Well-Posedness and Scattering for Wave Equations on Hyperbolic Spaces With Singular Data

Abstract

We consider wave equations with critical shift on the real hyperbolic space $H^{n}$ ( $n \geq 2$ ) in the framework of weak- $L^{p}$ spaces. First, we establish dispersive estimates on Lorentz spaces in the context of $H^{n}$ . Then, employing those estimates, we prove global well-posedness of solutions and an exponential asymptotic stability property. Moreover, we develop a scattering theory and construct wave operators in such singular framework.

Keywords

wave equations Klein-Gordon equations well-posedness scattering hyperbolic spaces

1. Introduction

In the present article, we are concerned with the wave equation on the real hyperbolic space $H^{n}$ $(n \geq 2)$ with critical shift:

{\begin{cases} \partial_{t}^{2} u (t, x) - Δ_{x} u (t, x) + c u (t, x) = F (u (t, x)), \\ u (0, x) = u_{0} (x), \partial_{t} u (t, x) = u_{1} (x), \end{cases}

(1.1)

where

Δ_{x}

stands for the Laplace-Beltrami operator associated with the hyperbolic metric, the shift

c = - (n - 1)^{2} / 4

, and the nonlinearity

F (u)

satisfies

F (0) = 0 and | F (u) - F (v) | \leq C (| u |^{b - 1} + | v |^{b - 1}) | u - v |, for b > 1.

(1.2)

Equation (1.1) is referred to as the shifted wave equation. In the case

c = 0

, equation (1.1) is called non-shifted wave equation (for more details see Section 2).

The wave and Klein-Gordon equations have been studied by several authors. In the sequel, we review some important results of the literature. Without making a complete list, we begin by recalling briefly results for equations on the Euclidean space $R^{n}$ . The existence of global solutions was analyzed by Belchev et al. (1999), Georgiev et al. (1997), Ginibre and Velo (1985), Zhou (1995), among others. The nonexistence of global solutions was studied by Sideris (1984). Time decays of solutions for wave equations were proved by Ginibre and Velo (1987a, 1987b), Pecher (1982) and Strauss (1968). In particular, Pecher (1982, 1984) obtained results on time decays and established the nonlinear small data scattering for wave and Klein-Gordon equations in $R^{3}$ . The scattering theory was also studied by using the time decays in the works (Ginibre & Velo, 1989; Lindblad & Sogge, 1995; Strauss, 1981). After that, Hidano (1998, 2000, 2001) obtained the small data scattering and blow-up theory for nonlinear wave equation on $R^{n}$ with $n = 3, 4$ by using the integral representation formula of solutions. In addition, the scattering for the two-dimensional energy-critical wave equation and Klein-Gordon equation with exponential-type nonlinearity were provided by Ibrahim et al. (2006, 2009). Concerning the nonlinear wave equations, Strichartz estimates were obtained by Georgiev et al. (1997), Ginibre and Velo (1989), and Keel and Tao (1998). Moreover, we quote the results on blow-up (Glassey, 1981) and the life-span of solutions (Li & Yu, 1991; Li & Zhou, 1995).

The well-posedness and scattering theory for wave equations are obtained in Sobolev spaces by first proving dispersive and Strichartz type estimates in suitable norms. By following this general spirit, Metcalfe and Taylor considered wave equations on the three-dimensional hyperbolic space $H^{3}$ by Metcalfe and Taylor (2011, 2012). They proved dispersive and Strichartz estimates for the associated linear problem and then obtained the global well-posedness of solutions in $C^{\infty} (R \times H^{3})$ with smooth compactly supported initial-data $u_{0}, u_{1} \in C_{0}^{\infty} (H^{3})$ . For general dimension $n \geq 2$ , Anker et al. (2012) and Anker and Pierfelice (2014) have succeeded to establish dispersive and Strichartz estimates and obtained the local well-posedness in $C ([- T, T]; H^{s, τ} (H^{n})) \cap C^{1} ([- T, T]; H^{s, τ - 1} (H^{n}))$ by Anker et al. (2012) and the global well-posedness in $C (R; H^{σ} (H^{n})) \cap C^{1} (R; H^{σ - 1} (H^{n})) \cap L^{p} (R; L^{q} (H^{n}))$ by Anker and Pierfelice (2014). By employing dispersive and Strichartz estimates obtained by Anker et al. and French (2012) studied the scattering for (1.1) with the scattering data space $H^{σ, 2} \oplus H^{σ - 1, 2}$ . By the same process, Zhang (2015) established the global well-posedness and scattering theory for wave equations on nontrapping asymptotically conic manifolds. Tataru (2001) improved the dispersive estimates for the wave operator in $H^{n} (n \geq 2)$ and used these estimates to obtain the global well-posedness of the wave equation ( $c = 0$ ) in $R^{n}$ with initial data $(u_{0}, u_{1}) \in H^{1} \times L^{2}$ . There are some related works about the Strauss conjecture and blow-up for wave equations on hyperbolic spaces by Sire et al. (2019) and Wang and Zhang (2025) and on asymptotically flat space-times by Lindblad et al. (2014) and Metcalfe and Wang (2017).

On the other hand, the existence and scattering theory for the wave equations on hyperbolic spaces, noncompact manifolds, and asymptotically Euclidean manifolds have been developed by Phillips et al. (1987), Debièvre et al. (1992), and Bony and Häfner (2009), respectively, by means of the translation representation for the unperturbed system and Mourre theory, that is, the spectral method. The scattering data spaces by Debièvre et al. (1992) and Phillips et al. (1987) are Hilbert spaces as the completions of $C_{0}^{\infty} (H^{n}) \times C_{0}^{\infty} (H^{n})$ under the energy norms. For the original reference about this method, we refer the readers to the book of Lax and Phillips (1967). In addition, the geometric scattering and inverse scattering for the wave equations on some manifolds such as asymptotically Euclidean and hyperbolic manifolds have been constructed by using the radiation fields (see, for instance, Friedlander, 2001; Sá Barreto, 2005, 2008, and references therein). In related works, the scattering theory for nonlinear conformal wave equations on global hyperbolic space-time was studied by Baez et al. (1990) and results on the hyperbolic spaces have been studied by Antoci (2000) and Banica et al. (2009, 2008) for $p$ -forms and Schrödinger equations.

Another approach to scattering theory in hyperbolic spaces was recently developed by Chen and Liu (2026), who constructed a theoretical framework for time-harmonic wave scattering by extending the notions of the Sommerfeld radiation condition and far-field pattern from Euclidean spaces to the hyperbolic setting. In particular, by considering specific time-harmonic waves, one can transform the wave equations (or Schrödinger equations) into the corresponding Helmholtz systems, which can be used to describe the scattering associated with these wave (or Schrödinger) equations. Based on this observation, Chen and Liu (2026) constructed the scattering theory in hyperbolic spaces as follows: first, they provided explicit formulas for the ingoing and outgoing Green functions of the Helmholtz operator in hyperbolic spaces, and proved that they are fundamental solutions of the Helmholtz equation in this setting. Then, they established the asymptotic behaviour of the Green functions and used it to formulate the ingoing and outgoing radiation conditions, which are analogues of the Sommerfeld radiation condition in the Euclidean setting (see, e.g., Li & Liu, 2023). In addition, they proved a Rellich-type theorem in the hyperbolic setting, which guarantees that the scattered field as well as its far-field pattern are uniquely defined. Finally, they obtained the scattering theory for the Helmholtz equation involving a potential.

The purpose of this article is to establish the local and global well-posedness, asymptotic stability and scattering for equation (1.1) in the framework of weak- $L^{p}$ spaces on $H^{n}$ . Moreover, we construct wave operators in such singular setting. The idea of studying the well-posedness and scattering in weak- $L^{p}$ spaces were initially developed by Cazenave et al. (2001) for Schrödinger equations in the Euclidean space $R^{n}$ by considering mixed space-time weak- $L^{p}$ , namely $L^{p, \infty} (R \times R^{n})$ with $p = (b - 1) (N + 2) / 2$ , and employing Strichartz-type estimates. Then, still considering Schrödinger equations in $R^{n},$ but employing dispersive-type estimates, the global well-posedness and asymptotic behavior of solutions were obtained by Ferreira et al. (2009) in a framework of time polynomial weighted spaces based on the $L^{(p, \infty)} (R^{n})$ with $p = b + 1$ , extending some results obtained in the $L^{p}$ -setting by Cazenave and Weissler (1998, 2000). In this direction, we also have well-posedness and scattering results for Boussinesq equations (Ferreira, 2011) and wave equations (de Almeida & Ferreira, 2017; Liu, 2009). In these works, the authors used $L^{(p, r)}$ - $L^{(p^{'}, r)}$ -dispersive estimates on $R^{n},$ where $1 / p + 1 / p^{'} = 1$ , $1 \leq r \leq \infty,$ and $L^{(p, r)}$ stands for the so-called Lorentz space. The weak- $L^{p}$ space corresponds to the case $r = \infty .$ The global well-posedness and scattering in $L^{(p, \infty)} (R^{n})$ via that approach require the use of suitable Kato-type classes. For example, in the case of the Schrödinger equation in $R^{n}$ , it is considered the time polynomial weighted space

H_{α} = {u measurable; sup_{t \in R} | t |^{α} {‖ u (t, \cdot) ‖}_{L^{(p, \infty)}} < \infty}

as well as the initial-data class

{u_{0} \in S^{'}; sup_{t \in R} | t |^{α} ‖ S (t) u_{0} ‖_{L^{(p, \infty)}} < \infty}

, where

S (t)

is the Schrödinger operator and the power

α

depends on

p

via a suitable relation. Note that in the hyperbolic space

H^{n}

, one can obtain dispersive estimates with decay rates that are faster than those in the Euclidean space

R^{n}

. This fact was mentioned in the works by Anker et al. (2012), Anker and Pierfelice (2014), Metcalfe and Taylor (2011, 2012), and Tataru (2001).

In this article, we employ the $L^{p}$ -dispersive estimates obtained by Tataru (2001) in order to establish $L^{(p, r)}$ - $L^{(p^{'}, r)}$ -dispersive estimates in the hyperbolic space $H^{n}$ , where the decays are of exponential type for large $| t |$ and polynomial for small $| t | .$ Using these estimates, we obtain a global well-posedness result in the mixed time weighted space $E_{α, \tilde{α}}$ , namely the set of all Bochner measurable $u : (- \infty, \infty) \to L^{(b + 1, \infty)}$ such that

sup_{| t | \geq t_{0}} e^{α | t |} {‖ u (\cdot, t) ‖}_{(b + 1, \infty)} + sup_{t \in (- t_{0}, t_{0})} {| t |}^{\tilde{α}} {‖ u (\cdot, t) ‖}_{(b + 1, \infty)} < \infty,

(1.3)

where

α, \tilde{α}

are suitable positive parameters (see Section 4 and Theorem 4.1). Note that the exponential weight for large times in the norm (1.3) arises from the asymptotic behavior of the wave group on hyperbolic spaces and constants

b, α

, and

\tilde{α}

must satisfy certain conditions to ensure the global existence of solutions (see Theorem 4.1 and Remark 4.2

(i i)

for details). In fact, we do not know whether these conditions are sharp w.r.t.

b

and the time-weights, but they seem difficult to improve in view of their intrinsic relations with the properties and decay rates of the wave group in the hyperbolic space. In view of the exponential weight in (1.3), we show an exponential asymptotic stability result for the global solutions (see Theorem 4.4), as well as an exponential scattering behavior (see Theorem 4.5). Finally, we give the construction of wave operator in Theorem 4.7.

From a structural standpoint, in the hyperbolic geometry there is no scaling map valid in weak- $L^{p}$ as in the case of $R^{n}$ . In fact, from the norm expressions (3.1) and (3.3), we can see that the presence of the exponential factor induced by the hyperbolic setting breaks the homogeneity of the norm, generating a further difficulty for the problem. Consequently, any type of scaling argument must be adapted with appropriate care and should be used only as a kind of intuitive guide for the arguments, bearing in mind that there is no full scaling invariance as in the Euclidean setting. On the other hand, regarding the index $p = b + 1$ in the functional space, it arises naturally from the dispersive estimate $L^{p^{'}} - L^{p}$ together with the degree $b$ of the power-like nonlinearity $F (u)$ , recalling that $p^{'}$ is the conjugate exponent of $p$ . In turn, a scaling-type balance in the norm for small $t$ is achieved through the time polynomial weight appearing in the norm itself; namely, the control is provided by a factor of the form $| t |^{\tilde{α}}$ .

In comparison with the wave equations in weak- $L^{p}$ spaces on the Euclidean space (see Liu, 2009), we obtain exponential decay (and hence exponential stability) for the global-in-time solutions on hyperbolic spaces in Theorem 4.1 $(i)$ , whereas on the Euclidean setting only polynomial decay is known (see Liu, 2009, Theorem 2.9). Thus, together, the present work and Liu (2009) provide complementary sets of results on well-posedness for wave equations in weak- $L^{p}$ spaces on Euclidean and curved (hyperbolic) settings, offering a more complete picture across these two geometries.

The article is organized as follows. Section 2 provides the Klein-Gordon equations on $H^{n}$ and the dispersive estimates in the $L^{p}$ -setting. Section 3 is devoted to the definition of Lorentz spaces $L^{(p, q)}$ on $H^{n}$ and the dispersive estimates of the wave operator in $L^{(p, q)}$ . In Section 4, we state and prove the main results.

2. Wave and Klein-Gordon Equations on Hyperbolic Space

This section is devoted to recall some facts about wave equations on hyperbolic spaces. For further details, the reader is referred to Anker et al. (2012), Metcalfe and Taylor (2011), and Tataru (2001).

Let $H^{n} = H^{n} (R)$ stand for a real hyperbolic manifold, where $n \geq 2$ is the dimension, endowed with a Riemannian metric $g$ . This space is realized via a hyperboloid in $R^{n + 1}$ by considering the upper sheet

{(x_{0}, x_{1}, \dots, x_{n}) \in R^{n + 1}; x_{0} \geq 1 and x_{0}^{2} - x_{1}^{2} - x_{2}^{2} - \dots - x_{n}^{2} = 1},

where the metric is given by

d g = - d x_{0}^{2} + d x_{1}^{2} + \dots + d x_{n}^{2} .

In geodesic polar coordinates, the hyperbolic manifold $(H^{n}, g)$ can be described as follows:

H^{n} = {(\cosh τ, ω \sinh τ), τ \geq 0, ω \in S^{n - 1}}

with

d g = d τ^{2} + (\sinh τ)^{2} d ω^{2},

where

d ω^{2}

is the canonical metric on the sphere

S^{n - 1}

. In these coordinates, the Laplace-Beltrami operator

Δ_{H^{n}}

H^{n}

can be expressed as follows:

Δ_{x} := Δ_{H^{n}} = \partial_{τ}^{2} + (n - 1) \coth τ \partial_{τ} + \sinh^{- 2} τ Δ_{S^{n - 1}} .

It is well-known that the spectrum of

- Δ_{x}

is the half-line

[ρ^{2}, \infty)

, where

ρ = (n - 1) / 2

We consider the shifted wave equation on $H^{n}$

{\begin{cases} \partial_{t}^{2} u (t, x) - Δ_{x} u (t, x) + c u (t, x) = F (u (t, x)), \\ u (0, x) = u_{0} (x), \partial_{t} u (t, x) = u_{1} (x), \end{cases}

(2.1)

where

c = - ρ^{2} = - (n - 1)^{2} / 4

and the nonlinearity

F (u (t, x))

satisfies

F (0) = 0 and | F (u) - F (v) | \leq C (| u |^{b - 1} + | v |^{b - 1}) | u - v |

(2.2)

with

b > 1

. Note that, in the case

c = 0

, then equation (2.1) is called the non-shifted wave equation.

Setting $D_{x} = \sqrt{- Δ_{x} + c}$ , Cauchy problem (2.1) takes the form

{\begin{cases} \partial_{t}^{2} u (t, x) + D_{x}^{2} u (t, x) = F (u (t, x)), \\ u (0, x) = u_{0} (x), \partial_{t} u (t, x) = u_{1} (x) . \end{cases}

(2.3)

In view of Duhamel’s principle, we can formally write (2.3) as

u (t) = \dot{W} (t) u_{0} + W (t) u_{1} + \int_{0}^{t} W (t - s) F (u (s)) d s,

(2.4)

where

W (t)

is the wave group. Recall that

W (t) = \frac{\sin (t D_{x})}{D_{x}} \ and \ \dot{W} (t) = \cos (t D_{x}) .

(2.5)

Solutions

u

of the integral equation (2.4) are referred to as mild solutions.

The dispersive and Strichartz estimates for the solutions of wave and Klein-Gordon equations on the hyperbolic manifolds were studied in many works such as Anker et al. (2012), Anker and Pierfelice (2014), Metcalfe and Taylor (2011, 2012), Tataru (2001), and references therein. Here, we recall the $L^{p} - L^{p^{'}}$ -dispersive estimates obtained by Tataru (2001).

Proposition 2.1

Let $n \geq 2$ , $2 \leq p \leq 2 (n + 1) / (n - 1)$ and $1 / p + 1 / p^{'} = 1$ . Then, there exists a constant $C > 0$ (independent of $t$ ) such that

{‖ W (t) g ‖}_{L^{p}} \leq C ϕ_{p} (t) {‖ g ‖}_{L^{p^{'}}},

(2.6)

for all

g \in L^{p^{'}} (H^{n}),

where

ϕ_{p} (t) = \frac{(1 + | t |)^{\frac{2}{p}}}{(\sinh | t |)^{\frac{n - 1}{2} (1 - \frac{2}{p})}}

Remark 2.2

(i)

We notice that Proposition 2.1 is a consequence of Tataru (2001, Theorem 3) by considering $2 s = ((n + 1) / 2) (1 - 2 / p) \leq 1$ . This condition is equivalent to $p \leq 2 (n + 1) / (n - 1)$ . Notably, inequality (2.6) is precisely the same as (Tataru, 2001, inequality (35)).

(ii)

Proceeding as in the proof of Theorem 3 by Tataru (2001), we also have the estimate

{‖ \frac{\dot{W} (t)}{D_{x}} g ‖}_{L^{p}} = {‖ \frac{\cos (t D_{x})}{D_{x}} g ‖}_{L^{p}} \leq C ϕ_{p} (t) {‖ g ‖}_{L^{p^{'}}},

(2.7)

for all

g \in L^{p^{'}} (H^{n})

3. Lorentz Spaces and Interpolation Estimates on Hyperbolic Spaces

A measurable function $f$ defined on $H^{n}$ belongs to the Lorentz space $L^{p, q} (H^{n})$ if the quantity

‖ f ‖_{L^{(p, q)} (H^{n})} = {\begin{cases} {(\int_{0}^{\infty} {[t^{\frac{1}{p}} f^{* *} (t)]}^{q} \frac{d t}{t})}^{\frac{1}{q}}, & if 1 < p < \infty, 1 \leq q < \infty, \\ sup_{t > 0} t^{\frac{1}{p}} f^{* *} (t), & if 1 < p \leq \infty, q = \infty, \end{cases}

(3.1)

is finite, where

f^{* *} (t) = \frac{1}{t} \int_{0}^{t} f^{*} (s) d s

and

\begin{aligned} f^{*} (t) = inf {s > 0 : m ({x \in H^{n} : | f (x) | > s}) \leq t}, for t > 0, \end{aligned}

with

m = d V o l_{g} = 2 (\sinh^{n - 1} τ) d τ d V o l_{S^{n - 1}}

denoting the hyperbolic volume measure on

H^{n}

. For simplicity, we denote the norm

‖ \cdot ‖_{L^{(p, q)} (H^{n})}

‖ \cdot ‖_{(p, q)}

in the rest of this article. The space

L^{(p, q)} (H^{n})

with the norm

‖ \cdot ‖_{(p, q)}

is a Banach space. In particular,

L^{p} (H^{n}) = L^{(p, p)} (H^{n})

and

L^{(p, \infty)}

is called weak-

L^{p}

space or Marcinkiewicz space.

If the function $f$ is a positive and radially decreasing function on $H^{n}$ , then its norm in the Lorentz space $L^{p, q} (H^{n})$ is given by the following equivalent expression (see Anker & Pierfelice, 2009, Corollary 3.3):

{‖ f ‖}_{(p, q)} ≃ {(\int_{0}^{1} (f (τ))^{q} r^{\frac{n q}{p} - 1} d τ)}^{\frac{1}{q}} + {(\int_{1}^{+ \infty} (f (τ))^{q} e^{\frac{q (n - 1)}{p} τ} d τ)}^{\frac{1}{q}},

(3.2)

for

1 ⩽ q < + \infty

, and the one in the weak-

L^{p}

space is equivalent to

{‖ f ‖}_{(p, \infty)} ≃ sup_{0 < τ < 1} τ^{\frac{n}{p}} f (τ) + sup_{τ ⩾ 1} e^{\frac{n - 1}{p} τ} f (τ) .

(3.3)

From (3.3), it is not difficult to see that weak-

L^{p}

spaces allow functions exhibiting strong singular behavior and outside the

L^{2}

-setting. Moreover, there is no scaling map valid in weak-

L^{p}

spaces as in the case of

R^{n}

. In fact, from (3.1) and (3.3), we can see that the presence of the exponential factor arising from the hyperbolic geometry breaks the homogeneity of the norm.

For $1 \leq q_{1} \leq p \leq q_{2} \leq \infty,$ we have the following inclusions:

L^{(p, 1)} (H^{n}) \subset L^{(p, q_{1})} (H^{n}) \subset L^{p} (H^{n}) \subset L^{(p, q_{2})} (H^{n}) \subset L^{(p, \infty)} (H^{n}) .

(3.4)

Let

1 < p_{1}, p_{2}, p_{3} \leq \infty

and

1 \leq r_{1}, r_{2}, r_{3} \leq \infty

be such that

1 / p_{3} = 1 / p_{1} + 1 / p_{2}

and

1 / r_{1} + 1 / r_{2} \geq 1 / r_{3}

. We have the Hölder inequality

{‖ f g ‖}_{(p_{3}, r_{3})} \leq C {‖ f ‖}_{(p_{1}, r_{1})} {‖ g ‖}_{(p_{2}, r_{2})},

(3.5)

where

C > 0

is a constant independent of

f

and

g

. Moreover, for

1 < p_{1} < p_{2} \leq \infty

0 < θ < 1

1 \leq r_{1}, r_{2}, r \leq \infty,

and

1 / p = (1 - θ) / p_{1} + θ / p_{2},

by reiteration theorem (see Bergh & Löfström, 1976, Theorem 3.5.3), we have the interpolation property

{(L^{(p_{1}, r_{1})} (H^{n}), L^{(p_{2}, r_{2})} (H^{n}))}_{θ, r} = L^{(p, r)} (H^{n}) .

(3.6)

Now we extend the dispersive estimates in Proposition 2.1 to the framework of Lorentz spaces.

Lemma 3.1

Let $n \geq 2$ , $1 \leq r \leq \infty$ , $2 < p < 2 (n + 1) / (n - 1)$ and $1 / p + 1 / p^{'} = 1$ . Then, there exists a constant $C > 0$ (independent of $t$ ) such that

{‖ W (t) g ‖}_{(p, r)} + {‖ \frac{\dot{W} (t)}{D_{x}} g ‖}_{(p, r)} \leq C ϕ_{p} (t) {‖ g ‖}_{(p^{'}, r)},

(3.7)

for all

g \in L^{p^{'}, r} (H^{n}),

where

ϕ_{p} (t) = \frac{(1 + | t |)^{\frac{2}{p}}}{(\sinh | t |)^{\frac{n - 1}{2} (1 - \frac{2}{p})}}

Proof.

The proof follows by means of an interpolation argument. For that, let $p_{1}, p_{2}, p \in (2, 2 (n + 1) / (n - 1))$ and $θ \in (0, 1)$ be such that $p^{- 1} = θ p_{1}^{- 1} + (1 - θ) p_{2}^{- 1} .$ Employing (2.6) and (2.7), and recalling that $L^{p} = L^{(p, p)},$ we have

{‖ W (t) g ‖}_{(p_{i}, p_{i})} + {‖ \frac{\dot{W} (t)}{D_{x}} g ‖}_{(p_{i}, p_{i})} \leq C_{i} ϕ_{p_{i}} (t) {‖ g ‖}_{(p_{i}^{'}, p_{i}^{'})},

(3.8)

where

C_{i} > 0

is independent of

t .

Recalling that the interpolation functor

(\cdot, \cdot)_{θ, r}

is of exponent

θ

and

(L^{p_{1}}, L^{p_{2}})_{θ, r} = L^{(p, r)}

(see Bergh & Löfström, 1976), estimate (3.8) leads us to

{‖ W (t) g ‖}_{(p, r)} + {‖ \frac{\dot{W} (t)}{D_{x}} g ‖}_{(p, r)} \leq K (C_{1} ϕ_{p_{1}} (t))^{θ} (C_{2} ϕ_{p_{2}} (t))^{1 - θ} {‖ g ‖}_{(p^{'}, r)},

(3.9)

where

K > 0

is independent of

t

. Next, taking

C = K (C_{1})^{θ} (C_{2})^{1 - θ}

and using

p^{- 1} = θ p_{1}^{- 1} + (1 - θ) p_{2}^{- 1},

note that

\begin{aligned} K (C_{1} ϕ_{p_{1}} (t))^{θ} (C_{2} ϕ_{p_{2}} (t))^{1 - θ} & = K {(C_{1})}^{θ} (C_{2})^{1 - θ} \frac{(1 + | t |)^{\frac{2 θ}{p_{1}}}}{(\sinh | t |)^{\frac{n - 1}{2} θ (1 - \frac{2}{p_{1}})}} \frac{(1 + | t |)^{\frac{2 (1 - θ)}{p_{2}}}}{(\sinh | t |)^{\frac{n - 1}{2} (1 - θ) (1 - \frac{2}{p_{2}})}} \\ = C \frac{(1 + | t |)^{\frac{2}{p}}}{(\sinh | t |)^{\frac{n - 1}{2} (1 - \frac{2}{p})}}, \end{aligned}

which, together with (3.9), gives the desired estimate.

4. Well-Posedness, Stability, and Scattering

In order to establish the well-posedness of the integral equation (2.4), we define some suitable functional spaces as follows. Let $T > 0$ , $1 < b < \infty$ , $η \geq 0$ , and denote by $E_{η}^{T}$ the set of all Bochner measurable $u : (- T, T) \to L^{(b + 1, \infty)} (H^{n})$ such that

{‖ u ‖}_{E_{η}^{T}} = sup_{t \in (- T, T)} | t |^{η} {‖ u ‖}_{(b + 1, \infty)} < \infty .

The pair

(E_{η}^{T}, ‖ \cdot ‖_{E_{η}^{T}})

is a Banach space.

Now, for a fixed $t_{0} \geq 1$ (see (4.10)), $α, \tilde{α} > 0$ , $1 < b < \infty$ , and $1 \leq d \leq \infty$ , consider the space $E_{α, \tilde{α}}^{d}$ of all Bochner measurable $u : (- \infty, \infty) \to L^{(b + 1, d)}$ satisfying

{‖ u ‖}_{E_{α, \tilde{α}}^{d}} = sup_{| t | \geq t_{0}} e^{α | t |} {‖ u (t) ‖}_{(b + 1, d)} + sup_{t \in (- t_{0}, t_{0})} {| t |}^{\tilde{α}} {‖ u (t) ‖}_{(b + 1, d)} < \infty .

(4.1)

The space

E_{α, \tilde{α}}^{d}

endowed with

‖ \cdot ‖_{E_{α, \tilde{α}}^{d}}

is a Banach space. Also, we consider the initial-data space

I_{0}^{d}

as the set of all pairs

(u_{0}, u_{1}) \in S^{'} (H^{n}) \times S^{'} (H^{n})

such that

{‖ (u_{0}, u_{1}) ‖}_{I_{0}^{d}} = {‖ \dot{W} (t) u_{0} + W (t) u_{1} ‖}_{E_{α, \tilde{α}}^{d}} < \infty .

(4.2)

In the case

d = \infty,

we use the notation

E_{α, \tilde{α}} = E_{α, \tilde{α}}^{\infty} \ \ \ and \ I_{0} = I_{0}^{\infty} .

(4.3)

4.1. Global Well-Posedness

Let $n \geq 2$ . To carry out our analysis, we consider the constants $b_{0}$ , $b$ , and $b_{c}$ satisfying $b_{0} < b < b_{c}$ , where $b_{c} = (n + 3) / (n - 1)$ is the conformal exponent and

b_{0} = \frac{n + 1 + \sqrt{(n + 1)^{2} + 8 (n - 1)}}{2 (n - 1)}

is the positive root of the equation

b (b - 1) (n - 1) = 2 (b + 1)

. Moreover, for

0 < σ < n - 1

, we define the constant

b_{σ} = \frac{n + 1 + σ + \sqrt{(n + 1 + σ)^{2} + 8 (n - 1 - σ)}}{2 (n - 1 - σ)} .

With these definitions in place, we now state the main result of this subsection in the following theorem.

Theorem 4.1 (Global-in-time solution)

(i)
(Well-posedness). Let $n \geq 2, b_{0} < b < b_{c}$ , and set $β = ((n - 1) / 2) (1 - 2 / (b + 1))$ . Consider $0 < σ < β$ such that $b_{σ} < b$ (which is possible since $b_{0} < b$ ) and define $\tilde{α} = ((n - 1) / (b^{2} - 1)) - ((n - 3) / 2 (b - 1))$ and $α = \tilde{α} + (σ / (b - 1))$ . If $(u_{0}, u_{1}) \in I_{0}$ with $‖ (u_{0}, u_{1}) ‖_{I_{0}} \leq ε$ for some $ε > 0$ small enough, then equation (2.4) has a unique global-in-time mild solution $u \in E_{α, \tilde{α}}$ such that $‖ u ‖_{E_{α, \tilde{α}}} \leq 2 ε$ .
(ii)
( $L^{p, q}$ -regularity). Let $1 \leq d \leq \infty$ and $0 \leq h < 1 - b α$ . Suppose further that
$Γ_{1, h}^{d} := sup_{| t | \geq t_{0}} e^{(α + h) | t |} {‖ \dot{W} (t) u_{0} + W (t) u_{1} ‖}_{(b + 1, d)} + sup_{t \in (- t_{0}, t_{0})} {| t |}^{\tilde{α} + h} {‖ \dot{W} (t) u_{0} + W (t) u_{1} ‖}_{(b + 1, d)} < \infty .$
(4.4)
Then, there exists $ε > 0$ such that the previous solution satisfies
$sup_{| t | \geq t_{0}} e^{(α + h) | t |} {‖ u (t) ‖}_{(b + 1, d)} + sup_{t \in (- t_{0}, t_{0})} {| t |}^{\tilde{α} + h} {‖ u (t) ‖}_{(b + 1, d)} < \infty,$
(4.5)
provided that $‖ (u_{0}, u_{1}) ‖_{I_{0}} \leq ε$ .

Remark 4.2
(i)
(Local-in-time solution). With an adaptation on the arguments, we can obtain a local version of the above well-posedness result, regardless of the initial-data size. More precisely, let $1 < b < (n + 1 + \sqrt{n^{2} + 10 n - 7}) / 2 (n - 1)$ and $0 < β = ((n - 1) / 2) (1 - 2 / (b + 1)) < 1$ . If $(D_{x} u_{0}, u_{1}) \in L^{(\frac{b + 1}{b}, \infty)} (H^{n}) \times L^{(\frac{b + 1}{b}, \infty)} (H^{n})$ , then there exists $0 < T < \infty$ , such that (2.4) has a unique solution $u \in E_{η}^{T}$ with $η = β$ . Moreover, if $(D_{x} u_{0}, u_{1}) \in L^{(\frac{b + 1}{b}, d)} (H^{n}) \times L^{(\frac{b + 1}{b}, d)} (H^{n})$ with $1 \leq d \leq \infty$ , then the solution $u$ satisfies
$sup_{t \in (- T, T)} | t |^{β} {‖ u ‖}_{(b + 1, d)} < \infty .$
(4.6)
(ii)
The restrictions imposed on the parameters $b$ , $α$ , and $\tilde{α}$ arise from technical considerations required to ensure the validity of the nonlinear estimates in Lemma 4.3 below which will be used to prove Theorem 4.1. In particular, the limiting values $b_{σ}$ and $b_{c}$ cannot be used in Theorem 4.1, since at these endpoints certain integrals appearing in the proof of Lemma 4.3 fail to converge, thereby preventing the nonlinear estimates from holding.

Before proving Theorem 4.1, we provide the following lemma with useful nonlinear estimates.
Lemma 4.3
Suppose that the conditions of Theorem 4.1 hold. Set
$T (u) (t) = \int_{0}^{t} W (t - s) F (u (s)) d s .$
(4.7)
There exists a positive constant $K$ such that the following nonlinear estimate holds:
$‖ T (u) - T (v) ‖_{E_{α, \tilde{α}}} \leq K ‖ u - v ‖_{E_{α, \tilde{α}}} (‖ u ‖_{E_{α, \tilde{α}}}^{b - 1} + ‖ v ‖_{E_{α, \tilde{α}}}^{b - 1}),$
(4.8)
for all $u, v \in E_{α, \tilde{α}}$ . Moreover, for $0 \leq h < 1 - b α$ and $1 \leq d \leq \infty$ , there exists a positive constant $K_{d, h}$ such that the following nonlinear estimate holds:
$‖ T (u) - T (v) ‖_{E_{α + h, \tilde{α} + h}^{d}} \leq K_{d, h} ‖ u - v ‖_{E_{α + h, \tilde{α} + h}^{d}} (‖ u ‖_{E_{α, \tilde{α}}}^{b - 1} + ‖ v ‖_{E_{α, \tilde{α}}}^{b - 1}),$
(4.9)
for all $u, v \in E_{α, \tilde{α}} \cap E_{α + h, \tilde{α} + h}^{d}$ .
Proof.
First we prove the inequality (4.8). Note that we can choose $t_{0} \geq 1$ such that the function $ϕ_{p} (t)$ in (3.7) satisfies
$ϕ_{p} (t) \leq {\begin{cases} C {| t |}^{\frac{2}{p}} e^{- β_{p} | t |}, & for \ | t | \geq t_{0}, \\ C {| t |}^{- β_{p}}, & for \ 0 < | t | < t_{0}, \end{cases}$
(4.10)
where $β_{p} = ((n - 1) / 2) (1 - 2 / p)$ .

From the conditions on $σ, b, α$ , and $\tilde{α}$ , we have the relations
$\begin{aligned} 2 < b + 1 < \frac{2 (n + 1)}{n - 1}, \tilde{α} = \frac{n - 1}{b^{2} - 1} - \frac{n - 3}{2 (b - 1)}, α = \tilde{α} + \frac{σ}{b - 1}, \\ \ \\ 0 < β < 1, 0 < b \tilde{α} < b α < 1, and 0 < 1 - b α = (β - σ) - α . \end{aligned}$

In view of the time symmetry of the wave group (4.7) and estimates (3.7) and (4.10), we can assume $t > 0$ without loss of generality. Moreover, we consider three cases for the time variable $t$ when estimating the operator $T$ .

Case 1: For $t_{0} - δ \leq t \leq t_{0} + δ$ , where $0 < δ < t_{0}$ . In this case, letting $p = b + 1$ in (4.10), we have $β_{p} = β$ , $e^{α t} \sim t^{\tilde{α}}$ , and $t^{\frac{2}{b + 1}} e^{- β t} \sim t^{- β} .$ Then, it is not difficult to see that
$sup_{t \in (t_{0} - δ, t_{0} + δ)} (max {e^{α | t |}, {| t |}^{\tilde{α}}} {‖ T (u) - T (v) ‖}_{(b + 1, \infty)}) \leq C_{1} {‖ u - v ‖}_{E_{α, \tilde{α}}} ({‖ u ‖}_{E_{α, \tilde{α}}}^{b - 1} + {‖ v ‖}_{E_{α, \tilde{α}}}^{b - 1}) .$
(4.11)

Case 2: For $t \geq t_{0} + δ$ . First note that $1 - β - b \tilde{α} = - \tilde{α},$ $b \tilde{α} < 1$ and $b α < 1.$ Then, using (4.10), Lemma 3.1, Remark 2.2 (ii) and Hölder’s inequality, we can estimate
$\begin{aligned} {‖ T (u) - T (v) ‖}_{(b + 1, \infty)} & \leq \int_{0}^{t} {‖ W (t - s) [F (u) - F (v)] (s) ‖}_{(b + 1, \infty)} d s \\ = \int_{0}^{t} {‖ W (s) [F (u) - F (v)] (t - s) ‖}_{(b + 1, \infty)} d s \\ \leq C \int_{0}^{t} ϕ_{b + 1} (s) {‖ [F (u) - F (v)] (t - s) ‖}_{(\frac{b + 1}{b}, \infty)} d s \\ \leq C \int_{0}^{t} ϕ_{b + 1} (s) {‖ (u - v) (t - s) ‖}_{(b + 1, \infty)} ({‖ u (t - s) ‖}_{(b + 1, \infty)}^{b - 1} + {‖ v (t - s) ‖}_{(b + 1, \infty)}^{b - 1}) d s \\ \leq C \int_{t_{0}}^{t} s^{\frac{2}{b + 1}} e^{- \frac{n - 1}{2} (1 - \frac{2}{b + 1}) s} {‖ (u - v) (t - s) ‖}_{(b + 1, \infty)} ({‖ u (t - s) ‖}_{(b + 1, \infty)}^{b - 1} + {‖ v (t - s) ‖}_{(b + 1, \infty)}^{b - 1}) d s \\ + C \int_{0}^{t_{0}} s^{- \frac{n - 1}{2} (1 - \frac{2}{b + 1})} {‖ (u - v) (t - s) ‖}_{(b + 1, \infty)} ({‖ u (t - s) ‖}_{(b + 1, \infty)}^{b - 1} + {‖ v (t - s) ‖}_{(b + 1, \infty)}^{b - 1}) d s \\ = I_{1} + I_{2} . \end{aligned}$
(4.12)
Considering $p = b + 1$ and $β_{p} = β$ in (4.10), and taking $δ = t_{0} / 2$ , we estimate the integral $I_{1}$ as follows:
$\begin{aligned} I_{1} & = C \int_{t_{0}}^{t} s^{\frac{2}{b + 1}} e^{- \frac{n - 1}{2} (1 - \frac{2}{b + 1}) s} {‖ (u - v) (t - s) ‖}_{(b + 1, \infty)} ({‖ u (t - s) ‖}_{(b + 1, \infty)}^{b - 1} + {‖ v (t - s) ‖}_{(b + 1, \infty)}^{b - 1}) d s \\ = C \int_{t_{0}}^{t - δ} s^{\frac{2}{b + 1}} e^{- \frac{n - 1}{2} (1 - \frac{2}{b + 1}) s} {‖ (u - v) (t - s) ‖}_{(b + 1, \infty)} ({‖ u (t - s) ‖}_{(b + 1, \infty)}^{b - 1} + {‖ v (t - s) ‖}_{(b + 1, \infty)}^{b - 1}) d s \\ + C \int_{t - δ}^{t} s^{\frac{2}{b + 1}} e^{- \frac{n - 1}{2} (1 - \frac{2}{b + 1}) s} {‖ (u - v) (t - s) ‖}_{(b + 1, \infty)} ({‖ u (t - s) ‖}_{(b + 1, \infty)}^{b - 1} + {‖ v (t - s) ‖}_{(b + 1, \infty)}^{b - 1}) d s \\ \leq C sup_{t > δ} e^{α t} {‖ (u - v) (t) ‖}_{(b + 1, \infty)} (sup_{t > δ} e^{(b - 1) α t} {‖ u (t) ‖}_{(b + 1, \infty)}^{b - 1} + sup_{t > δ} e^{(b - 1) α t} {‖ v (t) ‖}_{(b + 1, \infty)}^{b - 1}) \\ \times \int_{t_{0}}^{t - δ} s^{\frac{2}{b + 1}} e^{- β s} e^{- b α (t - s)} d s \\ + C sup_{0 < t < δ} t^{\tilde{α}} {‖ (u - v) (t) ‖}_{(b + 1, \infty)} (sup_{0 < t < δ} t^{(b - 1) \tilde{α}} {‖ u (t) ‖}_{(b + 1, \infty)}^{b - 1} + sup_{0 < t < δ} t^{(b - 1) \tilde{α}} {‖ v (t) ‖}_{(b + 1, \infty)}^{b - 1}) \\ \times \int_{t - δ}^{t} s^{\frac{2}{b + 1}} e^{- β s} (t - s)^{- b \tilde{α}} d s \\ \leq C e^{- α t} {‖ u - v ‖}_{E_{α, \tilde{α}}} ({‖ u ‖}_{E_{α, \tilde{α}}}^{b - 1} + {‖ v ‖}_{E_{α, \tilde{α}}}^{b - 1}), \end{aligned}$
(4.13)
where we have used (4.11), $1 - (β - σ) - b α = - α$ , $0 < σ < β < 1$ , and
$\begin{aligned} \int_{t_{0}}^{t - δ} s^{\frac{2}{b + 1}} e^{- β s} e^{- b α (t - s)} d s & \leq \int_{t_{0}}^{t} s^{\frac{2}{b + 1}} e^{- β s} e^{- b α (t - s)} d s \leq \int_{t_{0}}^{t} (s^{\frac{2}{b + 1}} e^{- σ s}) e^{- (b - σ) s} e^{- b α (t - s)} d s \\ \leq C \int_{t_{0}}^{t} e^{- (β - σ) s} e^{- b α (t - s)} d s (because s^{\frac{2}{b + 1}} e^{- σ s} \leq C) \\ \leq C \int_{t_{0}}^{t} e^{(b α - α - 1) s} e^{- b α (t - s)} d s \leq C e^{- b α t} \int_{t_{0}}^{t} e^{(2 b α - α - 1) s} d s \\ \leq | \frac{C e^{- b α t}}{2 b α - α - 1} (e^{(2 b α - α - 1) t} - e^{(2 b α - α - 1) t_{0}}) | \\ \leq C e^{- α t} (because b α < 1) \end{aligned}$
and
$\begin{aligned} \int_{t - δ}^{t} s^{\frac{2}{b + 1}} e^{- β s} (t - s)^{- b \tilde{α}} d s & \leq \int_{t - δ}^{t} s^{\frac{2}{b + 1}} e^{- β s} (t - s)^{- b \tilde{α}} d s \leq C \int_{t - δ}^{t} e^{- (β - σ) s} (t - s)^{- b \tilde{α}} d s \\ \leq C \int_{t - δ}^{t} e^{(b α - α - 1) s} (t - s)^{- b \tilde{α}} d s = C \int_{0}^{δ} e^{(b α - α - 1) (t - s)} s^{- b \tilde{α}} d s \\ = C \int_{0}^{δ / 2} e^{(b α - α - 1) (t - s)} s^{- b \tilde{α}} d s + C \int_{δ / 2}^{δ} e^{(b α - α - 1) (t - s)} s^{- b \tilde{α}} d s \\ \leq C e^{(b α - α - 1) (t - \frac{δ}{2})} \int_{0}^{δ / 2} s^{- b \tilde{α}} d s + C (\frac{δ}{2})^{- b \tilde{α}} \int_{δ / 2}^{δ} e^{(b α - α - 1) (t - s)} d s \\ \leq C e^{- α t} (because b \tilde{α} < b α < 1) . \end{aligned}$
The remaining term $I_{2}$ can be estimated as follows:
$\begin{aligned} I_{2} & = C \int_{0}^{t_{0}} s^{- \frac{n - 1}{2} (1 - \frac{2}{b + 1})} {‖ (u - v) (t - s) ‖}_{b + 1, \infty} ({‖ u (t - s) ‖}_{b + 1, \infty}^{b - 1} + {‖ v (t - s) ‖}_{b + 1, \infty}^{b - 1}) d s \\ \leq C sup_{t > δ} e^{α t} {‖ (u - v) (t) ‖}_{(b + 1, \infty)} (sup_{t > δ} e^{(b - 1) α t} {‖ u (t) ‖}_{(b + 1, \infty)}^{b - 1} + sup_{t > δ} e^{(b - 1) α t} {‖ v (t) ‖}_{(b + 1, \infty)}^{b - 1}) \\ \times \int_{0}^{t_{0}} s^{- β} e^{- b α (t - s)} d s \\ \leq C e^{- α t} {‖ u - v ‖}_{E_{α, \tilde{α}}} ({‖ u ‖}_{E_{α, \tilde{α}}}^{b - 1} + {‖ v ‖}_{E_{α, \tilde{α}}}^{b - 1}), \end{aligned}$
(4.14)
where we have used (4.11) and the inequality
$\begin{aligned} \int_{0}^{t_{0}} s^{- β} e^{- b α (t - s)} d s & = \int_{0}^{t_{0} / 2} s^{- β} e^{- b α (t - s)} d s + \int_{t_{0} / 2}^{t_{0}} s^{- β} e^{- b α (t - s)} d s \\ \leq e^{- b α (t - \frac{t_{0}}{2})} \frac{1}{1 - β} {(\frac{t_{0}}{2})}^{1 - β} + {(\frac{t_{0}}{2})}^{- β} \frac{1}{b α} (e^{- b α (t - t_{0})} - e^{- b α (t - \frac{t_{0}}{2})}) \\ \leq C e^{- α t}, \end{aligned}$
because $t > t_{0}$ , $b > 1$ , and $0 < β < 1$ . Combining (4.12), (4.13), and (4.14), we arrive at
$sup_{t \geq t_{0} + δ} e^{α t} {‖ T (u) - T (v) ‖}_{(b + 1, \infty)} \leq C_{2} {‖ u - v ‖}_{E_{α, \tilde{α}}} ({‖ u ‖}_{E_{α, \tilde{α}}}^{b - 1} + {‖ v ‖}_{E_{α, \tilde{α}}}^{b - 1}) .$
(4.15)

Case 3: For $0 < t \leq t_{0}$ . Employing (4.10), Lemma 3.1, Remark 2.2 (ii), and Hölder’s inequality, we obtain that
$\begin{aligned} {‖ T (u) - T (v) ‖}_{(b + 1, \infty)} & \leq \int_{0}^{t} {‖ W (t - s) [F (u) - F (v)] (s) ‖}_{(b + 1, \infty)} d s \\ \leq C \int_{0}^{t} ϕ_{b + 1} (s) {‖ [F (u) - F (v)] (t - s) ‖}_{(\frac{b + 1}{b}, \infty)} d s \\ \leq C \int_{0}^{t} ϕ_{b + 1} (s) {‖ (u - v) (t - s) ‖}_{(b + 1, \infty)} ({‖ u (t - s) ‖}_{(b + 1, \infty)}^{b - 1} + {‖ v (t - s) ‖}_{(b + 1, \infty)}^{b - 1}) d s \\ \leq C \int_{0}^{t} s^{- \frac{n - 1}{2} (1 - \frac{2}{b + 1})} {‖ (u - v) (t - s) ‖}_{(b + 1, \infty)} ({‖ u (t - s) ‖}_{(b + 1, \infty)}^{b - 1} + {‖ v (t - s) ‖}_{(b + 1, \infty)}^{b - 1}) d s \\ \leq C \int_{0}^{t} s^{- β} (t - s)^{- b \tilde{α}} d s sup_{t \in (0, t_{0})} t^{\tilde{α}} {‖ (u - v) (t) ‖}_{(b + 1, \infty)} \\ \times (sup_{t \in (0, t_{0})} t^{(b - 1) \tilde{α}} {‖ u (t) ‖}_{(b + 1, \infty)}^{b - 1} + sup_{t \in (0, t_{0})} t^{(b - 1) \tilde{α}} {‖ v (t) ‖}_{(b + 1, \infty)}^{b - 1}) \\ \leq C_{3} t^{- \tilde{α}} {‖ u - v ‖}_{E_{α, \tilde{α}}} ({‖ u ‖}_{E_{α, \tilde{α}}}^{b - 1} + {‖ v ‖}_{E_{α, \tilde{α}}}^{b - 1}), \end{aligned}$
(4.16)
where we have used in (4.16) that $\int_{0}^{t} s^{- β} (t - s)^{- b \tilde{α}} d s = C t^{- \tilde{α}} .$ Therefore, we get
$sup_{t \in (0, t_{0})} t^{\tilde{α}} {‖ T (u) - T (v) ‖}_{(b + 1, \infty)} \leq C_{3} {‖ u - v ‖}_{E_{α, \tilde{α}}} ({‖ u ‖}_{E_{α, \tilde{α}}}^{b - 1} + {‖ v ‖}_{E_{α, \tilde{α}}}^{b - 1}) .$
(4.17)
Using (4.11), (4.15), (4.17) and letting $K = C_{1} + C_{2} + C_{3},$ we obtain the first nonlinear estimate (4.8).

Now, for $0 \leq h < 1 - b α,$ $1 \leq d \leq \infty$ , and let $t_{0}$ be as in (4.10). Proceeding similarly to the proof of the first nonlinear estimate (4.8), we have that: for $| t | \in [t_{0} - δ, t_{0} + δ]$ ,
$sup_{| t | \in [t_{0} - δ, t_{0} + δ]} (max {e^{(α + h) | t |}, | t |^{\tilde{α} + h}} {‖ T (u) - T (v) ‖}_{(b + 1, d)}) \leq C_{4} {‖ u - v ‖}_{E_{α + h, \tilde{α} + h}^{d}} ({‖ u ‖}_{E_{α, \tilde{α}}}^{b - 1} + {‖ v ‖}_{E_{α, \tilde{α}}}^{b - 1}),$
(4.18)
and for $t \geq t_{0} + δ,$
$\begin{aligned} {‖ T (u) - T (v) ‖}_{(b + 1, d)} \\ \leq C sup_{t > δ} e^{(α + h) t} {‖ (u - v) (t) ‖}_{(b + 1, d)} (sup_{t > δ} e^{(b - 1) α t} {‖ u (t) ‖}_{(b + 1, \infty)}^{b - 1} + sup_{t > δ} e^{(b - 1) α t} {‖ v (t) ‖}_{(b + 1, \infty)}^{b - 1}) \\ \times \int_{t_{0}}^{t - δ} s^{\frac{2}{b + 1}} e^{- β s} e^{(- b α - h) (t - s)} d s \\ + C sup_{0 < t < δ} t^{\tilde{α} + h} {‖ (u - v) (t) ‖}_{(b + 1, d)} (sup_{0 < t < δ} t^{(b - 1) \tilde{α}} {‖ u (t) ‖}_{(b + 1, \infty)}^{b - 1} + sup_{0 < t < δ} t^{(b - 1) \tilde{α}} {‖ v (t) ‖}_{(b + 1, \infty)}^{b - 1}) \\ \times \int_{t - δ}^{t} s^{\frac{2}{b + 1}} e^{- β s} (t - s)^{(- b \tilde{α} - h)} d s \\ + C sup_{t > δ} e^{(α + h) t} {‖ (u - v) (t) ‖}_{(b + 1, d)} (sup_{t > δ} e^{(b - 1) α t} {‖ u (t) ‖}_{(b + 1, \infty)}^{b - 1} + sup_{t > δ} e^{(b - 1) α t} {‖ v (t) ‖}_{(b + 1, \infty)}^{b - 1}) \\ \times \int_{0}^{t_{0}} s^{- β} e^{(- b α - h) (t - s)} d s . \end{aligned}$
Therefore, for $| t | \geq t_{0} + δ$ , we obtain that
$sup_{| t | \geq t_{0} + δ} e^{(α + h) | t |} {‖ T (u) - T (v) ‖}_{(b + 1, d)} \leq C_{5} {‖ u - v ‖}_{E_{α + h, \tilde{α} + h}^{d}} ({‖ u ‖}_{E_{α, \tilde{α}}}^{b - 1} + {‖ v ‖}_{E_{α, \tilde{α}}}^{b - 1}) .$
(4.19)
Similarly, for $| t | < t_{0}$ , it follows that
$sup_{| t | < t_{0}} | t |^{\tilde{α} + h} {‖ T (u) - T (v) ‖}_{(b + 1, d)} \leq C_{6} sup_{| t | < t_{0}} | t |^{\tilde{α} + h} {‖ u - v ‖}_{(b + 1, d)} (| t |^{(b - 1) \tilde{α}} {‖ u ‖}_{(b + 1, \infty)}^{b - 1} + | t |^{(b - 1) \tilde{α}} {‖ v ‖}_{(b + 1, \infty)}^{b - 1}) .$
(4.20)

Putting together estimates (4.18), (4.19), and (4.20), considering $K_{d, h} = C_{4} + C_{5} + C_{6},$ and recalling the space $E_{α + h, \tilde{α} + h}^{d}$ (see (4.1)), we obtain the second nonlinear estimate (4.9).

Proof of Theorem 4.1.

Item (i). For some $ε > 0$ , we are going to show that the operator $[\dot{W} (t) u_{0} + W (t) u_{1} + T (u)]$ is a contraction in the closed ball $B (0, 2 ε) = {u \in E_{α, \tilde{α}}; ‖ u ‖_{E_{α, \tilde{α}}} \leq 2 ε} \subset E_{α, \tilde{α}}$ provided that $‖ (u_{0}, u_{1}) ‖_{I_{0}} \leq ε$ . For that, set $y = \dot{W} (t) u_{0} + W (t) u_{1}$ and
$Φ (u) (t) = y + T (u),$
(4.21)
where $T (u)$ is given in (4.7). Using the nonlinear estimate (4.8), we have that
$\begin{aligned} {‖ Φ (u) ‖}_{E_{α, \tilde{α}}} & \leq {‖ y ‖}_{E_{α, \tilde{α}}} + \int_{0}^{t} {‖ W (t - s) [F (u (s))] d s ‖}_{E_{α, \tilde{α}}} \\ \leq {‖ y ‖}_{E_{α, \tilde{α}}} + K {‖ u ‖}_{E_{α, \tilde{α}}}^{b} \\ \leq ε + K 2^{b} ε^{b} \\ \leq 2 ε \end{aligned}$
and
$\begin{aligned} {‖ Φ (u) - Φ (v) ‖}_{E_{α, \tilde{α}}} & \leq \int_{0}^{t} {‖ W (t - s) [F (u (s)) - F (v (s))] d s ‖}_{E_{α, \tilde{α}}} \\ \leq K {‖ u - v ‖}_{E_{α, \tilde{α}}} ({‖ u ‖}_{E_{α, \tilde{α}}}^{b - 1} + {‖ v ‖}_{E_{α, \tilde{α}}}^{b - 1}) \\ \leq K 2^{b} ε^{b - 1} {‖ u - v ‖}_{E_{α, \tilde{α}}}, \end{aligned}$
for all $u, v \in B (0, 2 ε),$ provided that $K 2^{b} ε^{b - 1} < 1$ . It follows that $Φ : B (0, 2 ε) \to B (0, 2 ε)$ is a contraction. Then, in view of the Banach fixed point theorem, we can conclude the existence of a unique solution $u \in E_{α, \tilde{α}}$ for (2.4) such that $‖ u ‖_{E_{α, \tilde{α}}} \leq 2 ε .$

Item (ii). The solution $u$ obtained in item (i) can be approximated by the Picard sequence ${v_{m}}_{m \in N}$ defined as follows:
$v_{1} = \dot{W} (t) u_{0} + W (t) u_{1} and v_{m + 1} = \dot{W} (t) u_{0} + W (t) u_{1} + \int_{0}^{t} W (t - s) v_{m} (s) d s,$
(4.22)
where the limit is taken in the norm $‖ \cdot ‖_{E_{α, \tilde{α}}}$ . Also, by the proof of Item (i), we have that
${‖ v_{m} ‖}_{E_{α, \tilde{α}}} \leq 2 ε .$
(4.23)
In the sequel, we show that the sequence (4.22) is uniformly bounded in the norm $‖ \cdot ‖_{E_{α + h, \tilde{α} + h}^{d}}$ . For that, denote
$Γ_{m, h}^{d} = {‖ v_{m} ‖}_{E_{α + h, \tilde{α} + h}^{d}} .$
First, for the case $h = 0$ , applying nonlinear estimate (4.9) with $h = 0$ and $v = 0$ yields
$\begin{aligned} Γ_{m + 1, 0}^{d} & \leq sup_{| t | \geq t_{0}} e^{α | t |} {‖ \dot{W} (t) u_{0} + W (t) u_{1} ‖}_{(b + 1, d)} + sup_{t \in (- t_{0}, t_{0})} {| t |}^{\tilde{α}} {‖ \dot{W} (t) u_{0} + W (t) u_{1} ‖}_{(b + 1, d)} \\ + K_{d, 0} {‖ v_{m} ‖}_{E_{α, \tilde{α}}^{d}} {‖ v_{m} ‖}_{E_{α, \tilde{α}}}^{b - 1} \\ \leq Γ_{1, 0}^{d} + (K_{α, \tilde{α}, 0} {‖ v_{m} ‖}_{E_{α, \tilde{α}}}^{b - 1}) Γ_{m, 0}^{d} \\ \leq \frac{1}{1 - L_{d}} Γ_{1, 0}^{d}, \end{aligned}$
(4.24)
provided that $ε > 0$ satisfies
$L_{d} = K_{d, 0} 2^{b - 1} ε^{b - 1} < 1.$
(4.25)
For the case $h \neq 0$ , applying again nonlinear estimate (4.9), we can estimate
$\begin{aligned} Γ_{m + 1, h}^{d} & \leq sup_{| t | \geq t_{0}} e^{(α + h) | t |} {‖ \dot{W} (t) u_{0} + W (t) u_{1} ‖}_{(b + 1, d)} + sup_{t \in (- t_{0}, t_{0})} {| t |}^{(\tilde{α} + h)} {‖ \dot{W} (t) u_{0} + W (t) u_{1} ‖}_{(b + 1, d)} \\ + K_{d, h} {‖ v_{m} ‖}_{E_{α + h, \tilde{α} + h}^{d}} {‖ v_{m} ‖}_{E_{α, \tilde{α}}}^{b - 1} \\ \leq Γ_{1, h}^{d} + (K_{d, h} {‖ v_{m} ‖}_{E_{α, \tilde{α}}}^{b - 1}) Γ_{m, h}^{d}, \end{aligned}$
(4.26)
for all $m \in N$ . Taking $ε > 0$ such that $L_{d, h} = K_{d, h} 2^{b - 1} ε^{b - 1} < 1$ and using (4.4), the R.H.S. of (4.26) can be bounded by $Γ_{1, h}^{d} / (1 - L_{d, h}) < \infty$ and then we obtain the desired boundedness. Finally, the uniform boundedness of ${v_{m}}_{m \in N}$ in $E_{α + h, \tilde{α} + h}^{d}$ and the uniqueness of the limit in the sense of distributions yield the property (4.5).
4.2. Exponential Stability

Theorem 4.4
(Exponential stability) Under the same assumptions of Theorem 4.1. Suppose also that $u$ and $\tilde{u}$ belonging to $E_{α, \tilde{α}}$ are solutions of (2.4) obtained in Theorem 4.1 with initial data $(u_{0}, u_{1})$ and $({\tilde{u}}_{0}, {\tilde{u}}_{1})$ , respectively. Then, we have that
$lim_{| t | \to \infty} e^{(α + h) | t |} {‖ \dot{W} (t) (u_{0} - {\tilde{u}}_{0}) + W (t) (u_{1} - {\tilde{u}}_{1}) ‖}_{(b + 1, d)} = 0$
(4.27)
if and only if
$lim_{| t | \to \infty} e^{(α + h) | t |} {‖ u (t) - \tilde{u} (t) ‖}_{(b + 1, d)} = 0.$
(4.28)
In particular, condition (4.27) holds provided that $D_{x} (u_{0} - {\tilde{u}}_{0}) \in L^{(\frac{b + 1}{b}, d)} (H^{n})$ and $(u_{1} - {\tilde{u}}_{1}) \in L^{(\frac{b + 1}{b}, d)} (H^{n})$ .
Proof.
First assuming (4.27), we prove (4.28). Without loss of generality, assume also that $t > 0$ . Then, we have that
$\begin{aligned} e^{(α + h) t} {‖ u (t) - \tilde{u} (t) ‖}_{(b + 1, d)} & \leq e^{(α + h) t} {‖ \dot{W} (t) (u_{0} - {\tilde{u}}_{0}) + W (t) (u_{1} - {\tilde{u}}_{1}) ‖}_{(b + 1, d)} \\ + e^{(α + h) t} \int_{0}^{t} {‖ W (t - s) (F (u) - F (\tilde{u})) (s) d s ‖}_{(b + 1, d)} d s . \end{aligned}$
(4.29)
Take $ε > 0$ such that $L_{d, h} = K_{d, h} 2^{b - 1} ε^{b - 1} < 1$ , where $K_{d, h}$ is given in the proof of Theorem 4.1 (ii) (see (4.26)). Therefore, it follows that
$\begin{aligned} sup_{t \geq t_{0}} e^{(α + h) t} {‖ u (t) ‖}_{(b + 1, d)} + sup_{t \in (0, t_{0})} t^{\tilde{α} + h} {‖ u (t) ‖}_{(b + 1, d)} & \leq \frac{1}{1 - L_{d, h}} Γ_{1, h}^{d}, \end{aligned}$
(4.30)

$\begin{aligned} sup_{t \geq t_{0}} e^{(α + h) t} {‖ \tilde{u} (t) ‖}_{(b + 1, d)} + sup_{t \in (0, t_{0})} t^{\tilde{α} + h} {‖ \tilde{u} (t) ‖}_{(b + 1, d)} & \leq \frac{1}{1 - L_{d, h}} Γ_{1, h}^{d} . \end{aligned}$
(4.31)
provided that $‖ (u_{0}, u_{1}) ‖_{I_{0}}, ‖ ({\tilde{u}}_{0}, {\tilde{u}}_{1}) ‖_{I_{0}} \leq ε$ . For $t \geq t_{0} + δ$ , using $‖ u ‖_{E_{α, \tilde{α}}}, ‖ \tilde{u} ‖_{E_{α, \tilde{α}}} \leq 2 ε$ , we can handle the R.H.S. of (4.29) as follows:
$\begin{aligned} e^{(α + h) t} \int_{0}^{t} {‖ W (t - s) (F (u) - F (\tilde{u})) (s) d s ‖}_{(b + 1, d)} d s \\ \leq C 2^{b} ε^{b - 1} e^{(α + h) t} \int_{0}^{t_{0}} s^{- β} e^{(- b α - h) (t - s)} e^{(α + h) (t - s)} {‖ u (t - s) - \tilde{u} (t - s) ‖}_{(b + 1, d)} d s \\ + C 2^{b} ε^{b - 1} e^{(α + h) t} \int_{t_{0}}^{t - δ} s^{\frac{2}{b + 1}} e^{- β s} e^{(- b α - h) (t - s)} e^{(α + h) (t - s)} {‖ u (t - s) - \tilde{u} (t - s) ‖}_{(b + 1, d)} d s \\ + C 2^{b} ε^{b - 1} e^{(α + h) t} \int_{t - δ}^{t} s^{\frac{2}{b + 1}} e^{- β s} (t - s)^{- b \tilde{α} - h} (t - s)^{\tilde{α} + h} {‖ u (t - s) - \tilde{u} (t - s) ‖}_{(b + 1, d)} d s \\ \leq C 2^{b} ε^{b - 1} e^{- (b - 1) α t} e^{(b α + h) t_{0}} \int_{0}^{t_{0}} s^{- β} d s ({‖ u ‖}_{E_{α + h, \tilde{α} + h}} + {‖ \tilde{u} ‖}_{E_{α + h, \tilde{α} + h}}) \\ + C 2^{b} ε^{b - 1} e^{- ((β - σ) - α - h) t_{0}} \int_{t_{0}}^{t - δ} e^{- (b - 1) α (t - s)} d s ({‖ u ‖}_{E_{α + h, \tilde{α} + h}} + {‖ \tilde{u} ‖}_{E_{α + h, \tilde{α} + h}}) (because s^{\frac{2}{b + 1}} e^{- σ s} \leq C) \\ + C 2^{b} ε^{b - 1} e^{- ((β - σ) - α - h) t} e^{β δ} \int_{t - δ}^{t} (t - s)^{- b \tilde{α} - h} d s ({‖ u ‖}_{E_{α + h, \tilde{α} + h}} + {‖ \tilde{u} ‖}_{E_{α + h, \tilde{α} + h}}), \end{aligned}$
(4.32)
where $t_{0}$ is as in (4.10), $δ = t_{0} / 2$ , and $β = ((n - 1) / 2) (1 - 2 / (b + 1))$ . Combining inequalities (4.29), (4.32), and (4.30) and (4.31), and condition (4.27), and using $b > 1$ and $β - σ > α + h$ , we obtain that
$\begin{aligned} \underset{t \to \infty}{lim sup} e^{(α + h) t} {‖ u (t) - \tilde{u} (t) ‖}_{(b + 1, d)} & \leq \underset{t \to \infty}{lim sup} e^{(α + h) t} {‖ \dot{W} (t) (u_{0} - {\tilde{u}}_{0}) + W (t) (u_{1} - {\tilde{u}}_{1}) ‖}_{(b + 1, d)} \\ + \underset{t \to \infty}{lim sup} e^{(α + h) t} \int_{0}^{t} {‖ W (t - s) (F (u) - F (\tilde{u})) (s) ‖}_{(b + 1, d)} d s \\ = \underset{t \to \infty}{lim sup} e^{(α + h) t} {‖ \dot{W} (t) (u_{0} - {\tilde{u}}_{0}) + W (t) (u_{1} - {\tilde{u}}_{1}) ‖}_{(b + 1, d)} \\ + C 2^{b} ε^{b - 1} e^{(b α + h) t_{0}} t_{0}^{1 - β} \frac{2}{1 - L_{d, h}} Γ_{1, h}^{d} e^{- (b - 1) α t} \\ + C 2^{b} ε^{b - 1} \frac{2}{1 - L_{d, h}} Γ_{1, h}^{d} e^{- (b - 1) α t} (e^{- ((β - σ) - b α - h) t_{0}} - e^{- ((β - σ) - b α - h) (t - δ)}) \\ + C 2^{b} ε^{b - 1} \frac{2}{1 - L_{d, h}} Γ_{1, h}^{d} e^{β δ} δ^{1 - b \tilde{α} - h} e^{- ((β - σ) - α - h) t} \\ \to 0 + 0 + 0 + 0 = 0, \end{aligned}$
which implies (4.28). Next, for the reciprocal assertion, assume (4.28). Then, using the same bounds for
$e^{(α + h) t} \int_{0}^{t} {‖ W (t - s) (F (u) - F (\tilde{u})) (s) d s ‖}_{(b + 1, d)} d s,$
we arrive at
$\begin{aligned} \underset{t \to \infty}{lim sup} e^{(α + h) t} {‖ \dot{W} (t) (u_{0} - {\tilde{u}}_{0}) + W (t) (u_{1} - {\tilde{u}}_{1}) ‖}_{(b + 1, d)} \\ \leq \underset{t \to \infty}{lim sup} e^{(α + h) t} {‖ u (t) - \tilde{u} (t) ‖}_{(b + 1, d)} + \underset{t \to \infty}{lim sup} e^{(α + h) t} \int_{0}^{t} {‖ W (t - s) (F (u) - F (\tilde{u})) (s) ‖}_{(b + 1, d)} d s \\ = 0 + 0 = 0, \end{aligned}$
which gives the desired conclusion.
4.3. Scattering and Wave Operator

In this part, we analyze the scattering property of the global mild solution obtained in Theorem 4.1.

Theorem 4.5

(Scattering) Suppose the same conditions of Theorem 4.1. Consider also the solution $u$ of (2.4) obtained in Theorem 4.1 with initial data $(u_{0}, u_{1})$ . Then, there exist $(u_{0}^{\pm}, u_{1}^{\pm}) \in I_{0}$ satisfying

\begin{aligned} {‖ u (t) - u^{+} (t) ‖}_{(b + 1, d)} & = O (e^{- b (α + h) t}), as t \to + \infty, \end{aligned}

(4.33)

\begin{aligned} {‖ u (t) - u^{-} (t) ‖}_{(b + 1, d)} & = O (e^{- b (α + h) | t |}), as t \to - \infty, \end{aligned}

(4.34)

where

u^{\pm}

are the unique solutions of the associated linear problem

u^{\pm} = \dot{W} (t) u_{0}^{\pm} + W (t) u_{1}^{\pm} .

Proof.

We show only the property (4.33). The proof of (4.34) is left to the reader. We start by defining

u_{1}^{+} = u_{1} + \int_{0}^{\infty} W (- s) F (u (s)) d s and u_{0}^{+} = u_{0} .

For

t > 0

, consider

u^{+}

given by

u^{+} = \dot{W} (t) u_{0} + W (t) u_{1} + \int_{0}^{\infty} W (t - s) F (u (s)) d s,

and note that

u - u^{+} = \int_{t}^{\infty} W (s - t) F (u (s)) d s = \int_{0}^{\infty} W (s) F (u (t + s)) d s .

(4.35)

Using (4.35) with

t > t_{0}

and recalling

0 < β = ((n - 1) / 2) (1 - 2 / (b + 1)) < 1,

we have that

\begin{aligned} {‖ u - u^{+} ‖}_{(b + 1, d)} = {‖ \int_{0}^{\infty} W (s) F (u (t + s)) d s ‖}_{(b + 1, d)} \\ \leq (\int_{0}^{t_{0}} s^{- β} e^{- b (α + h) (s + t)} d s + \int_{t_{0}}^{\infty} s^{\frac{2}{b + 1}} e^{- β s} e^{- b (α + h) (s + t)} d s) \\ \times {(sup_{t \geq t_{0}} e^{(α + h) t} {‖ u (t) ‖}_{(b + 1, d)})}^{b} \\ \leq C {(Γ_{1, h}^{d})}^{b} (\int_{0}^{t_{0}} s^{- β} e^{- b (α + h) (s + t)} d s + \int_{t_{0}}^{\infty} s^{\frac{2}{b + 1}} e^{(- 1 - σ - α - b h) (s + t)} d s) \\ (because 1 - β + σ - (b - 1) α = 0 \Rightarrow β + b (α + h) = 1 + σ + α + b h) \\ \leq C {(Γ_{1, h}^{d})}^{b} (e^{- b (α + h) t} \int_{0}^{t_{0}} s^{- β} e^{- b (α + h) s} d s + e^{(- β - b (α + h)) t} \int_{t_{0}}^{\infty} e^{(- 1 - α - b h) s} d s) \\ (because s^{\frac{2}{b + 1}} e^{- σ s} < C) \\ \leq C {(Γ_{1, h}^{d})}^{b} e^{- b (α + h) t} (\int_{0}^{t_{0}} s^{- β} d s + \int_{t_{0}}^{\infty} e^{(- 1 - α - b h) s} d s) \\ \leq C (\frac{t_{0}^{1 - β}}{1 - β} + \frac{e^{(1 + α + b h) t_{0}}}{1 + α + b h}) {(Γ_{1, h}^{d})}^{b} e^{- b (α + h) t} . \end{aligned}

This leads to the scattering behaviour (4.33). The proof is complete.

Remark 4.6

In view of Theorem 4.4, we can improve the scattering decay by replacing $O$ with $o$ in Theorem 4.5. Precisely, letting $u^{\pm}$ be as in Theorem 4.5, we have that

{‖ u (t) - u^{\pm} (t) ‖}_{(b + 1, d)} = o (e^{- b (α + h) | t |}), as t \to \pm \infty,

(4.36)

provided that

lim_{t \to \pm \infty} e^{(α + h) | t |} {‖ \dot{W} (t) u_{0} + W (t) u_{1} ‖}_{(b + 1, d)} = 0.

(4.37)

In fact, by (4.37), Theorem 4.4 gives

lim_{t \to \pm \infty} e^{(α + h) | t |} {‖ u (t) ‖}_{(b + 1, d)} = 0.

(4.38)

It follows that

\begin{aligned} {‖ u (t) - u^{+} (t) ‖}_{(b + 1, d)} \\ = {‖ \int_{0}^{\infty} W (s) F (u (s + t)) d s ‖}_{(b + 1, d)} \\ \leq 2^{b - 1} (Γ_{1, h}^{d})^{b - 1} (e^{- b (α + h) t} \int_{0}^{t_{0}} s^{- β} e^{- b (α + h) s} e^{(α + h) (s + t)} {‖ u (s + t) ‖}_{(b + 1, d)} d s \\ + e^{(- β - b (α + h)) t} \int_{t_{0}}^{\infty} s^{\frac{2}{b + 1}} e^{- (1 - σ - α - b h) s} e^{(α + h) (s + t)} {‖ u (s + t) ‖}_{(b + 1, d)} d s) \\ \leq C 2^{b - 1} (Γ_{1, h}^{d})^{b - 1} (e^{- b (α + h) t} \int_{0}^{t_{0}} s^{- β} e^{- b (α + h) s} e^{(α + h) (s + t)} {‖ u (s + t) ‖}_{(b + 1, d)} d s \\ + e^{(- β - b (α + h)) t} \int_{t_{0}}^{\infty} e^{- (1 - α - b h) s} e^{(α + h) (s + t)} {‖ u (s + t) ‖}_{(b + 1, d)} d s) (because s^{\frac{2}{b + 1}} e^{- σ s} < C), \end{aligned}

and then

\begin{aligned} \underset{t \to \infty}{lim sup} e^{b (α + h) t} {‖ u (t) - u^{+} (t) ‖}_{(b + 1, h)} & \leq C 2^{b - 1} (Γ_{1, h}^{d})^{b - 1} (\int_{0}^{t_{0}} s^{- β} d s + \int_{t_{0}}^{\infty} e^{- (1 + α + b h) s} d s) \\ \times \underset{t \to \infty}{lim sup} e^{(α + h) t} {‖ u (t) ‖}_{(b + 1, d)} \\ \leq C 2^{b - 1} (Γ_{1, h}^{d})^{b - 1} (\frac{t_{0}^{1 - β}}{1 - β} + \frac{e^{- (1 + α + b h) t_{0}}}{1 + α + b h}) \\ \times \underset{t \to \infty}{lim sup} e^{(α + h) t} {‖ u (t) ‖}_{(b + 1, d)} \to 0, \end{aligned}

as desired. The case

t \to - \infty

is done by a similar way.

Finally, we establish the construction of wave operators for equation (2.3), that is, the construction of solutions with prescribed scattering states. In particular, for a given initial data $(f_{0}, f_{1})$ , we find a global solution $u$ which converges as $| t | \to \infty$ to the solution of the associated linear problem with initial data $(f_{0}, f_{1})$ . In the following theorem, we state and prove the construction of future wave operator, the past wave one is constructed by the same way.

Theorem 4.7

(Future wave operator) Suppose the same conditions of Theorem 4.1 and let $γ \in [0, \infty)$ . Then, for any $(f_{0}, f_{1}) \in S^{'} (H^{n}) \times S^{'} (H^{n})$ such that $(D_{x} f_{0}, f_{1}) \in L^{(\frac{b + 1}{b}, \infty)} (H^{n}) \times L^{(\frac{b + 1}{b}, \infty)} (H^{n})$ and

sup_{t \geq t_{0}} e^{γ t} {‖ \dot{W} (t) f_{0} + W (t) f_{1} ‖}_{(b + 1, \infty)} < \infty,

there exist

T_{0} = T_{0} (f_{0}, f_{1}) > t_{0}

and a solution

u

of the integral equation (2.4) on

[T_{0}, \infty)

satisfying

sup_{t > T_{0}} e^{γ t} {‖ u (t) ‖}_{(b + 1, \infty)} < \infty

and

\begin{aligned} lim_{t \to \infty} e^{γ t} {‖ u (t) - (\dot{W} (t) f_{0} + W (t) f_{1}) ‖}_{(b + 1, \infty)} = 0. \end{aligned}

v (t)

is also a solution of equation (2.4) on

[T_{0}, \infty)

satisfying

sup_{t > T_{0}} e^{γ t} {‖ v (t) ‖}_{(b + 1, \infty)} < \infty,

then there exists

T > T_{0}

such that

u = v

[T, \infty)

Proof.

For $T > t_{0}$ , we consider the space $E_{γ}^{\geq T}$ of all Bochner measurable $u : [T, \infty) \to L^{(b + 1, \infty)} (H^{n})$ satisfying

{‖ u ‖}_{E_{γ}^{\geq T}} = sup_{t \geq T} e^{γ t} {‖ u (t) ‖}_{(b + 1, \infty)} < \infty .

(4.39)

The space

E_{γ}^{\geq T}

endowed with

‖ \cdot ‖_{E_{γ}^{\geq T}}

is a Banach space.

Step 1: Construction of the solution. Let $R > 0$ and consider the closed ball

B^{\geq T} (0, R) = {u \in E_{γ}^{\geq T} : {‖ u ‖}_{E_{γ}^{\geq T}} \leq R} .

We define the mapping

Φ : B^{\geq T} (0, R) \to B^{\geq T} (0, R)

Φ (ω) = - \int_{t}^{\infty} W (t - s) F (ω (s) + \dot{W} (s) f_{0} + W (s) f_{1}) d s .

Observe that, if

ω

is a fixed point of the operator

Φ : B^{\geq T_{0}} (0, R) \to B^{\geq T_{0}} (0, R)

, for some

T_{0} > 0

which will be chosen later, then

u (t) = \dot{W} (t) f_{0} + W (t) f_{1} + ω (t)

satisfies

u \in E_{γ}^{\geq T_{0}}

and

u \in C ([T_{0}, \infty); S^{'} (H^{n}))

. Using the group properties of

W (t)

, we can show that

u (t) = \dot{W} (t - T_{0}) u (T_{0}) + W (t - T_{0}) \partial_{t} u (T_{0}) + \int_{T_{0}}^{t} W (t - s) F (u (s)) d s,

(4.40)

where

G^{t \geq T_{0}} (u) = \int_{T_{0}}^{t} W (t - s) F (u (s)) d s \in S^{'} (H^{n})

and equation (4.40) holds for a.e.

t \geq T_{0}

. The function

u

given by (4.40) is a solution of the integral equation (2.4) on

[T_{0}, \infty)

Step 2: The existence of unique fixed point $ω$ of $Φ$ . We prove this argument by establishing that $Φ$ is a contraction mapping. Indeed, by using (4.10), Lemma 3.1, Remark 2.2 (ii), and Hölder’s inequality, for $t \geq T > t_{0}$ , we estimate

\begin{aligned} {‖ Φ (ω) ‖}_{(b + 1, \infty)} = {‖ \int_{t}^{\infty} W (s - t) F (ω (s) + \dot{W} (s) f_{0} + W (s) f_{1}) d s ‖}_{(b + 1, \infty)} \\ = {‖ \int_{0}^{\infty} W (s) F (ω (t + s) + \dot{W} (t + s) f_{0} + W (t + s) f_{1}) d s ‖}_{(b + 1, \infty)} \\ \leq (\int_{0}^{t_{0}} s^{- \frac{n - 1}{2} (1 - \frac{2}{b + 1})} e^{- b γ (s + t)} d s + \int_{t_{0}}^{\infty} s^{\frac{2}{b + 1}} e^{- \frac{n - 1}{2} (1 - \frac{2}{b + 1}) s} e^{- b γ (s + t)} d s) \\ \times {(sup_{t \geq T} e^{γ t} {‖ ω (t) + \dot{W} (t) f_{0} + W (t) f_{1} ‖}_{(b + 1, \infty)})}^{b} \\ \leq {(sup_{t \geq T} e^{γ t} {‖ ω (t) ‖}_{(b + 1, \infty)} + sup_{t \geq T} e^{γ t} {‖ \dot{W} (t) f_{0} + W (t) f_{1} ‖}_{(b + 1, \infty)})}^{b} \\ \times (e^{- b γ t} \int_{0}^{t_{0}} s^{- β} e^{- b γ s} d s + e^{- b γ t} \int_{t_{0}}^{\infty} (s^{\frac{2}{b + 1}} e^{- σ s}) e^{- (β - σ + b γ) s} d s) \\ \leq C {(sup_{t \geq T} e^{γ t} {‖ ω (t) ‖}_{(b + 1, \infty)} + sup_{t \geq T} e^{γ t} {‖ \dot{W} (t) f_{0} + W (t) f_{1} ‖}_{(b + 1, \infty)})}^{b} \\ \times e^{- γ t} e^{- (b - 1) γ T} (\int_{0}^{t_{0}} s^{- β} d s + \int_{t_{0}}^{\infty} e^{- b γ s} d s) (because β > σ and s^{\frac{2}{b + 1}} e^{- σ s} < C) \\ \leq {(sup_{t \geq T} e^{γ t} {‖ ω (t) ‖}_{(b + 1, \infty)} + sup_{t \geq T} e^{γ t} {‖ \dot{W} (t) f_{0} + W (t) f_{1} ‖}_{(b + 1, \infty)})}^{b} e^{- γ t} C (T), \end{aligned}

where

\begin{aligned} C (T) & = e^{- (b - 1) γ T} (\int_{0}^{t_{0}} s^{- β} d s + \int_{t_{0}}^{\infty} e^{- b γ s} d s) \\ = e^{- (b - 1) γ T} (\frac{t_{0}^{1 - β}}{1 - β} + \frac{e^{b γ t_{0}}}{b γ}) \to 0 as T \to \infty . \end{aligned}

Hence

sup_{t \geq T} e^{γ t} {‖ Φ (ω) ‖}_{(b + 1, \infty)} \leq C (T) {(sup_{t \geq T} e^{γ t} {‖ ω (t) ‖}_{(b + 1, \infty)} + sup_{t \geq T} e^{γ t} {‖ \dot{W} (t) f_{0} + W (t) f_{1} ‖}_{(b + 1, \infty)})}^{b},

for

t \geq T \geq t_{0}

. This leads to

{‖ Φ (ω) ‖}_{E_{γ}^{\geq T}} \leq C (T) {(R + {‖ \dot{W} (t) f_{0} + W (t) f_{1} ‖}_{E_{γ}^{\geq T}})}^{b},

(4.41)

for all

t \geq T \geq t_{0}

and

ω \in B^{\geq T} (0, R)

. By the same way and using the property (1.2) of function

F

, for every

ω, \tilde{ω} \in B^{\geq T} (0, R)

, we can estimate

\begin{aligned} sup_{t \geq T} e^{γ t} {‖ Φ (ω) - Φ (\tilde{ω}) ‖}_{(b + 1, \infty)} & \leq C (T) {‖ ω - \tilde{ω} ‖}_{E_{γ}^{\geq T}} [{({‖ ω ‖}_{E_{γ}^{\geq T}} + {‖ \dot{W} (t) f_{0} + W (t) f_{1} ‖}_{E_{γ}^{\geq T}})}^{b - 1} \\ + {({‖ ω ‖}_{E_{γ}^{\geq T}} + {‖ \dot{W} (t) f_{0} + W (t) f_{1} ‖}_{E_{γ}^{\geq T}})}^{b - 1}] \\ \leq 2 C (T) {‖ ω - \tilde{ω} ‖}_{E_{γ}^{\geq T}} {(R + {‖ \dot{W} (t) f_{0} + W (t) f_{1} ‖}_{E_{γ}^{\geq T}})}^{b - 1} . \end{aligned}

(4.42)

From inequalities (4.41) and (4.42), we can choose

T_{0} = T_{0} (f_{0}, f_{1}) \geq t_{0}

such that

Φ

is a contraction mapping on

B^{\geq T_{0}} (0, R)

, which implies the existence of a unique fixed point

ω

Φ

Step 3: The convergence and uniqueness. Let $0 < η < T_{0}$ be fixed and $τ > 2 T_{0}$ . Hence, we have $\tilde{τ} = τ - η > T_{0}$ . By the same estimates in Step 2, we obtain that

\begin{aligned} e^{γ τ} {‖ u (τ) - (\dot{W} (τ) f_{1} + W (τ) f_{0}) ‖}_{(b + 1, \infty)} & = e^{γ τ} {‖ ω (τ) ‖}_{(b + 1, \infty)} \\ \leq sup_{t > \tilde{τ}} e^{γ t} {‖ Φ (ω) (t) ‖}_{(b + 1, \infty)} \\ \leq C (\tilde{τ}) {(R + sup_{t > T_{0}} e^{γ t} {‖ \dot{W} (t) f_{0} + W (t) f_{1} ‖}_{(b + 1, \infty)})}^{b} \\ \to 0, as \tilde{τ} \to \infty, \end{aligned}

due to

C (\tilde{τ}) \to 0

\tilde{τ} \to \infty

. Finally, the uniqueness assertion follows from standard arguments (see, e.g., Farah & Ferreira, 2012).

Remark 4.8

(i)

In this article, we establish the well-posedness, exponential stability, and scattering theory for the shifted wave equation (2.3) by exploiting the dispersive estimates for the wave group associated with the operator $D_{x} = \sqrt{- Δ_{x} + c}, c = - (n - 1)^{2} / 4$ , as obtained by Tataru (2001). For wave equations with $c > - (n - 1)^{2} / 4$ (i.e., the Klein-Gordon equations), the above results can also be derived by employing the dispersive estimates for the associated wave groups obtained by Anker and Pierfelice (2014) (see Theorems 4.2 and 4.3 by Anker & Pierfelice, 2014), although this approach yields stability and scattering with at most polynomial decay. The method is similar to our recent work for the dispersive Boussinesq equations (Ferreira & Xuan, 2024).

(ii)

In the previous works (Anker & Pierfelice, 2009, 2014), the authors employed Strichartz estimates on hyperbolic spaces to obtain the well-posedness and scattering for nonlinear Schrödinger equations and Klein-Gordon equations in $H^{1}$ - and $L^{2}$ - spaces. Meanwhile, our method employs dispersive estimates to establish these results for the shifted wave equation (2.3) in the framework of Lorentz spaces, allowing us to extend the class of admissible initial data to include singular data outside the $L^{2}$ -setting. Consequently, the present work together with (Anker & Pierfelice, 2009, 2014) provide complementary results on the well-posedness and scattering for wave equations, offering a more comprehensive understanding across both regular and singular initial data.

Footnotes

ORCID iD

Lucas C. F. Ferreira

Funding

L.C.F. Ferreira was supported by FAPESP (grant: 2020/05618-6) and CNPq (grant: 308799/2019-4 and 312484/2023-2), Brazil.

Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Anker

J.-P.

Pierfelice

(2009). Nonlinear Schrödinger equation on real hyperbolic spaces. Annales de l’Institut Henri Poincaré C Analyse Non Linéaire, 26(5), 1853–1869. 10.1016/j.anihpc.2009.01.009

Anker

J.-P.

Pierfelice

(2014). Wave and Klein-Gordon equations on hyperbolic spaces. Analysis & PDE, 7(4), 953–995. 10.2140/apde.2014.7.953

Anker

J.-P.

Pierfelice

Vallarino

(2012). The wave equation on hyperbolic spaces. Journal of Differential Equations, 252, 5613–5661. 10.1016/j.jde.2012.01.031

Antoci

(2000). Scattering theory for

p

-forms on hyperbolic real space. Istituto Lombardo (Rend. Sc.) A, 134, 71–85.

Baez

J. C.

Segal

I. E.

Zhou

Z.-F.

(1990). The global Goursat problem and scattering for nonlinear wave equations. Journal of Functional Analysis, 93, 239–269. 10.1016/0022-1236(90)90128-8

Banica

Carles

Duyckaerts

(2009). On scattering for NLS: From Euclidean to hyperbolic space. Discrete and Continuous Dynamical Systems, 24(4), 1113–1127. 10.3934/dcds.2009.24.1113

Banica

Carles

Staffilani

(2008). Scattering theory for radial nonlinear Schrödinger equations on hyperbolic space. Geometric and Functional Analysis, 18, 367–399. 10.1007/s00039-008-0663-x

Belchev

Kepka

Zhou

(1999). Global existence of solutions to nonlinear wave equations. Communications in Partial Differential Equations, 24, 2297–2331. 10.1080/03605309908821503

Bergh

Löfström

(1976). Interpolation spaces: An introduction (Grundlehren der Mathematischen Wissenschaften), Springer.

10.

Bony

J.-F.

Häfner

(2009). The semilinear wave equation on asymptotically Euclidean manifolds. Communications in Partial Differential Equations, 35(1), 23–67. 10.1080/03605300903396601

11.

Cazenave

Vega

Vilela

M. C.

(2001). A note on the nonlinear Schrödinger equation in weak-

L^{p}

space. Communications in Contemporary Mathematics, 3(1), 153–162. 10.1142/S0219199701000317

12.

Cazenave

Weissler

F. B.

(1998). Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations. Mathematische Zeitschrift, 228(1), 83–120. 10.1007/PL00004606

13.

Cazenave

Weissler

F. B.

(2000). Scattering theory and self-similar solutions for the nonlinear Schrödinger equation. SIAM Journal on Mathematical Analysis, 31(3), 625–650. 10.1137/S0036141099351309

14.

Chen

Liu

(2026). The Sommerfeld-Rellich framework for scattering on hyperbolic space: Far-field patterns and inverse problems [Preprint]. arXiv, https://doi.org/10.48550/arXiv.2310.16416.

15.

de Almeida

M. F.

Ferreira

L. C. F.

(2017). Time-weighted estimates in Lorentz spaces and self-similarity for wave equations with singular potentials. Analysis & PDE, 10(2), 423–438. 10.2140/apde.2017.10.423

16.

Debièvre

Hislop

H. D.

Sigal

I. M.

(1992). Scattering theory for the wave equation on non-compact manifolds. Reviews in Mathematical Physics, 4(4), 575–618. 10.1142/S0129055X92000236

17.

Farah

L. G.

Ferreira

L. C. F.

(2012). On the wave operator for the generalized Boussinesq equation. Proceedings of the American Mathematical Society, 140(9), 3055–3066. 10.1090/S0002-9939-2011-11131-6

18.

Ferreira

L. C. F.

(2011). Existence and scattering theory for Boussinesq type equations with singular data. Journal of Differential Equations, 250(5), 2372–2388. 10.1016/j.jde.2010.11.013

19.

Ferreira

L. C. F.

Vilamizar-Roa

E. J.

Silva

P. B. E.

(2009). On the existence of infinite energy solutions for nonlinear Schrödinger equations. Proceedings of the American Mathematical Society, 137(6), 1977–1987. 10.1090/S0002-9939-09-09773-1

20.

Ferreira

L. C. F.

Xuan

P. T.

(2024). Dispersive estimates and generalized Boussinesq equation on hyperbolic spaces with rough initial data [Preprint]. arXiv. https://doi.org/10.48550/arXiv.2410.20472.

21.

French

(2012). Scattering for nonlinear waves on hyperbolic space [PhD’s thesis, University of North Carolina at Chapel Hill]. https://doi.org/10.17615/6fp0-qw21.

22.

Friedlander

F. G.

(2001). Notes on the wave equation on asymptotically Euclidean manifolds. Journal of Functional Analysis, 184(1), 1–18. 10.1006/jfan.2000.3546

23.

Georgiev

Lindblad

Sogge

C. D.

(1997). Weighted Strichartz estimates and global existence for semilinear wave equations. American Journal of Mathematics, 119, 1291–1319. 10.1353/ajm.1997.0038

24.

Ginibre

Velo

(1985). The global Cauchy problem for the nonlinear Klein-Gordon equation. Mathematische Zeitschrift, 189, 487–505. 10.1007/BF01168155

25.

Ginibre

Velo

(1987a). Conformal invariance and time decay for nonlinear wave equations I. Annales de l’Institut Henri Poincaré, Physique Théorique, 47, 221–261.

26.

Ginibre

Velo

(1987b). Conformal invariance and time decay for nonlinear wave equations II. Annales de l’Institut Henri Poincaré, Physique Théorique, 47, 263–276.

27.

Ginibre

Velo

(1989). Scattering theory in the energy space for a class of non-linear wave equations. Communications in Mathematical Physics, 123, 535–573. 10.1007/BF01218585

28.

Glassey

R. T.

(1981). Finite-time blow-up for solutions of nonlinear wave equations. Mathematische Zeitschrift, 177, 323–340. 10.1007/BF01162066

29.

Hidano

(1998). Nonlinear small data scattering for the wave equation in

R^{4 + 1}

. Journal of the Mathematical Society of Japan, 50, 253–292. 10.2969/jmsj/05020253

30.

Hidano

(2000). Small data scattering and blow-up for a wave equation with a cubic convolution. Funkcialaj Ekvacioj Serio Internacia, 43, 559–588. https://doi.org/10.24546/0100500048

31.

Hidano

(2001). Scattering problem for the nonlinear wave equation in the finite energy and conformal charged. Journal of Functional Analysis, 187(2), 274–307. 10.1006/jfan.2001.3801

32.

Ibrahim

Majdoub

Masmoudi

(2006). Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity. Communications on Pure and Applied Mathematics, 59, 1639–1658. 10.1002/cpa.20127

33.

Ibrahim

Majdoub

Masmoudi

Nakanishi

(2009). Scattering for the two-dimensional energy-critical wave equation. Duke Mathematical Journal, 150, 287–329. 10.1215/00127094-2009-053

34.

Keel

Tao

(1998). Endpoint Strichartz estimates. American Journal of Mathematics, 120, 955–980. 10.1353/ajm.1998.0039

35.

Lax

P. D.

Phillips

R. S.

(1967). Scattering theory (Pure and Applied Mathematics, Vol. 26, pp. xii + 276). Academic Press..

36.

Liu

(2023). Numerical methods for inverse scattering problems. Springer.

37.

T. T.

(1991). Life-span of classical solutions to fully nonlinear wave equations. Communications in Partial Differential Equations, 16, 909–940. 10.1080/03605309108820785

38.

T. T.

Zhou

(1995). A note on the life-span of classical solutions to nonlinear wave equations in four space dimensions. Indiana University Mathematics Journal, 44, 1207–1248. http://doi.org/10.1512/iumj.1995.44.2026

39.

Lindblad

Metcalfe

Sogge

C. D.

Tohaneanu

Wang

(2014). The Strauss conjecture on Kerr black hole backgrounds. Mathematische Annalen, 359, 637–661. 10.1007/s00208-014-1006-x

40.

Lindblad

Sogge

C. D.

(1995). On existence and scattering with minimal regularity for semilinear wave equations. Journal of Functional Analysis, 130, 357–426. 10.1006/jfan.1995.1075

41.

Liu

(2009). Remarks on infinite energy solutions of nonlinear wave equations. Methods & Applications, 71, 4231–4240. 10.1016/j.na.2009.02.112

42.

Metcalfe

Taylor

(2011). Nonlinear waves on 3D hyperbolic space. Transactions of the American Mathematical Society, 363(7), 3489–3529. 10.1090/S0002-9947-2011-05122-6

43.

Metcalfe

Taylor

(2012). Dispersive wave estimates on 3D hyperbolic space. Proceedings of the American Mathematical Society, 140(11), 3861–3866. 10.1090/S0002-9939-2012-11534-5

44.

Metcalfe

Wang

(2017). The Strauss conjecture on asymptotically flat space-times. SIAM Journal on Mathematical Analysis, 49(6), 4579–4594. 10.1137/16M1074886

45.

Pecher

(1982). Decay of solutions of nonlinear wave equations in three space dimensions. Journal of Functional Analysis, 46, 221–229. 10.1016/0022-1236(82)90035-0

46.

Pecher

(1984). Nonlinear small data scattering for the wave and Klein-Gordon equation. Mathematische Zeitschrift, 185, 261–270. 10.1007/BF01181697

47.

Phillips

Wiskott

Woo

(1987). Scattering theory for the wave equation on a hyperbolic manifold. Journal of Functional Analysis, 74(2), 346–398. 10.1016/0022-1236(87)90030-9

48.

Sá Barreto

(2005). Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds. Duke Mathematical Journal, 129(3), 407–480. 10.1215/S0012-7094-05-12931-3

49.

Sá Barreto

(2008). A support theorem for the radiation fields on asymptotically Euclidean manifolds. Mathematical Research Letters, 15(5), 973–991. 10.4310/MRL.2008.v15.n5.a11

50.

Sideris

T. C.

(1984). Nonexistence of global solutions to semilinear wave equations in high dimensions. Journal of Differential Equations, 52, 378–406. 10.1016/0022-0396(84)90169-4

51.

Sire

Sogge

C. D.

Wang

(2019). The Strauss conjecture on negatively curved backgrounds. Discrete and Continuous Dynamical Systems, 39(12), 7081–7099. 10.3934/dcds.2019296

52.

Strauss

W. A.

(1968). Decay and asymptotics for

◻ u = F (u)

. Journal of Functional Analysis, 2, 409–457. 10.1016/0022-1236(68)90004-9

53.

Strauss

W. A.

(1981). Nonlinear scattering theory at low energy. Journal of Functional Analysis, 41, 110–133. 10.1016/0022-1236(81)90063-X

54.

Tataru

(2001). Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation. Transactions of the American Mathematical Society, 353(2), 795–807. 10.1090/S0002-9947-00-02750-1

55.

Wang

Zhang

(2025). Wave equations with logarithmic nonlinearity on hyperbolic spaces. Transactions of the American Mathematical Society, 378, 2253–2269. 10.1090/tran/9404

56.

Zhang

(2015). Strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds. Advances in Mathematics, 271, 91–111. 10.1016/j.aim.2014.11.013

57.

Zhou

(1995). Cauchy problem for semilinear wave equations in four space dimensions with small initial data. Journal of Partial Differential Equations, 8(2), 135–144. 10.4208/jpde.v8.n2.5