Exponential stability for the coupled Klein–Gordon–Schrödinger equations with locally distributed damping in unbounded domains

Abstract

The following coupled damped Klein–Gordon–Schrödinger equations are considered $\begin{array}{l} i ψ_{t} + Δ ψ + i α b (x) (| ψ |^{2} + 1) ψ = ϕ ψ χ_{Ω_{R}} in R^{n} \times (0, \infty) (α > 0), \\ ϕ_{t t} - Δ ϕ + ϕ + a (x) ϕ_{t} = | ψ |^{2} χ_{Ω_{R}} in R^{n} \times (0, \infty), \end{array}$ where $Ω_{R} = R^{n} ∖ B_{R} = {| x | ⩾ R}$ and $χ_{Ω_{R}}$ represents the characteristic function of $Ω_{R}$ . Assuming that $a, b \in W^{1, \infty} (R^{n})$ are nonnegative functions such that $a (x) ⩾ a_{0} > 0$ in $Ω_{R}$ and $b (x) ⩾ b_{0} > 0$ in $Ω_{R}$ , the exponential decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous ones given by Cavalcanti et al. in the references (NoDEA, Nonlinear Differ. Equ. Appl. 15 (2008) 91–113), (NoDEA, Nonlinear Differ. Equ. Appl. 7 (2000) 285–307), (Communications on Pure and Applied Analysis 17 (2018) 2039–2061) and (Evolution Equations and Control Theory 8 (2019) 847–865).

Keywords

Klein–Gordon–Schrödinger localized damping unbounded domains

1. Introduction

We consider the following model of Klein–Gordon–Schrödinger equations with locally distributed damping in unbounded domains $\begin{matrix} (1.1) & \{\begin{matrix} i ψ_{t} + Δ ψ + i α b (x) (| ψ |^{2} + 1) ψ = ϕ ψ χ_{Ω_{R}} in R^{n} \times (0, \infty), \\ ϕ_{t t} - Δ ϕ + ϕ + a (x) ϕ_{t} = | ψ |^{2} χ_{Ω_{R}} in R^{n} \times (0, \infty), \\ ψ (0) = ψ_{0} \in H^{2} (R^{n}), \\ ϕ (0) = ϕ_{0} \in H^{2} (R^{n}), \\ ϕ_{t} (0) = ϕ_{1} \in H^{1} (R^{n}), \end{matrix} \end{matrix}$ where $n ⩽ 3$ and $Ω_{R} = R^{n} ∖ B_{R} = {| x | ⩾ R}$ . In what follows, α is a positive constant and $χ_{Ω_{R}}$ represents the characteristic function, that is, $χ = 1$ in $Ω_{R}$ and $χ = 0$ in $B_{R}$ . We consider $a, b \in W^{1, \infty} (R^{n})$ nonnegative functions such that $\begin{array}{rcl} a (x) ⩾ a_{0} > 0 in Ω_{R}, and b (x) ⩾ b_{0} > 0 in Ω_{R} . \end{array}$

Uniform decay rate estimates to problem (1.1) has been considered in the previous results due to Cavalcanti et al. [6,7]. While in [7] a full damping was in place in both equations, in contrast, in [6] a full damping has been considered in the Schröndinger equation but just a localized damping has been considered for the wave equation. In the article [1] the authors generalize both previous results just considering two localized dampings in both equations, namely: $\begin{array}{l} i ψ_{t} + Δ ψ + i α b (x) {(- Δ)}^{\frac{1}{2}} b (x) ψ = ϕ ψ χ_{ω} in Ω \times (0, \infty) (α > 0), \\ ϕ_{t t} - Δ ϕ + a (x) ϕ_{t} = | ψ |^{2} χ_{ω} in Ω \times (0, \infty), \end{array}$ making use of the multipliers method combined with integral inequalities of energy and a regularizing effect due to Aloui in [3]. The article [2] generalize the previous results by considering the weaker damped structure $i α b (x) (| ψ |^{2} + 1) ψ$ instead of $i α b (x) {(- Δ)}^{\frac{1}{2}} b (x) ψ$ assumed in [1] making use of the observability inequality of both the linear wave (see [5]) and the Schrödinger (see [11] and [22]) equations, combined with other tools in order to prove exponential decay as done in [10].

The main purpose of the present article is to generalize substantially all previous results due to Cavalcanti et al. given in the references [1,6,7] and [2] from bounded domains to unbounded ones. The techniques presented here differ substantially from those used in the bounded case, which bring new difficulties. To overcome these difficulties afformentioned , we make use of the techniques considered in [24] combined with the smoothing effect property proved in Constantin and Saut [8].

Problem (1.1) (see [23]) has its origin in the canonical model of the Yukawa interaction of conserved complex nucleon field ψ with neutral real meson field ϕ given by $\begin{matrix} (1.2) & \{\begin{matrix} i ψ_{t} + Δ ψ = ϕ ψ in Ω \times (0, \infty), \\ ϕ_{t t} - Δ ϕ + μ^{2} ϕ = | ψ |^{2} in Ω \times (0, \infty), \\ ψ = ϕ = 0 on Γ \times (0, \infty), \\ ψ (0) = ψ_{0}, ϕ (0) = ϕ_{0}, ϕ_{t} (0) = ϕ_{1} . \end{matrix} \end{matrix}$

Here, ψ is a complex scalar nucleon field while ϕ is a real scalar meson one and the positive constant μ represents the mass of a meson, for simplicity we will consider $μ = 1$ .

It is important to note that problem (1.2) is not naturally dissipative. So, the introduction of the dissipative mechanisms given by the damping terms are necessary to force the energy to decay to zero when t goes to infinity. In fact, the dissipative K-G-S equation has been widely studied, see for example the following references: [9,16,18,20,21] and references therein. The majority of works in the literature deal with linear dissipative terms acting in both equations, except for the works [17] and [7].

We would like to mention other papers in connection with problem (1.2), namely: Fukuda and Tsutsumi [12–15], Bachelot and Chadam [4] and Hayashi and W. Von Wahl [19]. In the above articles the unique global existence to problem (1.2) is established and some conservation laws are verified.

Our paper is organized as follows. In Section 2 we give the precise assumptions and state our main result, in Section 3 we give an idea of the proof of existence and in Section 4 we give the proof of the main theorem.

2. Main result

In what follows let us consider the Hilbert space $L^{2} (R^{n})$ of complex valued functions on $R^{n}$ endowed with the inner product $\begin{array}{l} (u, v) = \int_{R^{n}} u (x) \overline{v (x)} d x, \end{array}$ and the corresponding norm $\begin{matrix} ‖ u ‖_{2}^{2} = (u, u) . \end{matrix}$

We also consider the Sobolev space $H^{1} (R^{n})$ endowed with the scalar product $\begin{matrix} {(u, v)}_{H^{1} (R^{n})} = (u, v) + (\nabla u, \nabla v) . \end{matrix}$

The following assumptions are made:

Assumption 2.1.
We assume that $a, b \in W^{1, \infty} (R^{n})$ are nonnegative functions such that $\begin{array}{l} a (x) ⩾ a_{0} > 0 in Ω_{R}, \\ b (x) ⩾ b_{0} > 0 in Ω_{R} . \end{array}$

The energy associated to problem (1.1) is defined by $\begin{array}{rcl} (2.1) & E (t) : = \frac{1}{2} \int_{R^{n}} ({| ψ (x, t) |}^{2} + {| \nabla ϕ (x, t) |}^{2} + {| ϕ (x, t) |}^{2} + {| ϕ_{t} (x, t) |}^{2}) d x . \end{array}$

Now, we are in position to state our main results.
Theorem 2.1.
Given ${ψ_{0}, ϕ_{0}, ϕ_{1}} \in {H^{2} (R^{n})}^{2} \times H^{1} (R^{n})$ and assuming that Assumption 2.1 holds with $α ⩾ \frac{3}{2 a_{o} b_{0}}$ , then, there exists a unique regular solution to problem ( 1.1 ) such that $\begin{array}{l} ψ \in L^{\infty} (0, \infty; H^{2} (R^{n})), ψ^{'} \in L^{\infty} (0, \infty; L^{2} (R^{n})), \\ ϕ \in L^{\infty} (0, \infty; H^{2} (R^{n})), ϕ^{'} \in L^{\infty} (0, \infty; H^{1} (R^{n})), and \\ ϕ^{″} \in L^{\infty} (0, \infty; L^{2} (R^{n})) . \end{array}$

Setting $H : = {H^{2} (R^{n})}^{2} \times H^{1} (R^{n})$ , in the next theorem, below, we provide a local uniform decay of the energy. Indeed, we shall consider the initial data taken in bounded sets of $H$ , namely, $‖ {ψ_{0}, ϕ_{0}, ϕ_{1}} ‖_{H} ⩽ L$ , where L is a positive constant.This is strongly necessary due to the nonlinear character of system (1.1) and since the energy $E (t)$ is not naturally a nonincreasing function of the parameter t. Thus, the constants, C and γ which appear in (2.2) will depend on $L > 0$ . Now, we are in position to state our stabilization result.
Theorem 2.2.
Assume that the assumption of Theorem 2.1 holds. Then, there exist $C, γ$ positive constants such that the following decay rate holds $\begin{array}{rcl} (2.2) & E (t) ⩽ C e^{- γ t} E (0) for all t ⩾ 0 \end{array}$ for every regular solution of problem ( 1.1 ) in the class given in previous theorem, provided the initial data are taken in bounded sets of $H$ .
Remark 2.1.
The assumption $α ⩾ \frac{3}{2 a_{0} b_{0}}$ is required in the proof of the existence. On the other hand to take α sufficiently large is natural to guarantee the dissipativity of the system.

The next theorem proved in [8] will be used in the proof of Lemma 4.6.

Consider the equations and systems of the form $\begin{array}{l} (2.3) & u_{t} + i P (D) u = F, \\ (2.4) & u (0, x) = u_{0} (x), \end{array}$ where $u = u (t, x), t \in R, x \in R^{n}, D = \frac{1}{i} (\frac{\partial}{\partial x_{1}}, \dots, \frac{\partial}{\partial x_{n}})$ , and $P (D) u$ is defined via a real symbol $p (ξ)$ in the scalar case (or a matrix with real entries in the case of systems), $\begin{matrix} (2.5) & P (D) u = \int_{R^{n}} e^{2 i π ⟨ x, ξ ⟩} p (ξ) (F_{2} u) (ξ) d ξ, \end{matrix}$ where $F_{2}$ is the Fourier transform with respect to the x variables.

The assumptions on $p (ξ)$ , reflecting the dispersive nature of (2.3) are that, roughly speaking, $p (ξ)$ behaves like $| ξ |^{m}$ for $| ξ | ⟶ \infty$ , with $m > 1$ : $\begin{matrix} (2.6) & p \in L_{loc}^{\infty} (R^{n}, R) \end{matrix}$ and is continuously differentiable for $| ξ | > R$ , for some $R ⩾ 0$ .

There exist $m > 1, c_{1} > 0, c_{2} > 0$ such that $\begin{array}{l} (2.7) & | p (ξ) | ⩽ c_{1} {(1 + | ξ |)}^{m}, \forall ξ \in R^{n}, \\ (2.8) & | (\partial p / \partial ξ_{j}) (ξ) | ⩾ c_{2} {(1 + | ξ |)}^{m - 1} | ξ_{j} | / | ξ |, \forall ξ \in R^{n}, | ξ | > R, j = 1, \dots, n . \end{array}$

We introduce $χ : R^{n + 1} ⟶ R$ of the type $\begin{matrix} χ (t, x) = χ_{0} (t) χ_{1} (x_{1}) \dots χ_{n} (x_{n}), \end{matrix}$ where $x = (x_{1}, \dots, x_{n})$ and $χ_{j} \in C_{0}^{\infty} (R), j = 0, 1, \dots, n$ .

Also let $1 ⩽ q ⩽ 2$ and α be real numbers such that $\begin{array}{l} (2.9) & 2 α < m - 1 - ((2 - q) / q) n, if 1 ⩽ q < 2, \\ (2.10) & 2 α ⩽ m - 1, if q = 2 . \end{array}$ Theorem 2.3.
We assume that p satisfies ( 2.6 ), ( 2.7 ), and ( 2.8 ). Let $s ⩾ - (m - 1) / 2, u_{0} \in H^{s} (R^{n}), F \in L_{loc}^{1} (R; H^{s, q} (R^{n})), 1 ⩽ q ⩽ 2$ . Let α be given by ( 2.9 ). Then the solution u of ( 2.3 ), ( 2.4 ) belongs to $L^{2} (- T, T; H_{u; loc}^{s + α} (R^{n}))$ for every $T > 0$ . Moreover, for every $χ \in C_{0}^{\infty} (R^{n})$ of the form above and which is supported in $[- T, T] \times R^{n}$ , one has $\begin{array}{l} (2.11) & {(\int_{R^{n}} χ^{2} (t, x) {| {(I - Δ)}^{\frac{s + α}{2}} u (t, x) |}^{2} d x d t)}^{\frac{1}{2}} \\ (2.12) & ⩽ C_{χ} (‖ u_{0} ‖_{H^{s} (R^{n}}) + ‖ F ‖_{L^{1} (- T, T; H^{s, q} (R^{n})}) \end{array}$ with $C_{χ}$ depending only on $χ, q, α, n, m$ and can be estimated as follows: $\begin{matrix} C_{χ} ⩽ C [\prod_{j = 0}^{n} ‖ χ_{j} ‖_{2} + ‖ χ_{0} ‖_{\infty} \sum_{j = 1}^{n} ‖ χ_{j} ‖_{2} ‖ χ_{1} ‖_{\infty} \dots ‖ {\tilde{χ}}_{j} ‖_{\infty} \dots ‖ χ_{n} ‖_{\infty}] . \end{matrix}$

3. Existence and uniqueness

In this section we derive a priori estimates for the solutions of the Klein–Gordon–Schrödinger system (1.1). In what follows, for simplicity, we will denote $u_{t} = u^{'}$ . Let us represent by ${ω_{ν}}$ a basis in $H^{2} (R^{n})$ and by $V_{m}$ the subspace of $H^{2} (R^{n})$ generated by the first m vectors of basis which is orthonormal in $L^{2} (R^{n})$ and by $\begin{array}{rcl} ψ_{m} (t) = \sum_{i = 1}^{m} g_{i m} (t) ω_{i}, ϕ_{m} (t) = \sum_{i = 1}^{m} h_{i m} (t) ω_{i}, \end{array}$ where ${ψ_{m} (t), ϕ_{m} (t)}$ is the solution to the following Cauchy problem $\begin{array}{rcl} (3.1) & \{\begin{matrix} (ψ_{m}^{'} (t), u) + i (\nabla ψ_{m} (t), \nabla u) + α (b (x) | ψ_{m} (t) |^{2} ψ_{m} (t), u) + α (b (x) ψ_{m} (t), u) \\ = - i (ϕ_{m} (t) ψ_{m} (t) χ_{ω}, u), \forall u \in V_{m}, \\ (ϕ_{m}^{″} (t), v) + (\nabla ϕ_{m} (t), \nabla v) + (a (x) ϕ_{m}^{'} (t), v) = (| ψ_{m} (t) |^{2} χ_{ω}, v), \forall v \in V_{m}, \\ ψ_{m} (0) = ψ_{0 m} \to ψ_{0}, ϕ_{m} (0) = ϕ_{0 m} \to ϕ_{0} in H^{2} (R^{n}), \\ ϕ_{m}^{'} (0) = ϕ_{1 m} \to ϕ_{1} in H^{1} (R^{n}) . \end{matrix} \end{array}$

The approximate system (3.1) is a finite system of ordinary differential equations which has a solution in $[0, t_{m} [$ . The extension of the solution to the whole interval $[0, T]$ , for all $T > 0$ , is a consequence of the first estimate we are going to obtain below.

A priori estimates:

The First Estimate: Considering $u = \overline{ψ_{m}}$ in the first equation of (3.1) and taking the real part, we obtain $\begin{matrix} (3.2) & \frac{1}{2} \frac{d}{d t} {‖ ψ_{m} (t) ‖}_{2}^{2} + α \int_{R^{n}} b (x) {| ψ_{m} (t) |}^{4} d x + α \int_{R^{n}} b (x) {| ψ_{m} (t) |}^{2} d x = 0 . \end{matrix}$

So, since $b (x) > b_{0}$ almost everywhere in $Ω_{R}$ , we have $\begin{array}{rcl} (3.3) & \frac{1}{2} \frac{d}{d t} {‖ ψ_{m} (t) ‖}_{2}^{2} + α b_{0} {‖ ψ_{m} (t) ‖}_{L^{4} (Ω_{R})}^{4} + α b_{0} {‖ ψ_{m} (t) ‖}_{L^{2} (Ω_{R})}^{2} ⩽ 0 . \end{array}$

Multiplying (3.3) by 2, integrating over $(0, t)$ , $t \in [0, t_{m}]$ , we obtain $\begin{array}{rcl} (3.4) & {‖ ψ_{m} (t) ‖}_{2}^{2} + 2 α b_{0} \int_{0}^{t} {‖ ψ_{m} (s) ‖}_{L^{4} (ω)}^{4} d s + 2 α b_{0} \int_{0}^{t} {‖ ψ_{m} (s) ‖}_{L^{2} (ω)}^{2} d s ⩽ ‖ ψ_{m 0} ‖_{2}^{2} . \end{array}$

Then, from convergence $ψ_{m} (0) = ψ_{0 m} \to ψ_{0}$ in $H^{2} (R^{n})$ , we have $\begin{matrix} (3.5) & (ψ_{m}) is bounded in L^{\infty} (0, \infty; L^{2} (R^{n})), \end{matrix}$ and there exists $C_{1}, C_{2} > 0$ , such that $\begin{array}{l} (3.6) & \int_{0}^{\infty} {‖ ψ_{m} (s) ‖}_{L^{4} (Ω_{R})}^{4} d s ⩽ C_{1} = C (‖ ψ_{0 m} ‖_{2}), \\ (3.7) & \int_{0}^{\infty} {‖ ψ_{m} (s) ‖}_{L^{2} (Ω_{R})}^{2} d s ⩽ C_{2} = C (‖ ψ_{0 m} ‖_{2}) . \end{array}$

Now, considering $v = ϕ_{m}^{'}$ in the second equation of (3.1) and making use of Hölder and Young inequalities, we deduce $\begin{array}{l} \frac{1}{2} \frac{d}{d t} [{‖ ϕ_{m}^{'} (t) ‖}_{2}^{2} + {‖ ϕ_{m} (t) ‖}_{2}^{2} + {‖ \nabla ϕ_{m} (t) ‖}_{2}^{2}] + \int_{R^{n}} a (x) {| ϕ_{m}^{'} (t) |}^{2} d x \\ (3.8) & ⩽ \frac{1}{2 a_{0}} {‖ ψ_{m} (t) ‖}_{L^{4} (Ω_{R})}^{4} + \frac{1}{2} \int_{R^{n}} a (x) | ϕ_{m}^{'} (t) | d x . \end{array}$

Thus, $\begin{array}{l} (3.9) & \frac{1}{2} \frac{d}{d t} [{‖ ϕ_{m}^{'} (t) ‖}_{2}^{2} + {‖ ϕ_{m} (t) ‖}_{2}^{2} + {‖ \nabla ϕ_{m} (t) ‖}_{2}^{2}] \\ (3.10) & + \frac{1}{2} \int_{R^{n}} a (x) {| ϕ_{m}^{'} (t) |}^{2} d x ⩽ \frac{1}{2 a_{0}} {‖ ψ_{m} (t) ‖}_{L^{4} (Ω_{R})}^{4} . \end{array}$

Multiplying (3.9) by 2, integrating over $(0, t)$ , $t \in [0, t_{m}]$ , observing that $ϕ_{m} (0) = ϕ_{0 m} \to ϕ_{0} in H^{2} (R^{n}), ϕ_{m}^{'} (0) = ϕ_{1 m} \to ϕ_{1} in H^{1} (R^{n})$ , we obtain $\begin{matrix} (3.11) & {‖ ϕ_{m}^{'} (t) ‖}_{2}^{2} + {‖ ϕ_{m} (t) ‖}_{2}^{2} + {‖ \nabla ϕ_{m} (t) ‖}_{2}^{2} ⩽ C + \frac{1}{2 a_{0}} \int_{0}^{t} {‖ ψ_{m} (s) ‖}_{L^{4} (Ω_{R})}^{4} d s, \end{matrix}$ where $C : = C (‖ ϕ_{1 m} ‖_{2}, ‖ \nabla ϕ_{0 m} ‖_{2})$ .

So, for (3.6) $\begin{matrix} (3.12) & {‖ ϕ_{m}^{'} (t) ‖}_{2}^{2} + {‖ ϕ_{m} (t) ‖}_{2}^{2} + {‖ \nabla ϕ_{m} (t) ‖}_{2}^{2} ⩽ C, \end{matrix}$ where $C : = C (‖ ψ_{0 m} ‖, ‖ ϕ_{1 m} ‖_{2}, ‖ \nabla ϕ_{0 m} ‖_{2})$ .

Thus, $\begin{array}{l} (3.13) & (ϕ_{m}) is bounded in L^{\infty} (0, \infty; H^{1} (R^{n})), \\ (3.14) & (ϕ_{m}^{'}) is bounded in L^{\infty} (0, \infty; L^{2} (R^{n})) . \end{array}$

The Second Estimate: Taking the derivative on time of the first equation in (3.1), considering $u = \overline{ψ_{m}^{'}}$ , we have $\begin{array}{l} (ψ_{m}^{″} (t), ψ_{m}^{'} (t)) + i (\nabla ψ_{m}^{'} (t), \nabla ψ_{m}^{'} (t)) + α ({[b (x) {| ψ_{m} (t) |}^{2} ψ_{m} (t)]}^{'}, ψ_{m}^{'} (t)) \\ (3.15) & + α ({[b (x) ψ_{m} (t)]}^{'}, ψ_{m}^{'} (t)) = - i ({[ϕ_{m} (t) ψ_{m} (t) χ_{Ω_{R}}]}^{'}, ψ_{m}^{'} (t)) . \end{array}$

Considering $I_{1} : = ({[b (x) | ψ_{m} (t) |^{2} ψ_{m} (t)]}^{'}, ψ_{m}^{'} (t))$ , we have $\begin{array}{rcl} I_{1} & : = & (2 b (x) {| ψ_{m} (t) |}^{2} ψ_{m}^{'} (t) + b (x) {(ψ_{m} (t))}^{2} ψ_{m}^{'} (t), ψ_{m}^{'} (t)) \\ (3.16) & = & 2 \int_{R^{n}} b (x) {| ψ_{m} (t) |}^{2} {| ψ_{m}^{'} (t) |}^{2} d x + \int_{R^{n}} b (x) {[ψ_{m} (t) \overline{ψ_{m}^{'} (t)}]}^{2} d x . \end{array}$

Observing that $\begin{matrix} 2 {[Re (z_{1} \overline{z_{2}})]}^{2} = | z_{1} |^{2} | z_{2} |^{2} + Re {[(z_{1} \overline{z_{2}})]}^{2}, \forall z_{1}, z_{2} \in C, \end{matrix}$ and thus taking the part real of (3.16) and since $b (x) ⩾ b_{0} > 0$ a.e. in $Ω_{R}$ , we deduce $\begin{array}{l} b_{0} \int_{Ω_{R}} {| ψ_{m} (x, t) |}^{2} {| ψ_{m}^{'} (x, t) |}^{2} d x + 2 b_{0} \int_{Ω_{R}} {[Re (ψ_{m} (x, t) \overline{ψ_{m}^{'} (x, t)})]}^{2} d x \\ (3.17) & ⩽ \int_{R^{n}} b (x) {| ψ_{m} (x, t) |}^{2} {| ψ_{m}^{'} (x, t) |}^{2} d x + 2 \int_{R^{n}} b (x) {[Re (ψ_{m} (x, t) \overline{ψ_{m}^{'} (x, t)})]}^{2} d x . \end{array}$

It follows from (3.15), (3.16) and (3.17) that $\begin{array}{l} Re [(ψ_{m}^{″} (t), ψ_{m}^{'} (t)) + i (\nabla ψ_{m}^{'} (t), \nabla ψ_{m}^{'} (t))] + α b_{0} \int_{Ω_{R}} {| ψ_{m} (x, t) |}^{2} {| ψ_{m}^{'} (x, t) |}^{2} d x \\ + 2 α b_{0} \int_{Ω_{R}} {[Re (ψ_{m} (x, t) \overline{ψ_{m}^{'} (x, t)})]}^{2} d x + α ({[b (x) ψ_{m} (t)]}^{'}, ψ_{m}^{'} (t)) \\ (3.18) & ⩽ Re [- i ({[ϕ_{m} (t) ψ_{m} (t) χ_{ω}]}^{'}, ψ_{m}^{'} (t))] \end{array}$ that is, $\begin{matrix} (3.19) & \begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ ψ_{m}^{'} (t) ‖}_{2}^{2} + α b_{0} \int_{Ω_{R}} {| ψ_{m} (x, t) |}^{2} {| ψ_{m}^{'} (x, t) |}^{2} d x \\ + 2 α b_{0} \int_{Ω_{R}} {[Re (ψ_{m} (x, t) \overline{ψ_{m}^{'} (x, t)})]}^{2} d x + α b_{0} {‖ ψ_{m}^{'} (t) ‖}_{L^{2} (Ω_{R})}^{2} \\ ⩽ \int_{Ω_{R}} | ϕ_{m}^{'} (x, t) | | ψ_{m} (x, t) | | ψ_{m}^{'} (x, t) | d x . \end{array} \end{matrix}$

Thus, using the inequalities of Hölder and Young to estimate the term on the right hand side of (3.19), we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ ψ_{m}^{'} (t) ‖}_{2}^{2} + (α b_{0} - \frac{1}{2 a_{0}}) \int_{Ω_{R}} {| ψ_{m} (x, t) |}^{2} {| ψ_{m}^{'} (x, t) |}^{2} d x \\ + 2 α b_{0} \int_{Ω_{R}} [Re {(ψ_{m} (x, t) \overline{ψ_{m}^{'} (x, t)}]}^{2} d x + α b_{0} {‖ ψ_{m}^{'} (t) ‖}_{L^{2} (Ω_{R})}^{2} \\ (3.20) & ⩽ \frac{1}{2} \int_{R^{n}} a (x) {| ϕ_{m}^{'} (x, t) |}^{2} d x . \end{array}$

Taking the derivative in t of the second equation in (3.1), considering $v = ϕ_{m}^{″}$ and taking the real part, we have $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ ϕ_{m}^{″} (t) ‖}_{2}^{2} + \frac{1}{2} \frac{d}{d t} {‖ ϕ_{m}^{'} (t) ‖}_{2}^{2} + \frac{1}{2} \frac{d}{d t} {‖ \nabla ϕ_{m}^{'} (t) ‖}_{2}^{2} + \int_{R^{n}} a (x) {| ϕ_{m}^{″} (x, t) |}^{2} d x \\ (3.21) & ⩽ 2 \int_{Ω_{R}} | ψ_{m} (x, t) | | ψ_{m}^{'} (x, t) | | ϕ_{m}^{″} (x, t) | d x . \end{array}$

So, making use of the Hölder’s generalized inequality combined with the inequality of Young to estimate the term on the right hand side of (3.21), we get $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ ϕ_{m}^{″} (t) ‖}_{2}^{2} + \frac{1}{2} \frac{d}{d t} {‖ ϕ_{m}^{'} (t) ‖}_{2}^{2} + \frac{1}{2} \frac{d}{d t} {‖ \nabla ϕ_{m}^{'} (t) ‖}_{2}^{2} + \frac{1}{2} \int_{R^{n}} a (x) {| ϕ_{m}^{″} (x, t) |}^{2} d x \\ (3.22) & ⩽ \frac{1}{a_{0}} \int_{Ω_{R}} {| ψ_{m} (x, t) |}^{2} {| ψ_{m}^{'} (x, t) |}^{2} d x . \end{array}$

Adding (3.20), (3.22), and (3.9) we have $\begin{array}{l} \frac{1}{2} \frac{d}{d t} [{‖ ψ_{m}^{'} (t) ‖}_{2}^{2} + {‖ ϕ_{m}^{'} (t) ‖}_{2}^{2} + {‖ \nabla ϕ_{m} (t) ‖}_{2}^{2} + {‖ ϕ_{m} (t) ‖}_{2}^{2} + {‖ ϕ_{m}^{″} (t) ‖}_{2}^{2} + {‖ \nabla ϕ_{m}^{'} (t) ‖}_{2}^{2}] \\ \times (α b_{0} - \frac{1}{2 a_{0}} - \frac{1}{a_{0}}) \int_{Ω_{R}} {| ψ_{m} (x, t) |}^{2} {| ψ_{m}^{'} (x, t) |}^{2} d x + \frac{1}{2} \int_{R^{n}} a (x) {| ϕ_{m}^{″} (x, t) |}^{2} d x \\ \times α b_{0} {‖ ψ_{m}^{'} (t) ‖}_{L^{2} (Ω_{R})}^{2} + 2 α b_{0} \int_{Ω_{R}} [Re {(ψ_{m} (x, t) \overline{ψ_{m}^{'} (x, t)}]}^{2} d x ⩽ \frac{1}{2 a_{0}} {‖ ψ_{m} (t) ‖}_{L^{4} (Ω_{R})}^{4} . \end{array}$

Thus, considering that $α ⩾ \frac{3}{2 a_{0} b_{0}}$ , we have $\begin{array}{rcl} (α b_{0} - \frac{1}{2 a_{0}} - \frac{1}{a_{0}}) \int_{Ω_{R}} {| ψ_{m} (x, t) |}^{2} {| ψ_{m}^{'} (x, t) |}^{2} d x ⩾ 0, \end{array}$ and we also have to $\begin{array}{rcl} 2 α b_{0} \int_{Ω_{R}} [Re {(ψ_{m} (x, t) \overline{ψ_{m}^{'} (x, t)}]}^{2} d x ⩾ 0, \frac{1}{2} \int_{R^{n}} a (x) {| ϕ_{m}^{″} (x, t) |}^{2} d x ⩾ 0 . \end{array}$

So, $\begin{array}{l} \frac{1}{2} \frac{d}{d t} [{‖ ψ_{m}^{'} (t) ‖}_{2}^{2} + {‖ ϕ_{m}^{'} (t) ‖}_{2}^{2} + {‖ \nabla ϕ_{m} (t) ‖}_{2}^{2} + {‖ ϕ_{m} (t) ‖}_{2}^{2} + {‖ ϕ_{m}^{″} (t) ‖}_{2}^{2} + {‖ \nabla ϕ_{m}^{'} (t) ‖}_{2}^{2}] \\ (3.23) & \times α b_{0} {‖ ψ_{m}^{'} (t) ‖}_{L^{2} (Ω_{R})}^{2} ⩽ \frac{1}{2 a_{0}} {‖ ψ_{m} (t) ‖}_{L^{4} (Ω_{R})}^{4} . \end{array}$

Multiplying (3.23) by 2, integrating over $(0, t), t \in [0, T]$ , we infer $\begin{array}{l} {‖ ψ_{m}^{'} (t) ‖}_{2}^{2} + {‖ ϕ_{m}^{'} (t) ‖}_{2}^{2} + {‖ \nabla ϕ_{m} (t) ‖}_{2}^{2} + {‖ ϕ_{m} (t) ‖}_{2}^{2} + {‖ ϕ_{m}^{″} (t) ‖}_{2}^{2} + {‖ \nabla ϕ_{m}^{'} (t) ‖}_{2}^{2} \\ + 2 α b_{0} \int_{0}^{t} {‖ ψ_{m}^{'} (s) ‖}_{L^{2} (Ω_{R})}^{2} d s \\ ⩽ {‖ ψ_{m}^{'} (0) ‖}_{2}^{2} + {‖ ϕ_{m}^{'} (0) ‖}_{2}^{2} + {‖ \nabla ϕ_{m} (0) ‖}_{2}^{2} + {‖ ϕ_{m}^{″} (0) ‖}_{2}^{2} + {‖ \nabla ϕ_{m}^{'} (0) ‖}_{2}^{2} \\ (3.24) & + \frac{1}{a_{0}} \int_{0}^{t} {‖ ψ_{m} (s) ‖}_{L^{4} (Ω_{R})}^{4} d s . \end{array}$

Making $u = ψ_{m}^{'} (0)$ in the first equation of (3.1) and $v = ϕ_{m}^{″} (0)$ in the second equation and observing the convergences in (3.1), it is verified that there is $C > 0$ such that $‖ ψ_{m}^{'} (0) ‖^{2} + ‖ ϕ_{m}^{″} (0) ‖^{2} < C$ . Then, from (3.6) we obtain $\begin{matrix} (3.25) & {‖ ψ_{m}^{'} (t) ‖}_{2}^{2} + {‖ ϕ_{m}^{″} (t) ‖}_{2}^{2} + {‖ \nabla ϕ_{m}^{'} (t) ‖}_{2}^{2} ⩽ C, \end{matrix}$ where $C : = C (‖ ψ_{0} ‖_{2}, ‖ \nabla ψ_{0} ‖_{2}, ‖ Δ ϕ_{0} ‖_{2}, ‖ \nabla ϕ_{0} ‖_{2})$ . Thus, $\begin{array}{l} (3.26) & (ψ_{m}^{'}) is bounded in L^{\infty} (0, \infty; L^{2} (R^{n})), \\ (3.27) & (ϕ_{m}^{'}) is bounded in L^{\infty} (0, \infty; H^{1} (R^{n})), \\ (3.28) & (ϕ_{m}^{″}) is bounded in L^{\infty} (0, \infty; L^{2} (R^{n})) . \end{array}$

The Third Estimate: In (3.1) considering $u = \overline{ψ_{m}}$ and taking the imaginary part, we obtain $\begin{matrix} (3.29) & {‖ \nabla ψ_{m} (t) ‖}_{2}^{2} = - Im (ψ_{m}^{'} (t), ψ_{m} (t)) - \int_{Ω_{R}} ϕ_{m} (x, t) {| ψ_{m} (x, t) |}^{2} d x . \end{matrix}$

Making use of the Hölder’s inequality, we deduce $\begin{array}{l} (3.30) & | Im (ψ_{m}^{'} (t), ψ_{m} (t)) | ⩽ C, \\ (3.31) & | - \int_{Ω_{R}} ϕ_{m} (x, t) {| ψ_{m} (x, t) |}^{2} d x | ⩽ C . \end{array}$

So, from (3.14), (3.25), (3.30) and (3.31) we conclude $\begin{array}{rcl} {‖ \nabla ψ_{m} (t) ‖}_{2} ⩽ C for all t > 0, \end{array}$ where $C = C (‖ \nabla ψ_{0} ‖_{2}, ‖ Δ ϕ_{0} ‖_{2}, ‖ \nabla ϕ_{0} ‖_{2})$ .

Thus, $\begin{matrix} (3.32) & ψ_{m} is bounded in L^{\infty} (0, \infty; H^{1} (R^{n})) . \end{matrix}$

The rest of the proof follows the same basic steps as those ones of ([7], Theorem 2.1), with the difference that in the passage to the limit we define $\begin{matrix} B_{i} = {x \in R^{n}; ‖ x ‖_{R^{n}} ⩽ i}, i \in N, R^{n} \cap B_{i} = Ω_{i}, \end{matrix}$ where $R^{n} \subset ⋃_{i \in N} Ω_{i}$ , and we use the Aubin–Lions compactness theorem for each compact $Ω_{i}$ .

Uniqueness: Let ${ψ_{1}, ϕ_{1}}$ and ${ψ_{2}, ϕ_{2}}$ solutions do problem (1.1). Then, the uniqueness follows defining $z = ψ_{1} - ψ_{2}$ and $w = ϕ_{1} - ϕ_{2}$ and repeating verbatim the same arguments already used in the first estimate.

4. Uniform decay rates

In this section we work with regular solutions ${ψ (t), ϕ (t), ϕ_{t} (t)}$ to problem (1.1), that is, those ones that lie in $H^{2} (R^{n}) \times H^{2} (R^{n}) \times H^{1} (R^{n})$ and taking ${ψ_{0}, ϕ_{0}, ϕ_{1}} \in H^{2} (R^{n}) \times H^{2} (R^{n}) \times H^{1} (R^{n}) : = H$ , such that $‖ {ψ_{0}, ϕ_{0}, ϕ_{1}} ‖_{H} ⩽ L$ , where $L > 0$ . So, multiplying the first equation of (1.1) by $\overline{ψ}$ and integrating over $R^{n}$ , we have $\begin{matrix} (4.1) & \int_{R^{n}} ψ^{'} \overline{ψ} d x + i \int_{R^{n}} | \nabla ψ |^{2} d x + α \int_{R^{n}} b (x) (| ψ |^{4} + | ψ |^{2}) d x = - i \int_{Ω_{R}} ϕ | ψ |^{2} d x . \end{matrix}$

Taking the real part in (4.1) we obtain $\begin{matrix} (4.2) & \frac{1}{2} \frac{d}{d t} {‖ ψ (t) ‖}_{2}^{2} + α \int_{R^{n}} b (x) (| ψ |^{4} + | ψ |^{2}) d x = 0 . \end{matrix}$

Multiplying the second equation by $ϕ^{'}$ , integrating over $R^{n}$ and making use of Green formula, we deduce that $\begin{matrix} (4.3) & \frac{1}{2} \frac{d}{d t} ({‖ ϕ^{'} ‖}_{2}^{2} + ‖ ϕ ‖_{2}^{2} + ‖ \nabla ϕ ‖_{2}^{2}) d x + \int_{R^{n}} a (x) {| ϕ^{'} |}^{2} d x = \int_{Ω_{R}} | ψ |^{2} ϕ^{'} d x . \end{matrix}$

Adding (4.2) and (4.3) we obtain $\begin{array}{rcl} (4.4) & E^{'} (t) + α \int_{R^{n}} b (x) (| ψ |^{4} + | ψ |^{2}) d x + \int_{R^{n}} a (x) {| ϕ^{'} |}^{2} d x = \int_{Ω_{R}} | ψ |^{2} ϕ^{'} d x . \end{array}$

Next, we will analyze the last term on the RHS of (4.4). We have, from Assumption 2.1 and making use of the Cauchy-Schwarz inequality that $\begin{array}{rcl} | \int_{Ω_{R}} | ψ |^{2} ϕ^{'} d x | & ⩽ & {(\int_{Ω_{R}} | ψ |^{4} d x)}^{\frac{1}{2}} {(\int_{Ω_{R}} {| ϕ^{'} |}^{2} d x)}^{\frac{1}{2}} \\ ⩽ & b_{0}^{- \frac{1}{2}} {(\int_{Ω_{R}} b_{0} | ψ |^{4} d x)}^{\frac{1}{2}} a_{0}^{- \frac{1}{2}} {(\int_{Ω_{R}} a_{0} {| ϕ^{'} |}^{2} d x)}^{\frac{1}{2}} \\ ⩽ & b_{0}^{- \frac{1}{2}} {(\int_{Ω_{R}} b (x) | ψ |^{4} d x)}^{\frac{1}{2}} a_{0}^{- \frac{1}{2}} {(\int_{Ω_{R}} a (x) {| ϕ^{'} |}^{2} d x)}^{\frac{1}{2}} \\ (4.5) & ⩽ & \frac{1}{2 a_{0} b_{0}} \int_{R^{n}} b (x) | ψ |^{4} d x + \frac{1}{2} \int_{R^{n}} a (x) {| ϕ^{'} |}^{2} d x . \end{array}$

Combining (4.4) and (4.5) considering α large enough such that $β : = α - \frac{1}{2 a_{0} b_{0}} > 0$ it holds that $\begin{array}{rcl} E^{'} (t) & ⩽ & - β \int_{R^{n}} b (x) | ψ |^{4} d x - α \int_{R^{n}} b (x) | ψ |^{2} d x - \frac{1}{2} \int_{R^{n}} a (x) {| ϕ^{'} |}^{2} d x \\ (4.6) & ⩽ & - k [\int_{R^{n}} b (x) (| ψ |^{4} + | ψ |^{2}) d x + \int_{R^{n}} a (x) {| ϕ^{'} |}^{2} d x] d t, \end{array}$ where $k = min {\frac{1}{2}, β}$ .

Remark 4.1.

From (4.6) we deduce two facts: (i) the map $t \in (0, \infty) \mapsto E (t)$ is nonincreasing, and, in addition, (ii) we have the following inequality of the energy $\begin{array}{rcl} E (t_{2}) - E (t_{1}) ⩽ - k \int_{t_{1}}^{t_{2}} [\int_{R^{n}} a (x) {| ϕ^{'} |}^{2} d x + \int_{R^{n}} b (x) (| ψ |^{4} + | ψ |^{2}) d x] d t \end{array}$ for $0 ⩽ t_{1} ⩽ t_{2} < + \infty$ , which will be crucial in the proof.

In order to prove Theorem 2.2 we proceed in several steps.

Lemma 4.1.

There exists a positive constant $C > 0$ such that $\begin{matrix} (4.7) & \frac{1}{2} \int_{0}^{T} \int_{Ω_{R}} [| \nabla ϕ |^{2} + | ϕ |^{2} + {| ϕ^{'} |}^{2} + | ψ |^{2}] d x d t ⩽ C {2 E (0) + \int_{0}^{T} \int_{B_{2 R} ∖ B_{R}} | ϕ |^{2} d x d t} \end{matrix}$ for every $T > 0$ and for all regular solution of ( 1.1 ).

Proof.

Multiplying the first equation of problem (1.1) by $\overline{ψ}$ , integrating in $R^{n} \times (0, T)$ and taking the real part, we obtain $\begin{matrix} (4.8) & {[\int_{R^{n}} \frac{| ψ |^{2}}{2} d x]}_{0}^{T} + α \int_{0}^{T} \int_{R^{n}} b (x) (| ψ |^{4} + | ψ |^{2}) d x d t = 0 . \end{matrix}$

Now, multiplying the second equation of problem (1.1) by $φ (x) ϕ$ with $φ \in W^{2, \infty} (R^{n})$ and integrating by parts over $R^{n} \times (0, T)$ , we have that $\begin{array}{l} \int_{0}^{T} \int_{R^{n}} φ [| \nabla ϕ |^{2} + | ϕ |^{2}] d x d t & = \int_{0}^{T} \int_{R^{n}} (\frac{Δ φ}{2} | ϕ |^{2} + φ {| ϕ^{'} |}^{2}) d x d t \\ - {[\int_{R^{n}} [(ϕ^{'} + a (x) \frac{ϕ}{2}) φ ϕ] d x]}_{0}^{T} \\ (4.9) & + \int_{R^{n} ∖ B_{R}} | ψ |^{2} φ ϕ d x d t . \end{array}$

We apply (4.9) with $φ \in W^{2, \infty} (R^{n})$ verifying $\begin{matrix} (4.10) & 0 ⩽ φ ⩽ 1 in R^{n}; φ = 0 in B_{R}; φ = 1 in R^{n} ∖ B_{2 R} \end{matrix}$ it follows the lemma. □

Lemma 4.2.

Let be $q = (q_{1}, \dots, q_{n}) \in W^{1, \infty} {(R^{n})}^{n}$ . For every $T, r > 0$ and every regular solution of ( 1.1 ) the following identity holds in $B_{r} \times (0, T)$ $\begin{array}{l} \frac{1}{2} \int_{0}^{T} \int_{B_{r}} div (q) [{| ϕ^{'} |}^{2} - | \nabla ϕ |^{2} - | ϕ |^{2}] d x d t + \sum_{j, k = 1}^{n} \int_{0}^{T} \int_{B_{r}} \frac{\partial ϕ}{\partial x_{j}} \frac{\partial q_{k}}{\partial x_{j}} \frac{\partial ϕ}{\partial x_{k}} d x d t \\ + \int_{0}^{T} \int_{B_{r}} a (x) ϕ^{'} (q \cdot \nabla ϕ) d x d t - \int_{0}^{T} \int_{B_{r}} | ψ |^{2} χ_{Ω_{R}} (q \cdot \nabla ϕ) d x d t \\ + α \int_{0}^{T} \int_{B_{r}} b (x) (| ψ |^{4} + | ψ |^{2}) d x d t \\ = - {[\int_{B_{r}} \frac{| ψ |^{2}}{2} + ϕ^{'} (q \cdot \nabla ϕ) d x d t]}_{0}^{T} \\ (4.11) & + \frac{1}{2 r} \int_{0}^{T} \int_{S_{r}} (q \cdot x) [{| ϕ^{'} |}^{2} - | \nabla ϕ |^{2} - | ϕ |^{2}] d Γ d t + \int_{0}^{T} \int_{S_{r}} \frac{\partial ϕ}{\partial ν} (q \cdot \nabla ϕ) d Γ d t . \end{array}$

Proof.

It suffices to multiply the second equation of (1.1) by $q \cdot \nabla ϕ$ and to integrate by parts on $B_{r} \times (0, T)$ . □

Lemma 4.3.

Let be $ξ \in W^{1, \infty} (R^{n})$ . For every $T, r > 0$ and every regular solution of ( 1.1 ) the following identity holds in $B_{r} \times (0, T)$ $\begin{array}{l} \int_{0}^{T} \int_{B_{r}} ξ [| \nabla ϕ |^{2} + ϕ^{2}] d x d t & = \int_{0}^{T} \int_{B_{r}} [ξ {| ϕ^{'} |}^{2} - (\nabla ϕ \cdot \nabla ξ) ϕ] d x d t \\ \times \int_{0}^{T} \int_{B_{r}} \frac{\partial ϕ}{\partial ν} ξ ϕ d Γ d t - {[\int_{B_{r}} (ϕ^{'} + a (x) \frac{ϕ}{2}) ξ ϕ d x]}_{0}^{T} \\ (4.12) & + \int_{0}^{T} \int_{B_{r}} | ψ |^{2} χ_{Ω_{R}} ξ ϕ d x d t . \end{array}$

Proof.

It suffices to multiply the first equation of (1.1) by $ξ ϕ$ and to integrate by parts on $B_{r} \times (0, T)$ . □

Lemma 4.4.

For every $r > R$ there is a positive constant $C_{r} > 0$ such that the following estimate holds, for all $T > 0$ and all regular solution of ( 1.1 ) $\begin{array}{l} \frac{1}{2} \int_{0}^{T} \int_{B_{r}} [| \nabla ϕ |^{2} + | ϕ |^{2} + {| ϕ^{'} |}^{2} + | ψ |^{2}] d x d t \\ (4.13) & ⩽ C_{r} (2 E (0) + ‖ ϕ ‖_{L^{2} (B_{2 r} \times (0, T))}^{2} + ‖ ψ ‖_{L^{2} (B_{2 r} \times (0, T))}^{2}) . \end{array}$

Proof.

Applying identity (4.11) with $q = x$ we deduce $\begin{array}{l} \frac{n}{2} \int_{0}^{T} \int_{B_{r}} [{| ϕ^{'} |}^{2} - | \nabla ϕ |^{2} - | ϕ |^{2}] d x d t + \int_{0}^{T} \int_{B_{r}} | \nabla ϕ |^{2} d x d t \\ = - \int_{0}^{T} \int_{B_{r}} a (x) ϕ^{'} (x \cdot \nabla ϕ) d x d t - α \int_{0}^{T} \int_{B_{r}} b (x) (| ψ |^{4} + | ψ |^{2}) \\ (4.14) & + A + B + \int_{0}^{T} \int_{B_{r}} | ψ |^{2} χ_{Ω_{R}} (x \cdot \nabla ϕ) d x d t, \end{array}$ with $\begin{matrix} (4.15) & A = - {[\int_{B_{r}} \frac{| ψ |^{2}}{2} + ϕ^{'} (x \cdot \nabla ϕ) d x d t]}_{0}^{T} \end{matrix}$ and $\begin{matrix} (4.16) & B = r \int_{0}^{T} \int_{S_{r}} (\frac{| ϕ^{'} |^{2}}{2} - \frac{| \nabla ϕ |^{2}}{2} + {| \frac{\partial ϕ}{\partial ν} |}^{2} - \frac{| ϕ |^{2}}{2}) d Γ d t . \end{matrix}$

On the other hand, applying (4.84) with $ξ = 1$ and multiplying by a positive constant $β \in (\frac{n - 2}{2}, \frac{n}{2})$ we get $\begin{array}{l} (4.17) & \begin{matrix} β \int_{0}^{T} \int_{B_{r}} [| \nabla ϕ |^{2} + ϕ^{2}] d x d t & = β \int_{0}^{T} \int_{B_{r}} {| ϕ^{'} |}^{2} d x d t + β \int_{0}^{T} \int_{S_{r}} \frac{\partial ϕ}{\partial ν} ϕ d Γ d t \\ - β {[\int_{B_{r}} [(ϕ^{'} + a (x) \frac{ϕ}{2}) ϕ] d x]}_{0}^{T} + \int_{0}^{T} \int_{B_{r}} | ψ |^{2} | χ_{Ω_{R}} ϕ d x d t . \end{matrix} \end{array}$ Adding (4.14) and (4.17) we obtain $\begin{array}{l} (\frac{n}{2} - β) \int_{0}^{T} \int_{B_{r}} {| ϕ^{'} |}^{2} d x d t + (1 + β - \frac{n}{2}) \int_{0}^{T} \int_{B_{r}} | \nabla ϕ |^{2} d x d t \\ + (β - \frac{n}{2}) \int_{0}^{T} \int_{B_{r}} | ϕ |^{2} d x d t \\ = \int_{0}^{T} \int_{B_{r}} | ψ |^{2} χ_{Ω_{R}} (x \cdot \nabla ϕ) d x d t \\ + \int_{0}^{T} \int_{B_{r}} | ψ |^{2} χ_{Ω_{R}} ϕ d x d t - \int_{0}^{T} \int_{B_{r}} a (x) ϕ^{'} (x \cdot \nabla ϕ) d x d t + B \\ (4.18) & + β \int_{0}^{T} \int_{S_{r}} \frac{\partial ϕ}{\partial ν} ϕ d Γ d t + D - α \int_{0}^{T} \int_{B_{r}} b (x) (| ψ |^{4} + | ψ |^{2}) d x d t, \end{array}$ where $\begin{matrix} (4.19) & D = A - β {[\int_{B_{r}} [(ϕ^{'} + a (x) \frac{ϕ}{2}) ϕ] d x]}_{0}^{T} . \end{matrix}$

We now remark that $\begin{array}{l} | \int_{0}^{T} \int_{B_{r}} a (x) ϕ^{'} (x \cdot \nabla ϕ) d x d t | & ⩽ ε \int_{0}^{T} \int_{B_{r}} | \nabla ϕ |^{2} d x d t \\ (4.20) & + \frac{r^{2} ‖ a ‖_{L^{\infty} (B_{r})}}{4 ε} \int_{0}^{T} \int_{B_{r}} a (x) {| ϕ^{'} |}^{2} d x d t \end{array}$ for all $ε > 0$ . $\begin{array}{l} | \int_{0}^{T} \int_{B_{r} ∖ B_{R}} | ψ |^{2} (x \cdot \nabla ϕ) d x d t | & ⩽ \frac{r^{2} b_{0}^{- 1}}{4 ε} \int_{0}^{T} \int_{B_{r} ∖ B_{R}} b (x) | ψ |^{4} d x \\ (4.21) & + ε \int_{0}^{T} \int_{B_{r}} | \nabla ϕ |^{2} d x d t \end{array}$ for all $ε > 0$ . $\begin{array}{l} | \int_{0}^{T} \int_{B_{r} ∖ B_{R}} | & ⩽ \frac{b_{0}^{- 1}}{2} \int_{0}^{T} \int_{B_{r} ∖ B_{R}} b (x) | ψ |^{4} d x d t \\ (4.22) & + \frac{1}{2} \int_{0}^{T} \int_{B_{r}} | ϕ |^{2} d x d t . \end{array}$

Combining (4.18), (4.20), (4.21) and (4.22), we have that $\begin{array}{l} \frac{1}{2} \int_{0}^{T} \int_{B_{r}} [| \nabla ϕ |^{2} + | ϕ |^{2} + {| ϕ^{'} |}^{2}] d x d t & ⩽ C {\int_{0}^{T} \int_{B_{r}} a (x) {| ϕ^{'} |}^{2} d x d t + | \int_{0}^{T} \int_{S_{r}} \frac{\partial ϕ}{\partial ν} ϕ d Γ d t | \\ + | B | + | D | + ‖ ϕ ‖_{L^{2} (B_{r} \times (0, T)} \\ (4.23) & + \int_{0}^{T} \int_{B_{r}} b (x) | ψ |^{4} d x d t + \int_{0}^{T} \int_{B_{r}} b (x) | ψ |^{2} d x d t} \end{array}$ for some constant $C > 0$ which does not depend on T.

We now estimate the integrals over $S_{r} \times (0, T)$ on the right hand side of (4.23). Applying the identities (4.11) and (4.12) with $ϕ \in W^{1, \infty} (B_{r})$ and $ϕ x$ respectively with $\begin{matrix} (4.24) & 0 ⩽ ϕ ⩽ 1 in B_{r}; ϕ = 0 in B_{r^{'}}; with R < r^{'} < r; ϕ = 1 on S_{r} \end{matrix}$ we obtain for some $C > 0$ which does not depend on T, $\begin{array}{l} | \int_{0}^{T} \int_{S_{r}} \frac{\partial ϕ}{\partial ν} ϕ d Γ d t | + | B | & ⩽ C {\int_{0}^{T} \int_{B_{r} ∖ B_{r^{'}}} [{| ϕ^{'} |}^{2} + | \nabla ϕ |^{2} + | ϕ |^{2}] d x d t \\ + \int_{0}^{T} \int_{B_{r} ∖ B_{r^{'}}} b (x) | ψ |^{4} d x d t \\ (4.25) & + \int_{0}^{T} \int_{B_{r}} b (x) (| ψ |^{4} + | ψ |^{2}) d x d t + | G |}, \end{array}$ where $\begin{matrix} G = {[\int_{B_{r}} \frac{| ψ |^{2}}{2} + ϕ^{'} x \cdot \nabla ϕ d x]}_{0}^{T} + {[\int_{B_{r}} [ϕ^{'} + a (x) \frac{ϕ}{2}] φ ϕ d x]}_{0}^{T} . \end{matrix}$

Finally we apply (4.12) in $B_{2 r} \times (0, T)$ with $φ \in W^{1, \infty} (B_{2 r})$ such that $\begin{matrix} (4.26) & 0 ⩽ φ ⩽ 1 in B_{r}; φ = 0 in B_{r} ∖ B_{r^{'}} with R < r < r^{'}; φ = 0 in S_{2 r}, \end{matrix}$ and $\begin{matrix} (4.27) & \frac{| \nabla φ |^{2}}{φ} \in L^{\infty} (B_{2 r}) . \end{matrix}$

We have that $\begin{array}{l} \int_{0}^{T} \int_{B_{2 r}} φ [\nabla ϕ |^{2} + | ϕ |^{2}] d x d t & ⩽ C {\int_{0}^{T} \int_{B_{2 r}} a (x) {| ϕ^{'} |}^{2} d x d t + \int_{0}^{T} \int_{B_{2 r}} | ϕ \nabla ϕ \cdot \nabla ϕ | d x d t \\ (4.28) & + \int_{0}^{T} \int_{B_{2 r} ∖ B_{R}} b (x) | ψ |^{4} d x d t + \int_{0}^{T} \int_{B_{2 r}} | ϕ |^{2} d x d t + | H |}, \end{array}$ with $\begin{matrix} H = - {[\int_{B_{2} r} (ϕ^{'} + a (x) \frac{ϕ}{2}) φ ϕ d x]}_{0}^{T} . \end{matrix}$

Note that, $\begin{array}{l} (4.29) & \int_{0}^{T} \int_{2 r} | ϕ \nabla φ \cdot \nabla ϕ | d x d t ⩽ ε \int_{0}^{T} \int_{B_{2 r}} φ | \nabla ϕ |^{2} d x d t + \frac{1}{4 ε} {‖ \frac{| \nabla φ |^{2}}{φ} ‖}_{\infty} \int_{0}^{T} \int_{B_{2 r}} | ϕ |^{2} d x d t . \end{array}$ Combining (4.28) and (4.29) and taking $ε > 0$ small enough we obtain $\begin{array}{l} \int_{0}^{T} \int_{B_{r} ∖ B_{r^{'}}} φ [| \nabla ϕ |^{2} + | ϕ |^{2}] d x d t & ⩽ C {\int_{0}^{T} \int_{R^{n}} a (x) {| ϕ^{'} |}^{2} d x d t \\ (4.30) & + \int_{0}^{T} \int_{R^{n}} b (x) | ψ |^{4} d x d t + \int_{0}^{T} \int_{B_{2 r}} | ϕ |^{2} d x d t + | H |} . \end{array}$

So from (4.23), (4.25) and (4.30) we deduce $\begin{array}{l} \frac{1}{2} \int_{0}^{T} \int_{B_{r}} (| \nabla ϕ |^{2} + | ϕ |^{2} + {| ϕ^{'} |}^{2}) d x d t & ⩽ C {\int_{0}^{T} \int_{R^{n}} a (x) {| ϕ^{'} |}^{2} d x d t \\ + \int_{0}^{T} \int_{R^{n}} b (x) | ψ |^{2} d x d t + \int_{0}^{T} \int_{R^{n}} b (x) | ψ |^{4} d x d t \\ (4.31) & + ‖ ϕ ‖_{L^{2} (B_{2 r \times (0, T)})} + | D | + | G | + | H |} . \end{array}$ Note that $\begin{matrix} (4.32) & | D | + | G | + | H | ⩽ C (E (T) + E (0)) . \end{matrix}$ So, adding the term $\frac{1}{2} \int_{0}^{T} \int_{B_{2 r}} | ψ |^{2} d x d t$ in both sides of (4.31) and combining with (4.32) and (4.6), we obtain (4.13). □

Lemma 4.5.

For all $T > 0$ , there exists a constant $C > 0$ such that $\begin{matrix} (4.33) & \int_{0}^{T} E (T) ⩽ C (E (0) + ‖ ϕ ‖_{L^{2} (B_{4 R} \times (0, T))}^{2} + ‖ ψ ‖_{L^{2} (B_{4 R} \times (0, T))}^{2}) . \end{matrix}$

Proof.

Rewriting (4.7) and (4.13) for $r = 2 R$ we deduce the lemma. □

Lemma 4.6.

Let $T_{0} > 0$ be a sufficiently large constant. Then, for every $T > T_{0}$ there exists a constant $C (T_{0}) > 0$ such that $\begin{array}{l} ‖ ϕ ‖_{L^{2} (B_{4 R} \times (0, T))} + ‖ ψ ‖_{L^{2} (B_{4 R} \times (0, T))} \\ (4.34) & ⩽ C (T_{0}) \int_{0}^{T} \int_{R^{n}} a (x) {| ϕ^{'} |}^{2} d x d t + \int_{0}^{T} \int_{R^{n}} b (x) (| ψ |^{4} + | ψ |^{2}) d x d t . \end{array}$

Proof.

In the order to prove estimate (4.34) we argue by contradiction. Let us suppose that it does not hold. Then, there exists a time $T > 0$ and a sequence of initial data ${ψ_{k} (0), ϕ_{k} (0), ϕ_{k}^{'} (0)}$ such that the corresponding sequence ${ψ_{k}, ϕ_{k}}$ , of solutions of (1.1) with $E_{k} (0)$ uniformly bounded to k that satisfy $\begin{array}{rcl} (4.35) & lim_{k \to \infty} \frac{‖ ϕ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))} + ‖ ψ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}}{\int_{0}^{T} \int_{R^{n}} a (x) | ϕ_{k}^{'} |^{2} d x d t + \int_{0}^{T} \int_{R^{n}} b (x) (| ψ_{k} |^{4} + | ψ_{k} |^{2}) d x d t} = + \infty . \end{array}$

Since $E_{k} (t) ⩽ E_{k} (0) ⩽ L$ , where L is a positive constant, we get a subsequence, still denoted by ${ψ_{k}, ϕ_{k}}$ satisfying $\begin{array}{l} (4.36) & ψ_{k}^{'} ⇀ ψ weakly* in L^{\infty} (0, T; L^{2} (R^{n})), \\ (4.37) & ϕ_{k} ⇀ ϕ weakly* in L^{\infty} (0, T; H_{0}^{1} (R^{n})), \\ (4.38) & ϕ_{k}^{'} ⇀ ϕ^{'} weakly* in L^{\infty} (0, T; L^{2} (R^{n})) . \end{array}$

From (4.37) we have to $ϕ_{k} ⇀ ϕ$ weakly* in $L^{\infty} (0, T; H_{0}^{1} (B_{4 R}))$ and, from ${ϕ_{k}}$ is limited in $L^{\infty} (0, T; H_{0}^{1} (B_{4 R}))$ and the fact that $H_{0}^{1} (B_{4 R})$ have compact immersion in $L^{2} (B_{4 R})$ follows from the Aubin–Lions Theorem which, $\begin{array}{l} (4.39) & ϕ_{k} \to ϕ in L^{2} (0, T; L^{2} (B_{4 R})), \\ (4.40) & ϕ_{k} \to ϕ a.e in B_{4 R} \times (0, T) . \end{array}$

Now from (4.35), (4.36), and (4.37) we deduce that, $\begin{array}{l} (4.41) & lim_{k \to \infty} \int_{0}^{T} \int_{R^{n}} a (x) {| ϕ_{k}^{'} |}^{2} d x d t = 0, \\ (4.42) & lim_{k \to \infty} \int_{0}^{T} \int_{R^{n}} b (x) | ψ |^{4} d x d t = 0, \\ (4.43) & lim_{k \to \infty} \int_{0}^{T} \int_{R^{n}} b (x) | ψ |^{2} d x d t = 0 . \end{array}$

And we also have to, $\begin{array}{rcl} (4.44) & lim_{k \to \infty} \int_{0}^{T} \int_{R^{n} ∖ B_{R}} | ψ_{k} |^{4} d x d t = 0 . \end{array}$

From now on we will focus our attention on the “coupled” wave equation $\begin{array}{rcl} (4.45) & ϕ_{k}^{″} - Δ ϕ_{k} + ϕ_{k} + a (x) ϕ_{k}^{'} = | ψ_{k} |^{2} χ_{Ω_{R}} in R^{n} \times (0, T) . \end{array}$

Let’s divide the prove into two cases:

(a) The case where $ϕ \neq 0$ .

Passing the limit when $k \to \infty$ in (4.45) and considering the above convergences, we have $\begin{matrix} \{\begin{matrix} ϕ^{″} - Δ ϕ + ϕ = 0 in L^{2} (0, T; H^{- 1} (R^{n})), \\ ϕ^{'} = 0 a.e. in R^{n} ∖ B_{R} \times (0, T), \end{matrix} \end{matrix}$ and for $ϕ^{'} = v$ , we get in the distributional sense $\begin{matrix} (4.46) & \{\begin{matrix} v^{″} - Δ v = 0 in D^{'} (R^{n} \times (0, T)), \\ v = 0 a.e in R^{n} ∖ B_{R} \times (0, T) . \end{matrix} \end{matrix}$

Therefore, it follows the Holmgren’s Theorem which $v \equiv 0$ , that is, $ϕ^{'} \equiv 0$ . Returning to (4.75) we get, $\begin{matrix} (4.47) & - Δ ϕ + ϕ = 0 in L^{2} (R^{n}) . \end{matrix}$

From this we conclude that $Δ ϕ \in L^{2} (R^{n})$ . Then $ϕ \in H^{2} (R^{n})$ and the following identity is valid, $\begin{matrix} (4.48) & - \int_{R^{n}} Δ ϕ ϕ d x + \int_{R^{n}} | ϕ |^{2} d x = 0 . \end{matrix}$

From Green’s Theorem it follows that, $\begin{matrix} (4.49) & \int_{R^{n}} | \nabla ϕ |^{2} d x + \int_{R^{n}} | ϕ |^{2} d x = 0, \end{matrix}$ implying that $ϕ = 0$ , which is a contradiction.

(b) The case where $ϕ \equiv 0$ .

Defining, $\begin{array}{l} (4.50) & c_{k} : = {[\int_{0}^{T} \int_{B_{4 R}} | ϕ_{k} |^{2} d x d t + \int_{0}^{T} \int_{B_{4 R}} | ψ_{k} |^{2} d x d t]}^{\frac{1}{2}}, \\ (4.51) & {\hat{ϕ}}_{k} = \frac{1}{c_{k}} ϕ_{k}, {\hat{ψ}}_{k} = \frac{1}{c_{k}} ψ_{k}, \end{array}$ we have $\begin{array}{rcl} (4.52) & \int_{0}^{T} \int_{B_{4 R}} | \hat{ϕ_{k}} |^{2} + | \hat{ψ} |^{2} d x d t = 1 . \end{array}$

In addition, we have $\begin{array}{rcl} {\hat{E}}_{k} (t) & = & \frac{1}{2} [\int_{R^{n}} | {\hat{ψ}}_{k} |^{2} d x + \int_{R^{n}} {| {\hat{ϕ}}_{k}^{'} |}^{2} d x + \int_{R^{n}} | \nabla {\hat{ϕ}}_{k} |^{2} d x + \int_{R^{n}} | {\hat{ϕ}}_{k} |^{2} d x] \\ = & \frac{1}{2 c_{k}^{2}} [\int_{R^{n}} | ψ_{k} |^{2} d x + \int_{R^{n}} {| ϕ_{k}^{'} |}^{2} d x + \int_{R^{n}} | \nabla ϕ_{k} |^{2} d x + \int_{R^{n}} | ϕ_{k} |^{2} d x], \end{array}$ that is, $\begin{matrix} (4.53) & {\hat{E}}_{k} (t) = \frac{E_{k} (t)}{c_{k}^{2}} . \end{matrix}$

On the other hand, integrating (4.4) from 0 to T we have $\begin{array}{rcl} E_{k} (T) & = & E_{k} (0) - α \int_{0}^{T} \int_{R^{n}} b (x) (| ψ_{k} |^{4} + | ψ_{k} |^{2}) d x \\ (4.54) & - \int_{0}^{T} \int_{R^{n}} a (x) {| ϕ_{k}^{'} |}^{2} d x + \int_{0}^{T} \int_{R^{n} ∖ B_{R}} | ψ_{k} |^{2} ϕ_{k}^{'} d x . \end{array}$

From the fact that $E_{k} (t) ⩾ E_{k} (T)$ for every $t \in [0, T]$ and considering (4.54), we have $\begin{array}{rcl} \int_{0}^{T} E_{k} (t) d t & ⩾ & T E_{k} (T) \\ = & T E_{k} (0) - α T \int_{0}^{T} \int_{R^{n}} b (x) (| ψ_{k} |^{4} + | ψ_{k} |^{2}) d x \\ (4.55) & - T \int_{0}^{T} \int_{R^{n}} a (x) {| ϕ_{k}^{'} |}^{2} d x + T \int_{0}^{T} \int_{R^{n} ∖ B} | ψ_{k} |^{2} ϕ_{k}^{'} d x . \end{array}$

From (4.33) and (4.55) we have $\begin{array}{l} T E_{k} (0) - α T \int_{0}^{T} \int_{R^{n}} b (x) (| ψ_{k} |^{4} + | ψ_{k} |^{2}) d x - T \int_{0}^{T} \int_{R^{n}} a (x) {| ϕ_{k}^{'} |}^{2} d x \\ + T \int_{0}^{T} \int_{R^{n} ∖ B} | ψ_{k} |^{2} ϕ_{k}^{'} d x ⩽ C E_{k} (0) + C ‖ ϕ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2} + C ‖ ψ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2}, \end{array}$ and rewriting, $\begin{matrix} T E_{k} (0) & ⩽ α T \int_{0}^{T} \int_{R^{n}} b (x) (| ψ_{k} |^{4} + | ψ_{k} |^{2}) d x + T \int_{0}^{T} \int_{R^{n}} a (x) {| ϕ_{k}^{'} |}^{2} d x d t \\ - T \int_{0}^{T} \int_{R^{n} ∖ B} | ψ_{k} |^{2} ϕ_{k}^{'} d x d t + C E_{k} (0) + C ‖ ϕ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2} + C ‖ ψ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2} \\ ⩽ α T \int_{0}^{T} \int_{R^{n}} b (x) (| ψ_{k} |^{4} + | ψ_{k} |^{2}) d x d t + T \int_{0}^{T} \int_{R^{n}} a (x) {| ϕ_{k}^{'} |}^{2} d x d t \\ + \frac{T}{2 a_{0} b_{0}} \int_{0}^{T} \int_{R^{n}} b (x) | ψ_{k} |^{4} d x d t + \frac{T}{2} \int_{0}^{T} \int_{R^{n}} a (x) {| ϕ_{k}^{'} |}^{2} d x d t \\ + C E_{k} (0) + C ‖ ϕ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2} + C ‖ ψ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2} . \end{matrix}$

Therefore $\begin{array}{rcl} T E_{k} (0) & ⩽ & (α + \frac{1}{2 a_{0} b_{0}} + 2 α + \frac{3}{2}) T [\int_{0}^{T} \int_{R^{n}} b (x) (| ψ_{k} |^{4} + | ψ_{k} |^{2}) d x d t \\ + \int_{0}^{T} \int_{R^{n}} a (x) {| ϕ_{k}^{'} |}^{2} d x d t] + C E_{k} (0) \\ (4.56) & + C ‖ ϕ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2} + C ‖ ψ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2} . \end{array}$

From the last inequality comes that for a T sufficiently large, we have $\begin{array}{rcl} (4.57) & E_{k} (0) & ⩽ & C (T, α, a_{0}, b_{0}) {\int_{0}^{T} \int_{R^{n}} b (x) (| ψ_{k} |^{4} + | ψ_{k} |^{2}) d x d t \\ + \int_{0}^{T} \int_{R^{n}} a (x) {| ϕ_{k}^{'} |}^{2} d x d t + ‖ ϕ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2} + ‖ ψ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2}} . \end{array}$

Recalling that $E_{k} (t) ⩽ E_{k} (0)$ for all $t \in [0, T]$ , applying inequality (4.57) and dividing both sides by $‖ ϕ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2} + ‖ ψ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2}$ , we get $\begin{array}{l} \frac{E_{k} (t)}{\int_{0}^{T} \int_{B_{4 R}} | ϕ_{k} |^{2} d x d t + \int_{0}^{T} \int_{B_{4 R}} | ψ_{k} |^{2} d x d t} \\ (4.58) & ⩽ C (T, α, a_{0}, b_{0}) {\frac{\int_{0}^{T} \int_{R^{n}} b (x) (| ψ_{k} |^{4} + | ψ_{k} |^{2}) d x d t + \int_{0}^{T} \int_{R^{n}} a (x) | ϕ_{k}^{'} |^{2} d x d t}{‖ ϕ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2} + ‖ ψ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2}} + 1}, \end{array}$ and from (4.35) we have $\begin{matrix} (4.59) & lim_{k \to \infty} \frac{\int_{0}^{T} \int_{R^{n}} b (x) (| ψ_{k} |^{4} + | ψ_{k} |^{2}) d x d t + \int_{0}^{T} \int_{R^{n}} a (x) | ϕ_{k}^{'} |^{2} d x d t}{‖ ϕ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2} + ‖ ψ_{k} ‖_{L^{2} (B_{4 R} \times (0, T))}^{2}} = 0 . \end{matrix}$ So from (4.58) there is $M > 0$ such that, $\begin{matrix} (4.60) & \frac{E_{k} (t)}{c_{k}^{2}} ⩽ C (T, a_{0}, b_{0}, α) (M + 1), \forall t \in [0, T] and \forall k \in N . \end{matrix}$

Consequently, from (4.53) and (4.60) it follows that $\begin{matrix} (4.61) & {\hat{E}}_{k} (t) ⩽ C (T, α, a_{0}, b_{0}) (M + 1), \forall t \in [0, T] and \forall k \in N . \end{matrix}$

Thus, in particular, from (4.59) we deduce $\begin{array}{l} (4.62) & lim_{k \to \infty} \int_{0}^{T} \int_{R^{n}} a (x) {| {\hat{ϕ}}_{k}^{'} |}^{2} d x d t = 0, \\ (4.63) & lim_{k \to \infty} \frac{\int_{0}^{T} \int_{R^{n}} b (x) | {\hat{ψ}}_{k} |^{4} d x d t}{c_{k}^{2}} = 0, \end{array}$ and $\begin{matrix} (4.64) & lim_{k \to \infty} \int_{0}^{T} \int_{R^{n}} b (x) | {\hat{ψ}}_{k} |^{2} d x d t = 0, \end{matrix}$ and from (4.61), for a substring of ${{\hat{ψ}}_{k}, {\hat{ϕ}}_{k}}$ , which we will still denote in the same way such that $\begin{array}{l} (4.65) & {\hat{ψ}}_{k} ⇀ \hat{ψ} weakly* in L^{\infty} (0, T; L^{2} (R^{n})), \\ (4.66) & {\hat{ϕ}}_{k} ⇀ \hat{ϕ} weakly* in L^{\infty} (0, T; H_{0}^{1} (R^{n})), \\ (4.67) & {\hat{ϕ}}_{k}^{'} ⇀ {\hat{ϕ}}^{'} weakly* in L^{\infty} (0, T; L^{2} (R^{n})) . \end{array}$

Also, ${\hat{ϕ}}_{k}$ satisfies the equation $\begin{matrix} (4.68) & {\hat{ϕ}}_{k}^{″} - Δ {\hat{ϕ}}_{k} + a (x) {\hat{ϕ}}_{k}^{'} = \frac{| ψ_{k} |^{2}}{c_{k}} χ_{Ω_{R}} em R^{n} \times (0, T) . \end{matrix}$

Again from the fact that ${{\hat{ϕ}}_{k}}$ are bounded in $L^{2} (0, T; H_{0}^{1} (B_{4 R}))$ and the immersion of $L^{2} (0, T; H^{1} (B_{4 R}))$ in $L^{2} (0, T; L^{2} (B_{4 R}))$ is compact, follows the Aubin–Lions Theorem which, $\begin{matrix} (4.69) & {\hat{ϕ}}_{k} \to \hat{ϕ} in L^{2} (0, T; L^{2} (B_{4 R})), \end{matrix}$ hence, $\begin{matrix} (4.70) & {\hat{ϕ}}_{k} \to \hat{ϕ} in B_{4 R} \times (0, T), a.e. \end{matrix}$

Note that in (4.62) is equivalent to saying that $\begin{matrix} \int_{0}^{T} \int_{R^{n}} {| a {(x)}^{\frac{1}{2}} {\hat{ϕ}}_{k}^{'} |}^{2} d x d t \to 0 . \end{matrix}$

Therefore, $\begin{matrix} (4.71) & a {(x)}^{\frac{1}{2}} {\hat{ϕ}}_{k}^{'} \to 0 in L^{2} (0, T; L^{2} (R^{n})) . \end{matrix}$

On the other hand, from (4.67) we have to ${\hat{ϕ}}_{k}^{'} ⇀ {\hat{ϕ}}^{'}$ in $L^{2} (0, T; L^{2} (R^{n}))$ , which tells us that $\begin{matrix} \int_{0}^{T} \int_{R^{n}} {| a {(x)}^{\frac{1}{2}} {\hat{ϕ_{k}}}^{'} |}^{2} u d x d t \to \int_{0}^{T} \int_{R^{n}} {| a {(x)}^{\frac{1}{2}} {\hat{ϕ}}^{'} |}^{2} u d x d t \end{matrix}$ for all $u \in L^{2} (0, T; L^{2} (R^{n}))$ . Then, $\begin{matrix} (4.72) & a {(x)}^{\frac{1}{2}} {\hat{ϕ}}_{k}^{'} ⇀ a {(x)}^{\frac{1}{2}} {\hat{ϕ}}^{'} . \end{matrix}$

From (4.71), (4.72) and the uniqueness of the weak limit, we get $a {(x)}^{\frac{1}{2}} {\hat{ϕ}}^{'} = 0$ . Multiplying the equality by $a^{\frac{1}{2}}$ , will have to $\begin{matrix} (4.73) & a (x) {\hat{ϕ}}^{'} = 0 in L^{2} (0, T; L^{2} (R^{n})), \end{matrix}$ and consequently $a (x) {\hat{ϕ}}^{'} = 0$ in $B_{4 R} \times (0, T)$ , a.e.

Therefore, $\begin{matrix} (4.74) & {\hat{ϕ}}^{'} = 0 a.e. in R^{n} ∖ B_{R} \times (0, T) . \end{matrix}$

From (4.65), (4.66), and (4.44), it follows that $\begin{matrix} lim_{k \to + \infty} \int_{0}^{T} \int_{R^{n} ∖ B_{R}} \frac{| ψ_{k} |^{4}}{c_{k}^{2}} d x d t = 0 . \end{matrix}$

Passing the limit when $k \to + \infty$ and taking the above convergence into account, we have $\begin{matrix} (4.75) & \{\begin{matrix} {\hat{ϕ}}^{″} - Δ \hat{ϕ} + \hat{ϕ} = 0 a.e. in D^{'} (B_{4 R} \times (0, T), \\ {\hat{ϕ}}^{'} = 0 a.e. in (B_{4 R} ∖ B_{R}) \times (0, T), \end{matrix} \end{matrix}$ and for $v = {\hat{ϕ}}^{'}$ , we get, in the distributional sense $\begin{matrix} (4.76) & \{\begin{matrix} v^{″} - Δ v = 0 in D^{'} (B_{4 R} \times (0, T)), \\ v = 0 a.e. in (B_{4 R} ∖ B_{R}) \times (0, T) . \end{matrix} \end{matrix}$

Applying Holmgren’s Theorem again, we get $v \equiv {\hat{ϕ}}^{'} \equiv 0$ . Returning to (4.75) we get for almost every $t \in (0, T)$ which $\begin{matrix} - Δ \hat{ϕ} + ϕ = 0 a.e. in D^{'} (B_{4 R} \times (0, T)) . \end{matrix}$

Composing the above equation with $\hat{ϕ}$ , integrating into $B_{4 R} \times (0, T)$ and using Green’s Theorem, we get $\begin{matrix} \int_{0}^{T} \int_{B_{4 R}} | \nabla \hat{ϕ} |^{2} d x d t + \int_{0}^{T} \int_{B_{4 R}} | \hat{ϕ} |^{2} d x d t = 0, \end{matrix}$ and so $\hat{ϕ} = 0$ a.e. in $B_{4 R}$ .

In addition, we also have that ${\hat{ψ}}_{k}$ satisfies the equation $\begin{matrix} (4.77) & \{\begin{matrix} i {\hat{ψ}}_{k}^{'} + Δ {\hat{ψ}}_{k} + i α b (x) | ψ_{k} |^{2} {\hat{ψ}}_{k} + i α b (x) {\hat{ψ}}_{k} = {\hat{ϕ}}_{k} {\hat{ψ}}_{k} χ_{Ω_{R}} in R^{n} \times (0, T), \\ {\hat{ψ}}_{k} (0) = {\hat{ψ}}_{k_{0}} in R^{n} \times (0, T) . \end{matrix} \end{matrix}$

Now we can use the smoothing effect of Constantin and Saut enunciated in Theorem 2.3, for that, define $\begin{matrix} (4.78) & F (ψ_{k}) = - α b (x) | ψ_{k} |^{2} {\hat{ψ}}_{k} - α b (x) {\hat{ψ}}_{k} - i {\hat{ϕ}}_{k} ψ_{k} \end{matrix}$ that way we have to $\begin{matrix} (4.79) & F (ψ_{k}) \to 0 in L^{1} (0, T; L^{2} (R^{n})) . \end{matrix}$

Thus, for each $χ \in C_{0}^{\infty} (R^{n + 1})$ as given in Theorem 2.3 supported on $[0, T] \times R^{n}$ , we have $\begin{array}{rcl} {(\int_{0}^{T} \int_{R^{n}} χ^{2} {| {(I - Δ)}^{\frac{1}{4}} {\hat{ψ}}_{k} |}^{2} d x d t)}^{\frac{1}{2}} & ⩽ & C_{χ} ({‖ {\hat{ψ}}_{k} (0) ‖}_{L^{2} (R^{n}))} + ‖ F ‖_{L^{1} (0, T; L^{2} (R^{n}))}) \\ ⩽ & {\tilde{C}}_{χ} . \end{array}$

We will now describe the construction of the function χ:

First, be $ε > 0$ and $χ_{0} \in C_{0}^{\infty} (R)$ such that $0 ⩽ χ_{0} ⩽ 1$ , where $\begin{matrix} (4.80) & χ_{0} (l) = \{\begin{matrix} 1 in [ε, T - ε], \\ 0 in] - \infty, \frac{ε}{2}] \cup [T - \frac{ε}{2}, + \infty [. \end{matrix} \end{matrix}$ Note that $sup χ_{0} \subset [\frac{ε}{2}, T - \frac{ε}{2}]$ .

Similarly, be $χ_{j} \in C_{0}^{\infty} (R)$ such that $0 ⩽ χ_{j} ⩽ 1$ , where $\begin{matrix} χ_{j} (l) = \{\begin{matrix} 1 in [ε, y_{j} - ε] : = K_{j}, \\ 0 in] - \infty, \frac{ε}{2}] \cup [y_{j} - \frac{ε}{2}, + \infty [. \end{matrix} \end{matrix}$

Note that $sup χ_{j} \subset [\frac{ε}{2}, y_{j} - \frac{ε}{2}]$ , with $j = 1, \dots, n$ .

Now, be $χ : R^{n + 1} \to R$ such that $\begin{matrix} χ (t, x) = χ_{0} (t) χ_{1} (x_{1}) \dots χ_{n} (x_{n}), \end{matrix}$ where $x = (x_{1}, x_{2}, \dots, x_{n})$ and $χ_{j} \in C_{0}^{\infty} (R), j = 0, 1, \dots, n$ defined as above. So we have $χ \in C_{0}^{\infty} (R^{n + 1})$ and $\begin{matrix} (4.81) & χ (t, x) = 1 in [ε, T - ε] \times K_{1} \times K_{2} \times \dots \times K_{n} . \end{matrix}$

Notice that, $\begin{array}{rcl} ‖ {\hat{ψ}}_{k} ‖_{L^{2} (0, T; H^{\frac{1}{2}} (R^{n}))} & = & \int_{0}^{T} ‖ {\hat{ψ}}_{k} ‖_{H^{\frac{1}{2}} (R^{n})} d t \\ = & \int_{0}^{T} \int_{R^{n}} {| {(I - Δ)}^{\frac{1}{4}} {\hat{ψ}}_{k} |}^{2} d x d t \\ = & \int_{0}^{ε} \int_{R^{n}} {| {(I - Δ)}^{\frac{1}{4}} {\hat{ψ}}_{k} |}^{2} d x d t \\ + \int_{ε}^{T - ε} [\int_{R ∖ K_{1} \times \dots \times R ∖ K_{n}} {| {(I - Δ)}^{\frac{1}{4}} {\hat{ψ}}_{k} |}^{2} d x \\ + \int_{K_{1} \times \dots \times K_{n}} χ^{2} {| {(I - Δ)}^{\frac{1}{4}} {\hat{ψ}}_{k} |}^{2} d x] d t \\ + \int_{T - ε}^{T} \int_{R^{n}} {| {(I - Δ)}^{\frac{1}{4}} {\hat{ψ}}_{k} |}^{2} d x d t \\ ⩽ & ε max_{t \in [0, T]} ‖ {\hat{ψ}}_{k} ‖_{H^{\frac{1}{2}} (R^{n})} \\ + \int_{ε}^{T - ε} \int_{R ∖ K_{1} \times \dots \times R ∖ K_{n}} {| {(I - Δ)}^{\frac{1}{4}} {\hat{ψ}}_{k} |}^{2} d x d t \\ + \int_{ε}^{T - ε} \int_{K_{1} \times \dots \times K_{n}} χ^{2} {| {(I - Δ)}^{\frac{1}{4}} {\hat{ψ}}_{k} |}^{2} d x d t \\ + ε max_{t \in [0, T]} ‖ {\hat{ψ}}_{k} ‖_{H^{\frac{1}{2}} (R^{n})} \\ ⩽ & 2 ε max_{t \in [0, T]} ‖ {\hat{ψ}}_{k} ‖_{H^{\frac{1}{2}} (R^{n})} + \int_{ε}^{T - ε} \int_{R^{n}} {| {(I - Δ)}^{\frac{1}{4}} {\hat{ψ}}_{k} |}^{2} d x d t \\ = & + \int_{ε}^{T - ε} \int_{R^{n}} χ^{2} {| {(I - Δ)}^{\frac{1}{4}} {\hat{ψ}}_{k} |}^{2} d x d t \\ ⩽ & 2 ε max_{t \in [0, T]} ‖ {\hat{ψ}}_{k} ‖_{H^{\frac{1}{2}} (R^{n})} + (T - 2 ε) max_{t \in [0, T]} ‖ {\hat{ψ}}_{k} ‖_{H^{\frac{1}{2}} (R^{n})} \\ + C_{χ}^{2} ({‖ {\hat{ψ}}_{k} (0) ‖}_{L^{2} (R^{n}))} + ‖ F ‖_{L^{1} (0, T; L^{2} (R^{n}))}) \\ ⩽ & T max_{t \in [0, T]} ‖ {\hat{ψ}}_{k} ‖_{H^{\frac{1}{2}} (R^{n})} + C_{χ}^{2} {(L + C_{1})}^{2}, \end{array}$ because $‖ {\hat{ψ}}_{k} ‖_{H^{\frac{1}{2}} (R^{n})} < L$ .

Therefore, $\begin{matrix} ‖ {\hat{ψ}}_{k} ‖_{L^{2} (0, T; H^{\frac{1}{2}} (R^{n}))} ⩽ \tilde{C}, \end{matrix}$ allowing us to say that $\begin{matrix} (4.82) & {{\hat{ψ}}_{k}} is bounded in L^{2} (0, T; H^{\frac{1}{2}} (R^{n})), \end{matrix}$ in particular, $\begin{matrix} (4.83) & {{\hat{ψ}}_{k}} is bounded in L^{2} (0, T; H^{\frac{1}{2}} (B_{4 R})) . \end{matrix}$

From (1.1), we have to for almost every $t \in (0, T)$ $\begin{matrix} {\hat{ψ}}_{k, t} = i Δ {\hat{ψ}}_{k} - i α b (x) | ψ_{k} |^{2} {\hat{ψ}}_{k} - i α b (x) {\hat{ψ}}_{k} - i {\hat{ϕ}}_{k} ψ_{k} χ_{Ω} \in H^{- \frac{1}{2}} (R^{n}) . \end{matrix}$

Consequently, $\begin{matrix} (4.84) & {{\hat{ψ}}_{k, t}} is bounded in L^{2} (0, T; H^{- \frac{1}{2}} (R^{n})), \end{matrix}$ in particular, $\begin{matrix} (4.85) & {{\hat{ψ}}_{k, t}} is bounded in L^{2} (0, T; H^{- \frac{1}{2}} (B_{4 R})) . \end{matrix}$

From (4.83), (4.85) and the imersion $H^{\frac{1}{2}} (B_{4 R}) \overset{c}{↪} L^{2} (B_{4 R}) ↪ H^{- \frac{1}{2}} (R^{n})$ , by Aubin–Lions theorem, we have $\begin{matrix} (4.86) & {\hat{ψ}}_{k} \to \hat{ψ} in L^{2} (0, T; L^{2} (B_{4 R})) . \end{matrix}$

Also, from (4.64) and (4.86), we have $\begin{matrix} (4.87) & \hat{ψ} \to \tilde{ψ} in L^{2} (0, T; L^{2} (B_{4 R})), \end{matrix}$ where $\begin{matrix} (4.88) & \tilde{ψ} = \{\begin{matrix} \hat{ψ} a.e in B_{R}, \\ 0 a.e in B_{4 R} ∖ B_{R} . \end{matrix} \end{matrix}$

Note that if $\hat{ψ} = 0$ then $\tilde{ψ} = 0$ , so from (4.69), (4.87), (4.52) and noting that $\hat{ϕ} = 0$ in $B_{4 R}$ we have a contradiction.

On the other hand, if $\hat{ψ} \neq 0$ , then passing the limit in (4.77), we have to (4.87), (4.63) and (4.64), which $\begin{matrix} \{\begin{matrix} i {\hat{ψ}}^{'} + Δ \hat{ψ} + ϕ = 0 a.e. in D^{'} (B_{4 R} \times (0, T), \\ \hat{ψ} = 0 a.e. in (B_{4 R} ∖ B_{R}) \times (0, T) . \end{matrix} \end{matrix}$

Applying Holmgren’s Theorem, we have to $\hat{ψ} = 0$ in $B_{4 R}$ . So, from (4.69), (4.87), and (4.52) and noting that $\hat{ϕ} = 0$ em $B_{4 R}$ we have a contradiction. The proof of the lemma is now complete. □

Combining the estimates (4.33) and (4.34) we obtain (2.2) for $T > T_{0}$ and some positive constant $C = C (T)$ . The proof of Theorem 2.2 is complete.

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