ρhL^{2}] L(α + β). It follows by (5.6) and (5.11) that

|E_{ν_{ϵ}}(t) − E_{ν}(t)| = ϵ |Ψ(t)| ≤ ϵC_{1}E_{ν}(t),

that is,

(1 − ϵC_{1})E_{ν}(t) ≤ E_{ν_{ϵ}}(t) ≤ (1 + ϵC_{1})E_{ν}(t).

Taking 0 ≤ ϵ ≤ ϵ_{0} = 1/2C_{1}, we have

\frac{E_{ν}(t)}{2} ≤ E_{ν_{ϵ}}(t) ≤ 2E_{ν}(t).  (5.12)

Deriving the function (5.7) and using the equations (4.46) and (4.47) , it follows that

\begin{matrix}Ψ′(t) = α(u_{xx}^{ν} (t) , xu_{x}^{ν} (t)) + αk((u^{ν} + v_{x}^{ν})_{x} (t) , x(u^{ν} + v_{x}^{ν}) (t)) \\ +αk ((u^{ν} + v_{x}^{ν}) (t) , u^{ν} (t)) + αkLu^{ν}(L, t)v_{t}^{ν}(L, t) − α(f_{ν}(u^{ν} (t)), xu_{x}^{ν} (t)) \\ +α(\frac{ρh^{3}}{12} u_{t}^{ν} (t) , xu_{xt}^{ν} (t)) − α(g_{ν}(v^{ν} (t)), xv_{x}^{ν} (t)) + α(ρhv_{t}^{ν} (t) , xv_{xt}^{ν} (t)) − β ∥u^{ν} (t)∥^{2} \\ −βu^{ν}(L, t)u_{t}^{ν}(L, t) − βk |(u^{ν} + v_{x}^{ν}) (t)|^{2} − β(f_{ν}(u^{ν} (t)), u^{ν} (t)) \\ +\frac{βρh^{3}}{12} |u_{t}^{ν} (t)|^{2} − βkv^{ν}(L, t)v_{t}^{ν}(L, t) − β(g_{ν}(v^{ν} (t)), v^{ν} (t)) + βρh |v_{t}^{ν} (t)|^{2} . \end{matrix} (5.13)

Now, we will analyze some terms that appear on the right side of (5.13). EJQTDE, 2008 No. 34, p. 22 • Analysis of α(u_{xx}^{ν} (t) , xu_{x}^{ν} (t)).

α(u_{xx}^{ν} (t) , xu_{x}^{ν} (t)) = \frac{α}{2}∫_{0}^{L}x\frac{d}{dx} |u_{x}^{ν} (t)|^{2} dx = \frac{αL}{2} |u_{t}^{ν}(L, t)|^{2} − \frac{α}{2}∫_{0}^{L} |u_{x}^{ν} (t)|^{2} dx. (5.14)

• Analysis of \frac{αρh^{3}}{12} (u_{t}^{ν} (t) , xu_{tx}^{ν} (t))

\begin{matrix}\frac{αρh^{3}}{12} (u_{t}^{ν} (t) , xu_{tx}^{ν} (t)) = \frac{αρh^{3}}{24} ∫_{0}^{L}x\frac{d}{dx} |u_{t}^{ν} (t)|^{2} dx = \frac{αρh^{3}L}{24} |u_{t}^{ν}(L, t)|^{2} \\ −\frac{αρh^{3}}{24} ∫_{0}^{L} |u_{t}^{ν} (t)|^{2} dx. \end{matrix}  (5.15)

• Analysis of αρh(v_{t}^{ν} (t) , xv_{tx}^{ν} (t))

αρh(v_{t}^{ν} (t) , xv_{tx}^{ν} (t)) = \frac{αρh}{2}∫_{0}^{L}x\frac{d}{dx} |v_{t}^{ν} (t)|^{2} dx = \frac{αρhL}{2} |v_{t}^{ν}(L, t)|^{2}−\frac{αρh}{2}∫_{0}^{L} |v_{t}^{ν} (t)|^{2} dx.

(5.16)

• Analysis of −βu^{ν}(L, t)u_{t}^{ν}(L, t)

−βu^{ν}(L, t)u_{t}^{ν}(L, t) ≤ βc_{0} |u_{t}^{ν}(L, t)| ∥u^{ν}(t)∥ ≤ C_{2} |u_{t}^{ν}(L, t)|^{2} + ξ ∥u^{ν}(t)∥^{2} , (5.17)

where C_{2} = β^{2}c_{0}^{2}/4ξ, c_{0} > 0 such that |u^{ν}(L, t)| ≤ c_{0} ∥u^{ν}(t)∥ and ξ > 0 a constant to be chosen.

• Analysis of −βkv^{ν}(L, t)v_{t}^{ν}(L, t)

\begin{matrix}−βkv^{ν}(L, t)v_{t}^{ν}(L, t) ≤ βkc_{0} |v_{t}^{ν}(L, t)| (|(u^{ν} + v_{x}^{ν}) (t)| + L ∥u^{ν} (t)∥) \\ ≤ \frac{β^{2}kc_{0}^{2}}{4ξ} |v_{t}^{ν}(L, t)|^{2} + ξk |(u^{ν} + v_{x}^{ν}) (t)|^{2} + \frac{β^{2}k^{2}c_{0}^{2}L^{2}}{4ξ} |v_{t}^{ν}(L, t)|^{2} + ξ ∥u^{ν} (t)∥^{2} \\ ≤ C_{3} |v_{t}^{ν}(L, t)|^{2} + ξk |(u^{ν} + v_{x}^{ν}) (t)|^{2} + ξ ∥u^{ν} (t)∥^{2} , \end{matrix} (5.18)

where C_{3} = (1 + kL^{2}) β^{2}kc_{0}^{2}/4ξ.

• Analysis of α(f_{ν}(u^{ν} (t)), xu_{x}^{ν} (t)) Observing that F is of class C^{1}(ℝ) and F (0) = 0, so

\begin{matrix}α(f_{ν}(u^{ν} (t)), xu_{x}^{ν} (t)) = α∫_{0}^{L}xf_{ν}(u^{ν} (t))u_{x}^{ν} (t) dx = α∫_{0}^{L}x\frac{d}{dx}F_{ν}(u^{ν} (x, t))dx \\ = αLF (u^{ν}(L, t)) − α∫_{0}^{L}F_{ν}(u^{ν} (x, t))dx. \end{matrix}  (5.19)

• Analysis of α(g_{ν}(v^{ν} (t)), xv_{x}^{ν} (t)) Noting that G belongs to class C^{1}(ℝ) and G(0) = 0, we obtain, in a manner analogous

to the last analysis, that

α(g_{ν}(v^{ν} (t)), xv_{x}^{ν} (t)) = αLG(v^{ν}(L, t)) − α∫_{0}^{L}G_{ν}(v^{ν} (x, t))dx.  (5.20)

• Analysis of α((u^{ν} + v_{x}^{ν})_{x} (t) , x(u^{ν} + v_{x}^{ν}) (t))

\begin{matrix}((u^{ν} + v_{x}^{ν})_{x} (t) , x(u^{ν} + v_{x}^{ν}) (t)) = \frac{α}{2}∫_{0}^{L}x\frac{d}{dx} |(u^{ν} + v_{x}^{ν}) (t)|^{2} dx \\ = \frac{αL}{2} |v_{t}(L, t)| − \frac{α}{2}∫_{0}^{L} |(u^{ν} + v_{x}^{ν}) (t)|^{2} dx. \end{matrix}  (5.21)

• Analysis of αk ((u^{ν} + v_{x}^{ν}) (t) , u^{ν} (t))

αk((u^{ν} + v_{x}^{ν}) (t) , u^{ν} (t)) ≤ \frac{αL}{2}k |(u^{ν} + v_{x}^{ν}) (t)|^{2} + \frac{αLk}{2} ∥u^{ν} (t)∥^{2} .  (5.22)

• Analysis of αkLu^{ν}(L, t)v_{t}^{ν}(L, t)

αkLv_{t}^{ν}(L, t)u^{ν}(L, t) ≤ c_{0}αkL |v_{t}^{ν}(L, t)| ∥u^{ν} (t)∥ ≤ C_{4} |v_{t}^{ν}(L, t)|^{2} + ξ ∥u^{ν} (t)∥^{2} , (5.23)

where C_{4} = c_{0}^{2}α^{2}k^{2}L^{2}/4ξ. Substituting (5.14) − (5.23) in (5.13) and using (5.9) , (5.10), we get

\begin{matrix}Ψ′(t) ≤ \frac{αL}{2} |u_{t}^{ν}(L, t)|^{2} − \frac{α}{2} ∥u^{ν} (t)∥^{2} + \frac{αkL}{2} |v_{t}(L, t)| − \frac{α}{2}k |(u^{ν} + v_{x}^{ν}) (t)|^{2} \\ +\frac{αL}{2}k |(u^{ν} + v_{x}^{ν}) (t)|^{2} + \frac{αkL}{2} ∥u^{ν} (t)∥^{2} + C_{4} |v_{t}(L, t)|^{2} + ξ ∥u^{ν} (t)∥^{2} \\ −αLF_{ν}(u^{ν}(L, t)) + \frac{αρh^{3}L}{24} |u_{t}^{ν}(L, t)|^{2} − \frac{αρh^{3}}{24} |u_{t}^{ν} (t)|^{2} − αLG_{ν}(v^{ν}(L, t)) \\ +\frac{αρhL}{2} |v_{t}^{ν}(L, t)|^{2} − \frac{αρh}{2} |v_{t}^{ν} (t)|^{2} − β ∥u^{ν} (t)∥^{2} + C_{2} |u_{t}^{ν}(L, t)|^{2} + ξ ∥u^{ν}(t)∥^{2} \\ −βk |(u^{ν} + v_{x}^{ν}) (t)|^{2} − γ_{1}∫_{0}^{L}F_{ν}(u^{ν} (x, t))dx + \frac{βρh^{3}}{12} |u_{t}^{ν} (t)|^{2} + C_{3} |v_{t}^{ν}(L, t)|^{2} \\ +ξk |(u^{ν} + v_{x}^{ν}) (t)|^{2} + ξ ∥u^{ν} (t)∥^{2} − γ_{2}∫_{0}^{L}G_{ν}(v^{ν} (x, t))dx + βρh |v_{t}^{ν} (t)|^{2} , \end{matrix}  (5.24)

which implies

Ψ′(t) ≤ − (C_{5} − 2ξ) E_{ν}(t) + C_{6} |u_{t}^{ν}(L, t)|^{2} + C_{7} |v_{t}^{ν}(L, t)|^{2} ,  (5.25)

EJQTDE, 2008 No. 34, p. 24 where C_{5}, C_{6} and C_{7} are the following positive constants:

C_{5} = min {α + 2β − αkL, α + 2β − αL, α − 2β, γ_{1}, γ_{2}} ,

C_{6} = \frac{αL}{2} + \frac{αρh^{3}}{24} + C_{2} and C_{7} = \frac{αkL}{2} + \frac{αρhL}{2} + C_{3} + C_{4}.

The positivity of C_{5} is guaranteed by (5.8) − (5.10).

Now, deriving (5.6) and, soon after, substituting the expressions (5.5) and (5.25), we obtain

E′_{ν_{ϵ}}(t) ≤ −ϵ (C_{5} − 2ξ) E_{ν}(t) − (1 − ϵC_{6}) |u_{t}^{ν}(L, t)|^{2} − (1 − ϵC_{7}) |v_{t}^{ν}(L, t)|^{2} .

Taking 0 < ϵ ≤ ϵ_{1} = min {1/C_{6}, 1/C_{7}} and 0 < ξ < C_{5}/2, we have

E′_{ν_{ϵ}}(t) + κE_{ν_{ϵ}} ≤ 0,

with κ = min {ϵ_{0}, ϵ_{1}} (C_{5} − 2ξ) > 0. In this way

E_{ν_{ϵ}}(t) ≤ E_{ν_{ϵ}}(0)e^{−κt}, ∀t ≥ 0.

Combining the last inequality with (5.12), we deduce

E_{ν}(t) ≤ 4E_{ν}(0)e^{−κt}, ∀t ≥ 0.  (5.26)

Since F_{ν} and G_{ν} are continuous, it follows by (4.50) and (4.51) the convergences

F_{ν}(u^{ν}(·, t)) → F_{ν}(u(·, t)) a. e. in (0, L) , ∀t ≥ 0,  (5.27)
G_{ν}(v^{ν}(·, t)) → G_{ν}(v(·, t)) a. e. in (0, L) , ∀t ≥ 0.  (5.28)

For other hand, since f_{ν} → f and g_{ν} → g uniformly on bounded sets of ℝ, then

F_{ν}(u(·, t)) → F (u(·, t)) a. e. in (0, L) , ∀t ≥ 0,  (5.29)
G_{ν}(v(·, t)) → G(v(·, t)) a. e. in (0, L) , ∀t ≥ 0.  (5.30)

According to (5.27) − (5.30), we can conclude that

F_{ν}(u^{ν}(·, t)) → F (u(·, t)) a. e. in (0, L) , ∀t ≥ 0,  (5.31)
G_{ν}(v^{ν}(·, t)) → G(v(·, t)) a. e. in (0, L) , ∀t ≥ 0.  (5.32)

From (5.26) we have the following estimates

∫_{0}^{L} F_{ν}(u^{ν}(x, t))dx ≤ 4E_{ν}(0), ∫_{0}^{L} G_{ν}(v^{ν}(x, t))dx ≤ 4E_{ν}(0).  (5.33)

Using the convergences (4.11) − (4.16), (4.26) and (4.27) in (5.33), it follows that

lim inf_{ν→∞} ∫_{0}^{L} F_{ν}(u^{ν} (x, t))dx ≤ 4E(0), lim inf_{ν→∞} ∫_{0}^{L} G_{ν}(v^{ν} (x, t))dx ≤ 4E(0).  (5.34)

By (5.31), (5.32), (5.34) and Fatou’s lemma, we have

∫_{0}^{L} F (u (x, t))dx ≤ lim inf_{ν→∞} ∫_{0}^{L} F_{ν}(u^{ν} (x, t))dx,

and

∫_{0}^{L} G(v (x, t))dx ≤ lim inf_{ν→∞} ∫_{0}^{L} G_{ν}(v^{ν} (x, t))dx.

In this way, taking the lim inf_{ν→∞} in (5.26), we can deduce the inequality (5.4).

References

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(Received April 17, 2008)