σ -Evolution models with low regular time-dependent non-effective structural damping

Abstract

In this work we study decay rates for a σ-evolution equation in $R^{n}$ under effects of a damping term represented by the action of a fractional Laplacian operator and a time-dependent coefficient, $b (t) {(- Δ)}^{θ} u_{t} (t, x)$ . We consider that b is ‘confined’ in the curve $g (t) = {(1 + t)}^{α} {ln}^{γ} (1 + t)$ for large $t ⩾ t_{0}$ and without any control on $\frac{d}{d t} b (t)$ .

Keywords

Wave equation plate equation fractional damping sharp decay rates non-effective damping multiplier method

1. Introduction

We consider, for $0 < θ ⩽ σ$ , the initial value problem for a σ-evolution equation with fractional damping in $R^{n}$ : $\begin{matrix} (1.1) & u_{t t} (t, x) + A^{σ} u (t, x) + b (t) A^{θ} u_{t} (t, x) = 0, (t, x) \in (0, \infty) \times R^{n} \end{matrix}$ with initial data $\begin{matrix} (1.2) & u (0, x) = u_{0} (x), u_{t} (0, x) = u_{1} (x), x \in R^{n}, \end{matrix}$ where $A : = - Δ = - \sum_{i = 1}^{n} \frac{\partial^{2}}{\partial x_{i}^{2}} \cdot$

The fractional power operator $A^{δ} : D (A^{δ}) \subset L^{2} (R^{n}) \to L^{2} (R^{n})$ ( $δ ⩾ 0$ ) with its domain $D (A^{δ}) = H^{2 δ} (R^{n})$ is defined by $\begin{matrix} A^{δ} v (x) : = F^{- 1} (| ξ |^{2 δ} F (v) (ξ)) (x), v \in H^{2 δ} (R^{n}), x \in R^{n}, \end{matrix}$ where $F$ denotes the usual Fourier transform in $L^{2} (R^{n})$ and $| \cdot |$ denotes the usual norm in $R^{n}$ . The operator $A^{δ}$ is nonnegative and self-adjoint in $L^{2} (R^{n})$ and the Schwartz space $S (R^{n})$ is dense in $H^{2 δ} (R^{n})$ . Note that $A^{1} = A$ and $A^{0} = I$ . For $β \in N^{n}$ , say, $β = (β_{1}, \dots, β_{n})$ we also define $D^{β} : = \frac{\partial^{| β |}}{\partial x_{1}^{β_{1}} \dots \partial x_{n}^{β_{n}}}$ , in which $| β | : = \sum_{j = 1}^{n} β_{j}$ . The results obtained in this paper can be applied to several initial value problems associated to second-order equations, as for example, wave equation, plate equation, among others.

We assume, for a sufficient large $t_{0} > 0$ , that $b \sim g$ in $[t_{0}, \infty)$ , in other words, there exist $a_{1} > 0$ and $a_{2} > 0$ such that $a_{1} g (t) ⩽ b (t) ⩽ a_{2} g (t)$ for all $t ⩾ t_{0}$ , in which $g (t) = {(1 + t)}^{α} {ln}^{γ} (1 + t)$ . In addition, we consider $b (t) ⩾ 0$ for $t \in [0, t_{0})$ and the following hypothesis:

Hypothesis A.
For $0 < θ ⩽ σ$ , let $α \in [- 1, 1)$ and $γ \in R$ satisfying one of the following:
$α \in (- 1, 1)$ and $σ (1 + α) < 2 θ$ ;

$α \in (- 1, 1)$ , $γ ⩽ 0$ , and $σ (1 + α) = 2 θ$ ;

$α = - 1$ and $γ ⩾ - 1$ . In particular $σ (1 + α) < 2 θ$ .

The case $(θ, α) = (σ, 1)$ is not included in Hypothesis A, since decay rates could depend, in general, on $a_{1}$ and $a_{2}$ . For suitable constants, our method could be applied achieving sharp decay rates, but in general is not possible, since our method abuse in the use of constants when applying the multipliers. That is, the technique is still valid but must be improved in order to achieve sharp decay rates.

We can consider b a simple function such as $b (t) = 2 + cos (t)$ (case $α = γ = 0$ ) or more complicated functions. To illustrate an interesting example for b, we can consider the following (for simplicity we assume $α < 0$ and $γ > - 1$ if $μ_{1} \neq 0$ , but similar examples can be made for the other cases): $\begin{matrix} (1.3) & \begin{matrix} b (t) : = & μ_{1} Γ (1 + γ, - α ln (1 + t)) + μ_{2} g (t) sin ({(1 + t)}^{η_{2}}) \\ + μ_{3} g (t) cos ({(1 + t)}^{η_{3}}) + μ_{4} g (t), \end{matrix} \end{matrix}$ for $t ⩾ t_{0}$ big enough, where $Γ (s, x) : = \int_{x}^{\infty} y^{s - 1} e^{- y} d y$ is the upper incomplete gamma function, $η_{i}, μ_{j} \in R$ for all $2 ⩽ i ⩽ 3$ and $1 ⩽ j ⩽ 4$ , and at least $μ_{1}$ or $μ_{4}$ is big enough. The definition for b in $[0, t_{0})$ can be anything that make b non-negative. To see why this function can be applied, we apply Lemma A.1 with $f (y) = y^{s - 1} e^{- y}$ and $ψ (y) = - 1$ to obtain: $Γ (s, x) \sim x^{s - 1} e^{- x}$ . Therefore $Γ (1 + γ, - α ln (1 + t)) \sim g (t)$ for $t ⩾ t_{0}$ big enough. Since $μ_{1}$ or $μ_{4}$ is big enough, we have $b \sim g$ . We can construct several examples with special functions of physical mathematics, like Bessel functions or W-Lambert functions (see Appendix B), just proceeding as made for b in equation (1.3). That is, asymptotically several special functions reduce to the case $b \sim g$ . To calculate decay rates for the solution of (1.1)–(1.2) for such b, is straightforward by applying Theorem 1.1.

If the assumption on $b (t)$ are more general, for instance, $\begin{matrix} a_{1} {(1 + t)}^{α_{1}} ⩽ b (t) ⩽ a_{2} {(1 + t)}^{α_{2}} \end{matrix}$ with $\begin{matrix} - 1 < α_{1} < α_{2}, σ (1 + α_{2}) < 2 θ \end{matrix}$ was still an open problem. To study it, is necessary to adapt the method we used here, due to the other technical difficulties which will appear in that case.

The asymptotic profile of (1.1)–(1.2) for $σ > 0$ , $θ \in (0, σ)$ , $b (t) = 2 μ {(1 + t)}^{α}$ , $μ > 0$ and $α \in (- 1, 1)$ , was investigated by D’Abbicco–Ebert in [8]. They proved an anomalous diffusion phenomena for this equation and introduced a classification based on it: the damping is said effective when the diffusion phenomenon holds and non-effective otherwise. This concept generalized the classification introduced by J. Wirth for $θ = 0$ in [25] (non-effective case) and [26] (effective case). Furthermore, D’Abbicco–Ebert reported that when $2 θ < σ (1 + α)$ the damping is effective and non-effective if $2 θ > σ (1 + α)$ . The case $2 θ = σ (1 + α)$ is treated as a critical case and they do not discuss. In addition, is expected that their work could be extended for a more general class of coefficient b in terms of the following limits:

If $\frac{1}{b} \notin L^{1}$ and ${lim}_{t \to \infty} t^{1 - \frac{2 θ}{σ}} b (t) = \infty$ , the damping is effective;

If $b \notin L^{1}$ and ${lim}_{t \to \infty} t^{1 - \frac{2 θ}{σ}} b (t) = 0$ , the damping is non-effective.

Going back to Hypothesis A and based on the last classification introduced, except for the case $σ (1 + α) = 2 θ$ and $γ = 0$ (critical instance), we have exactly the non-effective damping case. The effective case will be treated in a forthcoming paper [24]. Our classification however, is not motivated by whether the asymptotic profile of the solution of the problem has or not an diffusion phenomena, but rely on a new classification which will be motivated and introduced next. The connection between our new classification and the diffusion phenomena is an open question.

In the case of $b = 1$ and $σ = 1$ , equation (1.1)–(1.2) turns to a wave equation with fractional damping. This equation was approached by Ikehata–Natsume [13] using the energy method in Fourier space, a technique due to Umeda–Kawashima–Shizuta [23]. However, since the mentioned result was not optimal for $0 ⩽ θ < \frac{1}{2}$ , the method was improved by Charão–da Luz–Ikehata [2] by using integrable properties of the equation. In this context, the key inequality (together with other techniques) to find the (almost) optimal decay rates for that equation with $0 ⩽ θ < \frac{1}{2}$ is given by Lemma 3.2 of [2]: $\begin{matrix} (1.4) & | ξ |^{4 θ} | \hat{u} (t) |^{2} ≲ | ξ |^{4 θ} | {\hat{u}}_{0} |^{2} + | {\hat{u}}_{1} |^{2}, for all | ξ | ⩽ 1, \end{matrix}$ in which improves the standard inequality given by the energy equation ( $σ = 1$ ): $\begin{matrix} (1.5) & | ξ |^{2 σ} | \hat{u} (t) |^{2} ≲ | ξ |^{2 σ} | {\hat{u}}_{0} |^{2} + | {\hat{u}}_{1} |^{2}, for all | ξ | ⩽ 1 . \end{matrix}$ The comparison between the powers $2 σ = 2$ and $4 θ$ lead us to separate in two cases: $0 ⩽ θ < \frac{1}{2}$ and $\frac{1}{2} ⩽ θ ⩽ 1 = σ$ . The first case is when inequality (1.4) gives an improvement of inequality (1.5) and second case is when inequality (1.5) is sufficient to obtain the optimal decay rates.

The method developed by Charão–da Luz–Ikehata [2] was also applied in an abstract second order equation [6] and further in a plate equation with an increasing time-dependent coefficient [7]. In the last case, the decay rates using the energy method hold for a general increasing function but the equivalent inequality (1.4) was not sufficient to ensure the (almost) optimal decay rate for the particular case $\begin{matrix} (1.6) & b (t) = μ {(1 + t)}^{α}, μ > 0 and α \in (0, 1] \end{matrix}$ in some cases of θ. However, in the same work, they also considered the particular case (1.6), obtaining optimal decay rate by using the diagonalization procedure. The enhancement for this specific case cast doubts concerning the improvement of the standard inequality given by energy inequality (1.5) (with $σ = 2$ in the case of plate equation) for a time-dependent context. This conclusion lead us to consider the steps in diagonalization procedure to get some relevant information for a better understanding of the problem.

The diagonalization procedure was successfully used in several papers to obtain decay rates for equation (1.1)–(1.2). For instance, Wirth in [25] and [26] considered this equation with $σ = 1$ , $θ = 0$ and $b (t)$ allowing small oscillations but closely related to $μ {(1 + t)}^{α}$ with $α \in [- 1, 1)$ . A less restrictive oscillations for b, that is, less control on $\frac{d}{d t} b$ was obtained by Hirosawa–Wirth [11] but still not too much general as we would like. For $σ = 1$ , $θ \in (0, 1)$ and $b (t) = μ {(1 + t)}^{α}$ , diagonalization procedure was used by Lu–Reissig [18] (decreasing case) and by Reissig [19] (increasing case). More recently, the result was extended by Kainane–Reissig [15] and [16] for $σ > 1$ , $θ \in (0, σ)$ and b satisfying suitable conditions but very similar to $μ {(1 + t)}^{α} {ln}^{γ} (1 + t)$ and requiring a high control on $\frac{d}{d t} b$ .

All the mentioned papers not only show the interest in equation (1.1)–(1.2) but also reveal a good acceptance of the diagonalization procedure as a suitable method. However, it should noticed that this method usually requires considerable control on oscillations of b. On the other hand, the method due to Charão-da Luz–Ikehata [2,6] and [7] in general is not enough to obtain the optimal decay rates in the case of a time-dependent coefficients, moreover only $L^{2}$ norms estimates are possible.

In addition, is well known (see [9,10,20,21]) that oscillations in the coefficient can deteriorate or even destroy the decay structure of the equation: $\begin{matrix} (1.7) & u_{t t} - a^{2} (t) Δ u = 0 . \end{matrix}$ Without control over oscillations of $a^{2}$ , it is also possible to show results of blow-up of solution of equation (1.7) (see [4,5]). Under suitable conditions, defining $A (t) : = 1 + \int_{0}^{t} a (s) d s$ and $v (t, x) : = u (A^{- 1} (t), x)$ , equation (1.7) is transformed into: $\begin{matrix} (1.8) & v_{t t} - Δ v + \tilde{b} (t) v_{t} = 0, \end{matrix}$ where $\tilde{b} (t) : = \frac{a^{'} (A^{- 1} (t))}{a^{2} (A^{- 1} (t))}$ . Therefore, taking in account the results concerning equation (1.7) and its relation with (1.8), it was not clear if equation (1.1)–(1.2) admits decay rates allowing substantial oscillations for b and if it has influence in the decay.

In this context, in this paper we provide an answer to this question. Thereby, the objective of this work is to develop a method (in the spirit on the works cited above), to obtain sharp decay rates $L^{p} - L^{q}$ for the solution of (1.1)–(1.2), with $1 ⩽ p ⩽ 2 ⩽ q ⩽ \infty$ , considering only $b (t) \sim {(1 + t)}^{α} {ln}^{γ} (1 + t) = : g (t)$ and b non-negative, that is, no control in $\frac{d}{d t} b$ will be assumed. In particular, we will prove that $\frac{d}{d t} b$ has no influence in the decay rates. Going back to the relation between equations (1.7) and (1.8), our hypothesis $b \sim g$ does not contradict the results concerning the control over the coefficient of (1.7). Indeed, $\tilde{b} (t) : = \frac{a^{'} (A^{- 1} (t))}{a^{2} (A^{- 1} (t))}$ and $\tilde{b} \sim g$ implies in $a^{'} (A^{- 1} (t)) \sim a^{2} (A^{- 1} (t)) g (t)$ , that is, we still have some control in the oscillations of the function a.

Furthermore, it should be noticed that this work can be extended for a more general class of functions g, but for the sake of brevity, we avoid this extension since there is specific calculations required depending on g (for example, see Proposition 2.4). In addition, our method can be applied to other equations, for example, plate equation under effects of rotational inertia.

To develop our method, we go back to the origin of the energy method in Fourier space but at the same time considering the knowledge provided by the diagonalization procedure and the method due to Charão-da Luz–Ikehata. For this sake, we consider hyperbolic and elliptic zones similarly as considered in the diagonalization procedure, see for example [14,15] and [16]. In the diagonalization procedure the zones came from WKB analysis, in our case the zones comes together with an energy multiplier, that is, comes from an algebraic understanding of the problem (see Proposition 2.1, in which has been motivated by the method due to Charão–da Luz–Ikehata [2] and [6]). For each ξ such that $0 < | ξ | ⩽ R$ , we consider $ψ (ξ) : = | ξ |^{2 θ} g (t_{ξ})$ , where $t_{ξ}$ separates low zone from elliptic and hyperbolic zones (see Section 2 for further details). We introduce a new classification based on the comparison between $| ξ |^{σ}$ and $ψ (ξ)$ . The aim of this paper is to investigate the case $max {| ξ |^{σ}, ψ (ξ)} = | ξ |^{σ}$ for small frequency in which correspond to our assumptions made on θ, σ, α and γ in Hypothesis A. The remaining case requires an improvement of the energy method and will be treated in a forthcoming paper [24].

Throughout this paper, we do not discuss the existence of solution to (1.1)–(1.2). Therefore, in addition to the conditions aforementioned, we assume suitable condition on b, $u_{0}$ and $u_{1}$ that ensure existence of solutions u and $u_{t}$ that make possible the method described in this work.

The main Theorem of this paper is the following:
Theorem 1.1.
Let $n ⩾ 1$ and θ, σ, α, γ satisfying Hypothesis A . Let $1 ⩽ p ⩽ 2 ⩽ q ⩽ \infty$ , $β, μ \in N^{n}$ and β satisfying $| β | + n (\frac{1}{p} - \frac{1}{q}) > σ$ if $u_{1} \neq 0$ . Consider the function $\begin{matrix} (1.9) & φ (t) = \{\begin{matrix} {(1 + t)}^{1 + α} {ln}^{γ} (1 + t), & if α > - 1, \\ {ln}^{1 + γ} (1 + t), & if α = - 1 and γ > - 1, \\ ln (ln (1 + t)), & if α = - 1 and γ = - 1 . \end{matrix} \end{matrix}$

Then there exists $t_{0}^{} (θ, σ, α, γ, β, μ, t_{0}) ⩾ t_{0}$ , such that the solution* $u (t, x)$ of ( 1.1 )–( 1.2 ) satisfies, for all $t ⩾ t_{0}^{}$ :
If $0 < θ < σ$ and* $u_{0}, u_{1} \in L^{p} (R^{n})$ , then $\begin{array}{l} ‖ D^{β} u (t, \cdot) ‖_{L^{q}} ≲ φ {(t)}^{- \frac{| β | + n (\frac{1}{p} - \frac{1}{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}} + φ {(t)}^{- \frac{| β | - σ + n (\frac{1}{p} - \frac{1}{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}, \\ ‖ D^{μ} u_{t} (t, \cdot) ‖_{L^{q}} ≲ φ {(t)}^{- \frac{| μ | + σ + n (\frac{1}{p} - \frac{1}{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}} + φ {(t)}^{- \frac{| μ | + n (\frac{1}{p} - \frac{1}{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}} . \end{array}$

If $θ = σ$ , $ω > n (\frac{1}{p} - \frac{1}{q})$ , $m : = max {| μ |, | β | - σ} + ω$ and $[u_{0}, u_{1}] \in W^{m + σ, p} (R^{n}) \times W^{m, p} (R^{n})$ , then $\begin{array}{l} ‖ D^{β} u (t, \cdot) ‖_{L^{q}} ≲ φ {(t)}^{- \frac{| β | + n (\frac{1}{p} - \frac{1}{q})}{2 θ}} ‖ u_{0} ‖_{W^{| β | + ω, p}} + φ {(t)}^{- \frac{| β | - σ + n (\frac{1}{p} - \frac{1}{q})}{2 θ}} ‖ u_{1} ‖_{W^{| β | - σ + ω, p}}, \\ ‖ D^{μ} u_{t} (t, \cdot) ‖_{L^{q}} ≲ φ {(t)}^{- \frac{| μ | + σ + n (\frac{1}{p} - \frac{1}{q})}{2 θ}} ‖ u_{0} ‖_{W^{| μ | + σ + ω, p}} + φ {(t)}^{- \frac{| μ | + n (\frac{1}{p} - \frac{1}{q})}{2 θ}} ‖ u_{1} ‖_{W^{| μ | + ω, p}} . \end{array}$

It should be noticed however, that it is possible to improve, in the case $σ = θ$ , the regularity of the last inequality requiring $u_{0} \in W^{| μ | + ω, p} (R^{n})$ and $u_{1} \in W^{k + ω, p} (R^{n})$ , where $k = max {| μ | - σ, 0}$ . For the case $b = 1$ and $σ = 1$ , this result is already known (see [22]). The reason for assuming this technical hypothesis, it is because we avoid to separate high zone in elliptic and hyperbolic zones, as we have made in low frequency (see next section). In addition, other results are required to treat this case. Therefore, for the sake of brevity, we postpone this more complete approach for $θ = σ$ for a forthcoming paper.

The rest of the paper is devoted to the proof Theorem 1.1. Our plan is the following: in Section 2 the step-by-step pointwise estimate are derived in Fourier space, in Section 3 the integration in ξ and the proof of Theorem 1.1. Finally in the end of the paper we include an appendix with some useful tools.
2. Main estimates in the Fourier space

In this section, we can assume that the initial data are sufficiently smooth and apply the density argument. Let $u = u (t, x)$ be the corresponding solution of (1.1)–(1.2).

We take the Fourier transform in the both sides of (1.1). Then in the Fourier space one has the reduced equation: $\begin{matrix} (2.1) & {\hat{u}}_{t t} (t, ξ) + | ξ |^{2 σ} \hat{u} (t, ξ) + b (t) | ξ |^{2 θ} {\hat{u}}_{t} (t, ξ) = 0, (t, ξ) \in (0, \infty) \times R^{n} . \end{matrix}$ The corresponding initial data are given by $\begin{matrix} (2.2) & \hat{u} (0, ξ) = {\hat{u}}_{0} (ξ), {\hat{u}}_{t} (0, ξ) = {\hat{u}}_{1} (ξ), ξ \in R^{n} . \end{matrix}$

Throughout this section we shall omit the dependence in ξ inside the functions $\hat{u} (t) = \hat{u} (t, ξ)$ , ${\hat{u}}_{t} (t) = {\hat{u}}_{t} (t, ξ)$ and the energy density $E (t) = E (t, ξ)$ , which will be defined further ahead.

When we obtain important estimates in order to prove our results, we apply the multiplier method in Fourier space. Take $Z \subset [0, \infty) \times R^{n}$ and $K : Z \to [0, \infty)$ . We multiply both sides of (2.1) by ${\overline{\hat{u}}}_{t}$ and further by $K (t, ξ) \overline{\hat{u}}$ . Then, taking the real part of the resulting identities we have (formally): $\begin{matrix} (2.3) & \frac{1}{2} \frac{d}{d t} {| {\hat{u}}_{t} (t) |^{2} + | ξ |^{2 σ} | \hat{u} (t) |^{2}} + b (t) | ξ |^{2 θ} | {\hat{u}}_{t} (t) |^{2} = 0 \end{matrix}$ and $\begin{array}{l} \frac{d}{d t} {K (t, ξ) Re ({\hat{u}}_{t} (t) \overline{\hat{u}} (t))} + b (t) K (t, ξ) | ξ |^{2 θ} Re ({\hat{u}}_{t} (t) \overline{\hat{u}} (t)) + K (t, ξ) | ξ |^{2 σ} | \hat{u} (t) |^{2} \\ (2.4) & = K (t, ξ) | {\hat{u}}_{t} (t) |^{2} + {\frac{d}{d t} K (t, ξ)} Re ({\hat{u}}_{t} (t) \overline{\hat{u}} (t)), \end{array}$ for each $(t, ξ) \in Z$ that it makes sense. We define the energy density as: $\begin{matrix} E (t) : = \frac{1}{2} {| {\hat{u}}_{t} (t) |^{2} + | ξ |^{2 σ} | \hat{u} (t) |^{2}} \forall t ⩾ 0 . \end{matrix}$ By integrating the equation (2.3) in $[S, T]$ , it follows: $\begin{matrix} (2.5) & E (T) + \int_{S}^{T} b (s) | ξ |^{2 θ} | {\hat{u}}_{t} (s) |^{2} d s = E (S) . \end{matrix}$

Proposition 2.1.
Let $\hat{u} = \hat{u} (t, ξ)$ the solution of ( 2.1 )–( 2.2 ), $Z \subset [0, \infty) \times R^{n}$ and $K : Z \to [0, \infty)$ , where $K (t, \cdot)$ is mensurable and $K (\cdot, ξ)$ is a $C^{1}$ piecewise function for each $(t, ξ) \in Z$ . Suppose that there exist $λ_{1}$ , $λ_{2}$ and $λ_{3} ⩾ 0$ such that K and Z satisfies, for all $(t, ξ) \in Z$ :
If $(s_{1}, ξ), (s_{2}, ξ) \in Z$ then $(s, ξ) \in Z$ for all $s_{1} ⩽ s ⩽ s_{2}$ ;

$K (t, ξ) ⩽ λ_{1} b (t) | ξ |^{2 θ}$ ;

$b (t) K (t, ξ) | ξ |^{2 θ} ⩽ λ_{2} | ξ |^{2 σ}$ ;

${(\frac{d}{d t} K (t, ξ))}^{2} ⩽ λ_{3} b (t) K (t, ξ) | ξ |^{2 θ + 2 σ}$ .

Then, there exists $C > 0$ such that: $\begin{matrix} (2.6) & \int_{S}^{T} K (s, ξ) | ξ |^{2 σ} | \hat{u} (s) |^{2} d s ⩽ C E (S) \forall (S, ξ), (T, ξ) \in Z such that T > S . \end{matrix}$
Proof.
Fix $(S, ξ), (T, ξ) \in Z$ such that $T > S$ . Since inequality (2.6) is trivial if $ξ = 0$ , we suppose $ξ \neq 0$ . By hypothesis (0), $(s, ξ) \in Z$ for all $s \in [S, T]$ .

Observe that conditions (1) and (2) imply that $K (s, ξ) ⩽ \sqrt{λ_{1} λ_{2}} | ξ |^{σ}$ for all $(s, ξ) \in Z$ . Thus, $\begin{matrix} (2.7) & | K (s, ξ) Re ({\hat{u}}_{t} (s) \overline{\hat{u}} (s)) | ⩽ \sqrt{λ_{1} λ_{2}} | Re ({\hat{u}}_{t} (s) | ξ |^{σ} \overline{\hat{u}} (s)) | ⩽ \sqrt{λ_{1} λ_{2}} E (s) \forall s \in [S, T] . \end{matrix}$

By (1), (2) and by energy density (2.5): $\begin{matrix} (2.8) & \int_{S}^{T} K (s, ξ) | {\hat{u}}_{t} (s) |^{2} d s ⩽ λ_{1} \int_{S}^{T} b (s) | ξ |^{2 θ} | {\hat{u}}_{t} (s) |^{2} d s ⩽ λ_{1} E (S) \end{matrix}$ and $\begin{array}{rcl} | \int_{S}^{T} b (s) K (s, ξ) | ξ |^{2 θ} Re ({\hat{u}}_{t} (s) \overline{\hat{u}} (s)) d s | & ⩽ & λ_{2} \int_{S}^{T} b (s) | ξ |^{2 θ} | {\hat{u}}_{t} (s) |^{2} d s \\ + \frac{1}{4 λ_{2}} \int_{S}^{T} b (s) K^{2} (s, ξ) | ξ |^{2 θ} | \hat{u} (s) |^{2} d s \\ (2.9) & ⩽ & λ_{2} E (S) + \frac{1}{4} \int_{S}^{T} K (s, ξ) | ξ |^{2 σ} | \hat{u} (s) |^{2} d s . \end{array}$

Using hypothesis (3), we have: $\begin{array}{l} \int_{S}^{T} {\frac{d}{d t} K (s, ξ)} Re ({\hat{u}}_{t} (s) \overline{\hat{u}} (s)) d s \\ = \int_{S}^{T} b (s) | ξ |^{2 θ} Re ({\hat{u}}_{t} (s) {\frac{\frac{d}{d t} K (s, ξ)}{b (s) | ξ |^{2 θ}}} \overline{\hat{u}} (s)) d s \\ ⩽ λ_{3} \int_{S}^{T} b (s) | ξ |^{2 θ} | {\hat{u}}_{t} (s) |^{2} d s + \frac{1}{4 λ_{3}} \int_{S}^{T} \frac{{(\frac{d}{d t} K (s, ξ))}^{2}}{b (s) | ξ |^{2 θ}} | \hat{u} (s) |^{2} d s \\ (2.10) & ⩽ λ_{3} E (S) + \frac{1}{4} \int_{S}^{T} K (s, ξ) | ξ |^{2 σ} | \hat{u} (s) |^{2} d s . \end{array}$ By integrating equation (2.4) and applying inequalities (2.7)–(2.10) we have: $\begin{array}{rcl} \int_{S}^{T} K (s, ξ) | ξ |^{2 σ} | \hat{u} (s) |^{2} d s ⩽ \frac{C}{2} E (S) + \frac{1}{2} \int_{S}^{T} K (s, ξ) | ξ |^{2 σ} | \hat{u} (s) |^{2} d s, \end{array}$ with $C : = 2 (λ_{1} + 2 \sqrt{λ_{1} λ_{2}} + λ_{2} + λ_{3})$ . This finishes the proof. □
Definition 2.1.
We say that $K : Z \to [0, \infty)$ is a multiplier of energy in $Z \subset [0, \infty) \times R^{n}$ if K and Z satisfy the conditions of Proposition 2.1.

If we had a global multiplier of energy K with $Z = [0, \infty) \times R^{n}$ and at same time sharp, it would be possible to prove the main results of this work using equation (2.5), Proposition 2.1, Lemma 2.1 and Proposition 2.3. Even though global multiplier are possible, they usually do not lead us to sharp decay rates. Therefore, we will separate the problem in zones to find the sharp multiplier of energy in each region. It is worth to highlight that these zones are pretty similar to the zone in works that use the diagonalization procedure, see [14,15] or [16] for example.

Let φ as defined in Theorem 1.1. We fix $R > 0$ whose choice will be clear in the course of the paper. It will satisfy Remark 2.3 and, in addition, will satisfy the following restrictions if $α \in (- 1, 1)$ : fulfill inequalities (2.16) and (2.17) if $γ < 0$ , $R < 1$ if $γ = 0$ , and realizes inequality (2.18) if $γ > 0$ . Furthermore, except for the case $σ (1 + α) = 2 θ$ and $γ = 0$ , if g is increasing we assume also the inequality (2.19).

For each $ξ \in B_{R} : = {ξ \in R^{n} ∖ {0} : | ξ | ⩽ R}$ , we define $t_{ξ} : = φ^{- 1} (| ξ |^{- 2 θ})$ . We consider $ψ : B_{R} \to [0, \infty)$ defined by $\begin{matrix} (2.11) & ψ (ξ) : = | ξ |^{2 θ} g (t_{ξ}) . \end{matrix}$

We shall deal with the problem using the following separation zones:

High Zone: $Z^{high} : = {(t, ξ) \in [t_{0}, \infty) \times R^{n} : | ξ | ⩾ R}$ ;

Hyperbolic Zone: $Z_{hyp} : = {(t, ξ) \in [t_{0}, \infty) \times B_{R} : | ξ |^{2 θ} φ (t) ⩾ 1 and | ξ |^{σ - 2 θ} ⩾ g (t)}$ ;

Elliptic Zone: $Z_{ell} : = {(t, ξ) \in [t_{0}, \infty) \times B_{R} : | ξ |^{2 θ} φ (t) ⩾ 1 and | ξ |^{σ - 2 θ} ⩽ g (t)}$ ;

Low Zone: $Z_{low} : = {(t, ξ) \in [t_{0}, \infty) \times B_{R} : | ξ |^{2 θ} φ (t) ⩽ 1}$ .
Remark 2.1.
The number $t_{0}$ is chosen such that $φ (t_{0}) > 1$ , $g (t_{0}) ⩽ 1$ if g is non-increasing, $g (t_{0}) ⩾ 1$ if g is increasing, $b \sim g$ for $t ⩾ t_{0}$ , $| g^{'} (t) | ⩽ (\frac{| α | + | γ |}{1 + t}) g (t)$ for $t ⩾ t_{0}$ and such that φ and g are monotone (without change of monotonicity) for $t ⩾ t_{0}$ . Furthermore, throughout this paper we will assume $t_{0}$ big enough to ensure the application of the results of the Appendix.

The next proposition provides us the multiplier of energy in each zone, with exception to low zone. Actually, it is possible to find a multiplier of the energy in this zone, but it is not necessary. This is because we have the frequency variable satisfying: $| ξ | ⩽ φ {(t)}^{- \frac{1}{2 θ}}$ . Using boundness for $| \hat{u} |$ and $| {\hat{u}}_{t} |$ (given for example by equation (2.5)) and by integrating in ξ in this region, a natural decay rates appear due to the radius $φ {(t)}^{- \frac{1}{2 θ}}$ . This process is made in details in Proposition 3.1. Furthermore, we do not consider $Z_{ell}$ for $α = - 1$ because it is empty if $t_{0}$ is big enough.
Proposition 2.2.
We have the following multipliers of energy:
Hyperbolic Zone: In $Z_{hyp}$ , $K (t, ξ) : = g (t) | ξ |^{2 θ}$ ;

Elliptic Zone: In $Z_{ell}$ and $α \neq - 1$ , $K (t, ξ) : = \frac{1}{g (t)} | ξ |^{2 σ - 2 θ}$ ;

High Zone: In $Z^{high}$ , $K (t, ξ) = min {\frac{1}{g (t)}, g (t)} | ξ |^{min {2 σ - 2 θ, 2 θ}}$ .

Proof.
In each zone, we must verify conditions of Proposition 2.1. We notice that condition (0) is satisfied by the fact that g and φ are monotone. Also, we have $g \sim b$ and $| g^{'} (t) | ⩽ \frac{c_{0}}{(1 + t)} g (t)$ in $[t_{0}, \infty)$ . For $α \neq - 1$ , we have $φ (t) = (1 + t) g (t)$ and therefore: $\begin{matrix} (2.12) & \frac{1}{(1 + t)} ⩽ g (t) | ξ |^{2 θ}, \end{matrix}$ for all $(t, ξ) \in Z_{ell} \cup Z_{hyp}$ . For $α = - 1$ (and $θ \neq 0$ ), we have: $\begin{matrix} (2.13) & | ξ |^{σ} (1 + t) = {[| ξ |^{2 θ} {(1 + t)}^{\frac{2 θ}{σ}}]}^{\frac{σ}{2 θ}} ≳ {[| ξ |^{2 θ} φ (t)]}^{\frac{σ}{2 θ}} ≳ 1, \end{matrix}$ for all $(t, ξ) \in Z_{hyp}$ . In $Z^{high}$ similar inequality is also true: $\begin{matrix} (2.14) & | ξ |^{σ} (1 + t) ⩾ R^{σ} . \end{matrix}$

To verify condition (3) of Proposition 2.1, we use inequalities (2.12), (2.13) and (2.14).

First consider the hyperbolic zone: $\begin{array}{l} (1) K (t, ξ) = g (t) | ξ |^{2 θ} ≲ b (t) | ξ |^{2 θ}; \\ (2) b (t) K (t, ξ) | ξ |^{2 θ} = b (t) g (t) | ξ |^{4 θ} ≲ g {(t)}^{2} | ξ |^{4 θ} ≲ | ξ |^{2 σ} . \end{array}$

In hyperbolic zone for $α \neq - 1$ , $\begin{matrix} (3) {(\frac{d}{d t} K (t, ξ))}^{2} & = g^{'} {(t)}^{2} | ξ |^{4 θ} ≲ \frac{1}{{(1 + t)}^{2}} g {(t)}^{2} | ξ |^{4 θ} ≲ g {(t)}^{2} | ξ |^{4 θ} g {(t)}^{2} | ξ |^{4 θ} \\ ≲ g (t) K (t, ξ) | ξ |^{2 θ + 2 σ} ≲ b (t) K (t, ξ) | ξ |^{2 θ + 2 σ} . \end{matrix}$

In hyperbolic zone for $α = - 1$ , $\begin{matrix} (3) {(\frac{d}{d t} K (t, ξ))}^{2} & = g^{'} {(t)}^{2} | ξ |^{4 θ} ≲ \frac{1}{{(1 + t)}^{2}} g {(t)}^{2} | ξ |^{4 θ} ≲ g (t) K (t, ξ) | ξ |^{2 θ + 2 σ} \\ ≲ b (t) K (t, ξ) | ξ |^{2 θ + 2 σ} . \end{matrix}$

Now, let us consider the elliptic zone with $α \neq - 1$ : $\begin{array}{l} (1) K (t, ξ) = \frac{1}{g (t)} | ξ |^{2 σ - 2 θ} ⩽ g (t) | ξ |^{2 θ} ≲ b (t) | ξ |^{2 θ}; \\ (2) b (t) K (t, ξ) | ξ |^{2 θ} = \frac{b (t)}{g (t)} | ξ |^{2 σ} ≲ | ξ |^{2 σ}; \\ (3) {(\frac{d}{d t} K (t, ξ))}^{2} = \frac{g^{'} {(t)}^{2}}{g {(t)}^{4}} | ξ |^{4 σ - 4 θ} ≲ \frac{| ξ |^{4 σ}}{{(1 + t)}^{2} g {(t)}^{2} | ξ |^{4 θ}} ≲ | ξ |^{4 σ} ≲ b (t) K (t, ξ) | ξ |^{2 θ + 2 σ} . \end{array}$

Finally, consider $Z^{high}$ . In this region, the right side of the inequalities can depend on R:
$K (t, ξ) ≲ g (t) | ξ |^{2 θ} ≲ b (t) | ξ |^{2 θ}$ ;

$b (t) K (t, ξ) | ξ |^{2 θ} ≲ \frac{b (t)}{g (t)} | ξ |^{2 σ} ≲ | ξ |^{2 σ}$ ;

In this region, we have: $\begin{array}{l} {(\frac{d}{d t} g (t))}^{2} ≲ \frac{g {(t)}^{2}}{{(1 + t)}^{2}} ≲ g {(t)}^{2} | ξ |^{2 σ}; \\ {(\frac{d}{d t} \frac{1}{g (t)})}^{2} ≲ \frac{g^{'} {(t)}^{2}}{g {(t)}^{4}} ≲ \frac{1}{g {(t)}^{2} {(1 + t)}^{2}} ≲ \frac{1}{g {(t)}^{2}} | ξ |^{2 σ} . \end{array}$

Therefore, $\begin{array}{rcl} {(\frac{d}{d t} K (t, ξ))}^{2} & ≲ & min {\frac{1}{g {(t)}^{2}}, g {(t)}^{2}} | ξ |^{2 σ} | ξ |^{2 min {2 σ - 2 θ, 2 θ}} \\ ≲ & b (t) min {\frac{1}{g (t)}, g (t)} | ξ |^{min {2 σ - 2 θ, 2 θ}} | ξ |^{min {4 σ - 2 θ, 2 θ + 2 σ}} \\ ≲ & b (t) K (t, ξ) | ξ |^{2 θ + 2 σ} . \end{array}$ □

Note that in $Z^{high}$ we have: $\begin{array}{l} | ξ |^{min {2 θ, 2 σ - 2 θ}} = \{\begin{matrix} | ξ |^{2 θ} & if σ ⩾ 2 θ \\ | ξ |^{2 σ - 2 θ} & if σ < 2 θ, \end{matrix} \\ min {\frac{1}{g (t)}, g (t)} = \{\begin{matrix} \frac{1}{g (t)} & if g is increasing \\ g (t) & if g is non-increasing. \end{matrix} \end{array}$

This allow us to calculate the multiplier in $Z^{high}$ in each case. Moreover, this multiplier can be improved but we avoid this procedure. Even though the multiplier is not sharp, the decay rates obtained in the high zone are better than the decay rates of the another regions. For another class of equations this point must be considered and improved.

The following lemma plays a fundamental role in order to prove Proposition 2.3. This result is a suitable adaptation of some ideas of [17].
Lemma 2.1.
Let $E : [S_{0}, \infty) \to [0, \infty)$ differentiable and non-increasing, $f : [S_{0}, T_{0}) \to [0, \infty)$ continuous, where $S_{0} \in R$ and $S_{0} < T_{0} \in R \cup {\infty}$ . Suppose that there exists $C > 0$ such that $\begin{matrix} \int_{S}^{T_{0}} f (s) E (s) d s ⩽ C E (S), \forall S \in [S_{0}, T_{0}), \end{matrix}$ then, for every $0 < ϵ < 1$ holds: $E (t) ⩽ \frac{E (S_{0})}{1 - ϵ} e^{- \frac{ϵ}{C} \int_{S_{0}}^{t} f (s) d s}$ , for all $t \in [S_{0}, T_{0})$ .
Proof.
Define $ρ (t) : = \frac{1}{C} \int_{t}^{T_{0}} f (s) E (s) d s$ for $t \in [S_{0}, T_{0})$ . For $0 < ϵ < 1$ , consider the “Lyapunov” function $L (t) : = (1 - ϵ) E (t) + ϵ ρ (t)$ , for $t \in [S_{0}, T_{0})$ . Therefore: $\begin{matrix} L^{'} (t) = (1 - ϵ) E^{'} (t) + ϵ ρ^{'} (t) ⩽ - \frac{ϵ}{C} f (t) E (t) ⩽ - \frac{ϵ}{C} f (t) L (t), for t \in [S_{0}, T_{0}) . \end{matrix}$

That is, $L (t) ⩽ L (S_{0}) e^{- \frac{ϵ}{C} \int_{S_{0}}^{t} f (s) d s}$ for $t \in [S_{0}, T_{0})$ . Since $(1 - ϵ) E (t) ⩽ L (t) ⩽ E (t)$ , the result follows. □
Proposition 2.3.
For a fixed $ξ \in R^{n}$ and given zone Z, we define $S_{0} (ξ) = inf {s \in [t_{0}, \infty) : (s, ξ) \in Z}$ and $T_{0} (ξ) = sup {s \in [t_{0}, \infty) : (s, ξ) \in Z}$ . Then, there exists $C > 0$ independent of ξ such that:
$E (t) ≲ e^{- \frac{1}{C} | ξ |^{min {2 σ - 2 θ, 2 θ}} \int_{t_{0}}^{t} min {\frac{1}{g (s)}, g (s)} d s} E (0)$ for all $(t, ξ) \in Z^{high}$ .

If $Z_{hyp}$ has no zero measure, $E (t) ≲ e^{- \frac{1}{C} | ξ |^{2 θ} \int_{S_{0} (ξ)}^{t} g (s) d s} E (S_{0} (ξ))$ for all $S_{0} (ξ) ⩽ t < T_{0} (ξ)$ .

If $Z_{ell}$ has no zero measure, $E (t) ≲ e^{- \frac{1}{C} | ξ |^{2 σ - 2 θ} \int_{S_{0} (ξ)}^{t} \frac{1}{g (s)} d s} E (S_{0} (ξ))$ for all $S_{0} (ξ) ⩽ t < T_{0} (ξ)$ .

Proof.
Let $K : Z \to [0, \infty)$ a multiplier of energy in $Z \subset [0, \infty) \times R^{n}$ . By Proposition 2.1, we know that: $\begin{matrix} \int_{S}^{T} K (s, ξ) | ξ |^{2 σ} | \hat{u} (s) |^{2} d s ≲ E (S), \forall (S, ξ), (T, ξ) \in Z with T > S . \end{matrix}$ Further, by equation (2.5) and by property (1) of multiplier of energy: $\begin{matrix} \int_{S}^{T} K (s, ξ) | \hat{u_{t}} (s) |^{2} d s ≲ \int_{S}^{T} b (s) | ξ |^{2 θ} | \hat{u_{t}} (s) |^{2} d s ⩽ E (S), \forall (S, ξ), (T, ξ) \in Z with T > S . \end{matrix}$

Therefore $\int_{S}^{T} K (s, ξ) E (s) d s ≲ E (S)$ , for all $(S, ξ), (T, ξ) \in Z$ with $T > S$ . Fixing $ξ \in R^{n}$ , such that ${t \in [t_{0}, \infty) : (t, ξ) \in Z}$ has non-zero measure, we apply Lemma 2.1 and conclude: $\begin{matrix} (2.15) & E (t) ≲ e^{- \frac{1}{C} \int_{S_{0} (ξ)}^{t} K (s, ξ) d s} E (S_{0} (ξ)), \end{matrix}$ for all $S_{0} (ξ) ⩽ t < T_{0} (ξ)$ . Using Proposition 2.2 we know that the corresponding multiplier of energy in each zone, applying in inequality (2.15) we conclude the result. □
Remark 2.2.
In the last proposition, when $T_{0} (ξ) < \infty$ the inequalities also holds for $t = T_{0} (ξ)$ . Furthermore, the estimates for $E (t) = E (t, ξ)$ are uniform in ξ, that is, the constants in the right side of inequality does not depends on ξ.

A careful analysis of Proposition 2.3 makes us observe another interesting detail: in the first inequality the integral begins in $t_{0}$ while in the remaining inequalities the integral begins in $S_{0} (ξ)$ . Furthermore, the energy in the right side of first inequality is valued in zero, while in the other cases is valued in $S_{0} (ξ)$ . In this context, the ξ independence (in the range of integration of t and in the time variable of E) of the first inequality make the pointwise estimates in Fourier space for $Z^{high}$ ready to be integrated and conclude the estimates (see Proposition 3.3). From now, our idea is improve the estimates in $Z_{hyp}$ and $Z_{ell}$ , in such a way that it has the desired independence on ξ in time variable. Proposition 2.4 will be fundamental for this upgrade.

We have defined $t_{ξ}$ as the unique solution of $| ξ |^{2 θ} φ (t_{ξ}) = 1$ and $ψ (ξ) = | ξ |^{2 θ} g (t_{ξ})$ , and now we want to investigate $max {| ξ |^{σ}, ψ (ξ)}$ in $B_{R}$ . In this paper, we consider the case $max {| ξ |^{σ}, ψ (ξ)} = | ξ |^{σ}$ for small frequencies. For the remaining case, it is necessary an improvement of the estimates in elliptic zone using a substantially different method, this is the reason why we do not treat both cases in this work. Furthermore, this justify the new classification introduced based in that maximum. In this sense, the restrictions assumed in Proposition 2.4 comes from this classification.
Proposition 2.4.
Let $α > - 1$ , γ, θ and σ satisfying Hypothesis A . Thus, there exist small $R > 0$ such that $ψ (ξ) < | ξ |^{σ}$ for all ξ in $B_{R}$ , except for the case $γ = 0$ and $σ (1 + α) = 2 θ$ such that holds $ψ (ξ) = | ξ |^{σ}$ for all $ξ \in B_{R}$ .
Proof.
For $ξ \neq 0$ , let $t_{ξ}$ be the unique solution of the equation ${(1 + t_{ξ})}^{1 + α} {ln}^{γ} (1 + t_{ξ}) = | ξ |^{- 2 θ}$ . Applying Lemma B.4 with $τ = 1 + t_{ξ}$ , $μ = 1 + α$ , $β = γ$ and $λ = | ξ |^{- 2 θ}$ , we have for $α \neq - 1$ : $\begin{matrix} (1 + t_{ξ}) = \{\begin{matrix} {(\frac{1 + α}{| γ |})}^{\frac{γ}{1 + α}} | ξ |^{- \frac{2 θ}{1 + α}} {[- W_{- 1} (- \frac{1 + α}{| γ |} | ξ |^{\frac{2 θ}{| γ |}})]}^{- \frac{γ}{1 + α}} & if γ < 0, \\ | ξ |^{- \frac{2 θ}{1 + α}} & if γ = 0, \\ {(\frac{1 + α}{γ})}^{\frac{γ}{1 + α}} | ξ |^{- \frac{2 θ}{1 + α}} {[W_{0} (\frac{1 + α}{γ} | ξ |^{- \frac{2 θ}{γ}})]}^{- \frac{γ}{1 + α}} & if γ > 0, \end{matrix} \end{matrix}$ where $W_{0}$ , $W_{- 1}$ are the two real-valued branches of W-Lambert’s function (see Appendix B for further details concerning this special function). To carefully apply Lemma B.4, we need to consider the following condition: $\begin{matrix} (2.16) & R < {(\frac{(1 + α) e}{| γ |})}^{- \frac{| γ |}{2 θ}}, if γ < 0 . \end{matrix}$

Case $γ < 0$ : In this case $σ ⩽ \frac{2 θ}{1 + α}$ . Since $ψ (ξ) = \frac{| ξ |^{2 θ} φ (t_{ξ})}{1 + t_{ξ}} = \frac{1}{1 + t_{ξ}}$ , we have: $\begin{matrix} ψ (ξ) = {(\frac{1 + α}{| γ |})}^{\frac{| γ |}{1 + α}} | ξ |^{\frac{2 θ}{1 + α}} {[- W_{- 1} (- \frac{1 + α}{| γ |} | ξ |^{\frac{2 θ}{| γ |}})]}^{- \frac{| γ |}{1 + α}}, \end{matrix}$ for $ξ \in B_{R}$ , R small enough. By Corollary B.1, we have the limit: $\begin{matrix} lim_{r \to 0^{+}} r^{σ - \frac{2 θ}{1 + α}} {[- W_{- 1} (- \frac{1 + α}{| γ |} r^{\frac{2 θ}{| γ |}})]}^{\frac{| γ |}{1 + α}} = \infty, \end{matrix}$ therefore, there exists $R (γ, α, σ, θ) > 0$ such that: $\begin{matrix} (2.17) & | ξ |^{σ - \frac{2 θ}{1 + α}} {[- W_{- 1} (- \frac{1 + α}{| γ |} | ξ |^{\frac{2 θ}{| γ |}})]}^{\frac{| γ |}{1 + α}} > {(\frac{1 + α}{| γ |})}^{\frac{| γ |}{1 + α}}, \end{matrix}$ for all $0 < | ξ | ⩽ R$ . That is, $ψ (ξ) < | ξ |^{σ}$ for all $ξ \in B_{R}$ .

Case $γ = 0$ : $ψ (ξ) = \frac{1}{1 + t_{ξ}} = | ξ |^{\frac{2 θ}{1 + α}}$ .

If $σ (1 + α) = 2 θ$ , trivially $ψ (ξ) = | ξ |^{σ}$ for $ξ \in B_{R}$ . In the case $σ (1 + α) < 2 θ$ , we have for $R < 1$ , $ψ (ξ) < | ξ |^{σ}$ for all $0 < | ξ | ⩽ R$ .

Case $γ > 0$ : In this case, necessarily $σ < \frac{2 θ}{1 + α}$ . Since $ψ (ξ) = \frac{1}{1 + t_{ξ}}$ , we have: $\begin{matrix} ψ (ξ) = {(\frac{γ}{1 + α})}^{\frac{γ}{1 + α}} | ξ |^{\frac{2 θ}{1 + α}} {[W_{0} (\frac{1 + α}{γ} | ξ |^{- \frac{2 θ}{γ}})]}^{\frac{γ}{1 + α}}, \end{matrix}$ furthermore, by Corollary B.2 the following limit holds: $\begin{matrix} lim_{r \to 0^{+}} \frac{r^{σ - \frac{2 θ}{1 + α}}}{{[W_{0} (\frac{1 + α}{γ} r^{- \frac{2 θ}{γ}})]}^{\frac{γ}{1 + α}}} = + \infty . \end{matrix}$ In this case, there exists $R (γ, α, σ, θ) > 0$ such that: $\begin{matrix} (2.18) & \frac{| ξ |^{σ - \frac{2 θ}{1 + α}}}{{[W_{0} (\frac{1 + α}{γ} | ξ |^{- \frac{2 θ}{γ}})]}^{\frac{γ}{1 + α}}} > {(\frac{γ}{1 + α})}^{\frac{γ}{1 + α}}, \end{matrix}$ for $0 < | ξ | ⩽ R$ . Then, $ψ (ξ) < | ξ |^{σ}$ for all $ξ \in B_{R}$ . □

The last proposition is necessary to treat the estimates in elliptic zone, in special to deal with the separation line between elliptic zone and hyperbolic zone. As mentioned before of Proposition 2.2, the elliptic zone is empty if $α = - 1$ and therefore, even though the same proposition holds in this case, the result of Proposition 2.4 is not necessary.
Remark 2.3.
We know that φ is increasing (therefore $φ^{- 1}$ is also increasing). Thus, if $R < φ {(t_{0})}^{- \frac{1}{2 θ}}$ , we have $t_{ξ} = φ^{- 1} (| ξ |^{- 2 θ}) > φ^{- 1} (φ (t_{0})) = t_{0}$ , for all $ξ \in B_{R}$ . Furthermore, taking in account the definition of the zones, given $ξ \in B_{R}$ , $t ⩾ t_{ξ}$ if, and only if $(t, ξ) \in Z_{ell} \cup Z_{hyp}$ and $t_{0} ⩽ t ⩽ t_{ξ}$ if, and only if $(t, ξ) \in Z_{low}$ . This remark will be widely used in demonstrating the next proposition.

When $| ξ |^{σ - 2 θ}$ is in the domain of $g^{- 1}$ , we define $t_{1} (ξ) : = g^{- 1} (| ξ |^{σ - 2 θ})$ . The number $t_{1} (ξ)$ is precisely the point where occur a change between hyperbolic behaviour and elliptic behaviour. But sometimes $t_{1} (ξ)$ simply does not exists, which means that elliptic zone or hyperbolic zone has zero measure. Since we are interested in applying Proposition 2.3, we must care about the lower $S_{0} (ξ)$ and up bounds $T_{0} (ξ)$ limits of the proposition. Therefore, throughout the demonstration below, the existence of $t_{1} (ξ)$ is discussed only when $t_{1} (ξ)$ plays role to calculate $S_{0} (ξ)$ or $T_{0} (ξ)$ . Proposition 2.5.
There exists $C > 0$ such that the following estimates hold:
If $Z_{ell}$ has no zero measure, $E (t) ≲ e^{- \frac{1}{C} | ξ |^{2 σ - 2 θ} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} E (0)$ , for all $(t, ξ) \in Z_{ell}$ .

If $Z_{hyp}$ has no zero measure, $E (t) ≲ e^{- \frac{1}{C} | ξ |^{2 θ} \int_{t_{0}}^{t} g (s) d s} E (0)$ , for all $(t, ξ) \in Z_{hyp}$ .

Proof.
By Proposition 2.4, we know that $ψ (ξ) ⩽ | ξ |^{σ}$ for all $ξ \in B_{R}$ . Initially we consider the case $ψ (ξ) < | ξ |^{σ}$ for all $ξ \in B_{R}$ and the case $α = - 1$ .

Case $g$ increasing: In this case, we have $0 ⩽ α < 1$ (with $γ > 0$ if $α = 0$ ) thus $σ < 2 θ$ . Since g can be seen as a bijection between $[t_{0}, \infty)$ and $[g (t_{0}), \infty)$ , $t_{1} (ξ)$ is well defined if $| ξ |^{σ - 2 θ} ⩾ g (t_{0})$ for all $ξ \in B_{R}$ , that is, if: $\begin{matrix} (2.19) & R ⩽ g {(t_{0})}^{\frac{1}{σ - 2 θ}} . \end{matrix}$

Since $ψ (ξ) < | ξ |^{σ}$ for all $ξ \in B_{R}$ , it follows directly from the definition of ψ that $t_{ξ} < t_{1} (ξ)$ for all $ξ \in B_{R}$ . Therefore, for each fixed $ξ \in B_{R}$ , $[t_{0}, \infty) = [t_{0}, t_{ξ}] \cup [t_{ξ}, t_{1} (ξ)] \cup [t_{1} (ξ), \infty)$ , where $(s, ξ) \in Z_{low}$ for $s \in [t_{0}, t_{ξ}]$ , $(s, ξ) \in Z_{hyp}$ for $s \in [t_{ξ}, t_{1} (ξ)]$ and $(s, ξ) \in Z_{ell}$ when $s ⩾ t_{1} (ξ)$ . That is, $S_{0} (ξ) = t_{ξ}$ and $T_{0} (ξ) = t_{1} (ξ)$ in hyperbolic zone, $S_{0} (ξ) = t_{1} (ξ)$ and $T_{0} (ξ) = \infty$ in elliptic zone.

By Lemma A.2, $| ξ |^{2 θ} \int_{t_{0}}^{t_{ξ}} g (s) d s ≲ | ξ |^{2 θ} (1 + t_{ξ}) g (t_{ξ}) = | ξ |^{2 θ} φ (t_{ξ}) = 1$ . Applying Proposition 2.3 in the hyperbolic zone we have (for any $C ⩾ C_{2}$ ): $\begin{matrix} (2.20) & E (t) ≲ e^{- \frac{1}{C_{2}} | ξ |^{2 θ} \int_{t_{ξ}}^{t} g (s) d s} E (t_{ξ}) ≲ e^{- \frac{1}{C} | ξ |^{2 θ} \int_{t_{0}}^{t} g (s) d s} E (0), \end{matrix}$ for all $(t, ξ) \in Z_{hyp}$ . On the other hand, choosing $C_{1} ⩾ C_{2} \frac{(1 + α)}{(1 - α)}$ (with $C_{1}$ and $C_{2}$ greater than or equal to the constant appearing in Proposition 2.3) and using Lemma A.2, we have: $\begin{matrix} - \frac{1}{C_{1}} | ξ |^{2 σ - 2 θ} \int_{t_{1} (ξ)}^{t} \frac{1}{g (s)} d s - \frac{1}{C_{2}} | ξ |^{2 θ} \int_{t_{0}}^{t_{1} (ξ)} g (s) d s ⩽ - \frac{1}{C_{3}} | ξ |^{2 σ - 2 θ} \frac{(1 + t)}{g (t)} + C_{4}, \end{matrix}$ for all $t ⩾ t_{1} (ξ)$ . Thus, using inequality (2.20) in $t = t_{1} (ξ)$ by applying Proposition 2.3 in $Z_{ell}$ : $\begin{array}{rcl} E (t) & ≲ & e^{- \frac{1}{C 1} | ξ |^{2 σ - 2 θ} \int_{t_{1} (ξ)}^{t} \frac{1}{g (s)} d s} E (t_{1} (ξ)) \\ ≲ & e^{- \frac{1}{C_{1}} | ξ |^{2 σ - 2 θ} \int_{t_{1} (ξ)}^{t} \frac{1}{g (s)} d s} e^{- \frac{1}{C_{2}} | ξ |^{2 θ} \int_{t_{0}}^{t_{1} (ξ)} g (s) d s} E (0) \\ ≲ & e^{- \frac{1}{C_{3}} | ξ |^{2 σ - 2 θ} \frac{(1 + t)}{g (t)}} E (0) \\ ≲ & e^{- \frac{1}{C} | ξ |^{2 σ - 2 θ} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} E (0), \end{array}$ for all $(t, ξ) \in Z_{ell}$ .

Case $g$ decreasing: For $α \neq - 1$ , by definition of ψ and due the fact $ψ (ξ) < | ξ |^{σ}$ , follows $t_{1} (ξ) < t_{ξ}$ for all $ξ \in B_{R}$ . Furthermore, this implies $Z_{ell} = \emptyset$ . Applying Proposition 2.3 in hyperbolic zone: $\begin{matrix} E (t) ≲ e^{- \frac{1}{C} | ξ |^{2 θ} \int_{t_{ξ}}^{t} g (s) d s} E (t_{ξ}), \end{matrix}$ for all $(t, ξ) \in Z_{hyp}$ . By Lemma A.2, we have that $| ξ |^{2 θ} \int_{t_{0}}^{t_{ξ}} g (s) d s ≲ | ξ |^{2 θ} (1 + t_{ξ}) g (t_{ξ}) ≲ 1$ , and therefore: $\begin{matrix} E (t) ≲ e^{- \frac{1}{C} | ξ |^{2 θ} \int_{t_{0}}^{t} g (s) d s} E (0), \end{matrix}$ for all $(t, ξ) \in Z_{hyp}$ .

The case $α = - 1$ is similar, applying Corollary A.1 instead of Lemma A.2.

Case $g = 1$ : By Proposition 2.4, we have $σ < 2 θ$ . In this case $Z_{ell} = \emptyset$ . Since $| ξ |^{2 θ} (1 + t_{ξ}) = 1$ , applying Proposition 2.3 in $Z_{hyp}$ we have: $\begin{matrix} E (t) ≲ e^{- \frac{1}{C} | ξ |^{2 θ} \int_{t_{ξ}}^{t} g (s) d s} E (t_{ξ}) ≲ e^{- \frac{1}{C} | ξ |^{2 θ} \int_{t_{0}}^{t} g (s) d s} E (0) \end{matrix}$ for all $(t, ξ) \in Z_{hyp}$ .

Let us consider the case $ψ (ξ) = | ξ |^{σ}$ for all $ξ \in B_{R}$ , that is, by Proposition 2.4 is the case $σ (1 + α) = 2 θ$ , $γ = 0$ and $α \in (- 1, 1)$ . Furthermore, $| ξ |^{σ} = ψ (ξ)$ implies $t_{ξ} = t_{1} (ξ)$ . For g decreasing, the proof is the same as in the case $ψ (ξ) < | ξ |^{σ}$ . When $g = 1$ , the proof is again as in the case $ψ (ξ) < | ξ |^{σ}$ , the only difference is that $Z_{ell} = Z_{hyp}$ .

Finally, for the case g increasing, $Z_{hyp}$ has zero measure. Indeed, if $t ⩾ t_{ξ}$ then $g (t) ⩾ g (t_{ξ}) = g (t_{1} (ξ)) = | ξ |^{σ - 2 θ}$ . Thus $[t_{0}, \infty) = [t_{0}, t_{ξ}] \cup [t_{ξ}, \infty)$ , where $(s, ξ) \in Z_{low}$ if $s \in [t_{0}, t_{ξ}]$ and $(s, ξ) \in Z_{ell}$ if $s \in [t_{ξ}, \infty)$ . Using the Lemma A.2, we have: $\begin{array}{rcl} - \frac{1}{C_{1}} | ξ |^{2 σ - 2 θ} \int_{t_{ξ}}^{t} \frac{1}{g (s)} d s & ⩽ & - \frac{1}{C_{2}} | ξ |^{2 σ - 2 θ} \frac{(1 + t)}{g (t)} + \frac{1}{C_{2}} | ξ |^{2 σ - 2 θ} \frac{(1 + t_{ξ})}{g (t_{ξ})} \\ (2.21) & ≲ & - \frac{1}{C} | ξ |^{2 σ - 2 θ} \int_{t_{0}}^{t} \frac{1}{g (s)} d s + \frac{1}{C_{2}}, \end{array}$ because $| ξ |^{2 σ - 2 θ} \frac{(1 + t_{ξ})}{g (t_{ξ})} = ψ {(ξ)}^{2} \frac{(1 + t_{ξ})}{| ξ |^{2 θ} g (t_{ξ})} = | ξ |^{2 θ} g (t_{ξ}) (1 + t_{ξ}) = | ξ |^{2 θ} φ (t_{ξ}) = 1$ .

By applying Proposition 2.3 and inequality (2.21), we have for all $(t, ξ) \in Z_{ell}$ : $\begin{array}{rcl} E (t) ≲ e^{- \frac{1}{C 1} | ξ |^{2 σ - 2 θ} \int_{t_{ξ}}^{t} \frac{1}{g (s)} d s} E (t_{ξ}) ≲ e^{- \frac{1}{C} | ξ |^{2 σ - 2 θ} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} E (0) . \end{array}$ □

3. Proof of results

In this section we prove the main result of the paper. We have to apply the pointwise estimates in Fourier space of the previous section, fix the time variable and integrate ξ in $R^{n}$ . This procedure is make by considering the zone separation introduced in the beginning of Section 2 and a proof divided in several propositions to deal with each zone. During the step of integration in ξ, we often use results of Appendix A.

To proof the results, we shall consider in this section $\hat{q}$ conjugate of $q \in [2, \infty]$ , that is $\hat{q} \in [1, 2]$ and $\frac{1}{\hat{q}} + \frac{1}{q} = 1$ . Furthermore, $u_{0}, u_{1} \in L^{p} (R^{n})$ and s conjugate of $p \in [1, 2]$ , that is, $s \in [2, \infty]$ and therefore $s ⩾ \hat{q}$ . We define $r : = \infty$ if $p + \hat{q} = p \hat{q}$ and $r : = \frac{p}{p + \hat{q} - p \hat{q}} ⩾ 1$ if $p + \hat{q} \neq p \hat{q}$ . That is, r is conjugated of $\frac{s}{\hat{q}}$ , since $\frac{p + \hat{q} - p \hat{q}}{p} + \frac{\hat{q}}{s} = 1$ . In addition, we take $μ, β \in N^{n}$ and $φ = φ (t)$ given by (1.9) in Theorem 1.1. In this section several times will appear the expression $\frac{n}{p \hat{q}} (p + \hat{q} - p \hat{q})$ , which is equal to $n (\frac{1}{p} - \frac{1}{q})$ that rises in Theorem 1.1.

Proposition 3.1.

Consider the conditions above. Let $u (t, x)$ the solution of ( 1.1 )–( 1.2 ), then the following estimates hold for $t ⩾ t_{0}$ : $\begin{array}{rcl} \int_{Z_{low}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ ≲ φ {(t)}^{- \frac{\hat{q} | μ | + \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | μ | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} . \end{array}$

Let $| β | - σ + \frac{n}{p \hat{q}} (p + q - p \hat{q}) > 0$ if $u_{1} \neq 0$ and any $β \in N^{n}$ if $u_{1} = 0$ , then: $\begin{array}{rcl} \int_{Z_{low}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ ≲ φ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} . \end{array}$

Proof.

Let $τ : = φ {(t)}^{- \frac{1}{2 θ}}$ . Since $| {\hat{u}}_{t} (t) |^{\hat{q}} ≲ | ξ |^{\hat{q} σ} | {\hat{u}}_{0} |^{\hat{q}} + | {\hat{u}}_{1} |^{\hat{q}}$ , we have: $\begin{array}{rcl} \int_{Z_{low}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ & ≲ & \int_{Z_{low}} | ξ |^{\hat{q} | μ | + \hat{q} σ} | {\hat{u}}_{0} |^{\hat{q}} d ξ + \int_{Z_{low}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{1} |^{\hat{q}} d ξ \\ ≲ & {‖ | \cdot |^{\hat{q} | μ | + \hat{q} σ} ‖}_{L^{r} (B_{τ})} ‖ {\hat{u}}_{0} ‖_{L^{s}}^{\hat{q}} + {‖ | \cdot |^{\hat{q} | μ |} ‖}_{L^{r} (B_{τ})} ‖ {\hat{u}}_{1} ‖_{L^{s}}^{\hat{q}} . \end{array}$

Using Hausdorff–Young inequality (see [1]) and Lemma A.3 with $k = \hat{q} | μ | + \hat{q} σ$ for the first term on the right side of above inequality, $k = \hat{q} | μ |$ for the second term, we have $\begin{array}{rcl} \int_{Z_{low}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ ≲ φ {(t)}^{- \frac{\hat{q} | μ | + \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | μ | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}}, \end{array}$ for $t ⩾ t_{0}$ .

Let $| β | - σ + \frac{n}{p \hat{q}} (p + \hat{q} - p \hat{q}) > 0$ if $u_{1} \neq 0$ and any $β \in N^{n}$ if $u_{1} = 0$ . Thus, $\begin{array}{rcl} \int_{Z_{low}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ & ≲ & \int_{Z_{low}} | ξ |^{\hat{q} | β |} | {\hat{u}}_{0} |^{\hat{q}} d ξ + \int_{Z_{low}} | ξ |^{\hat{q} | β | - \hat{q} σ} | {\hat{u}}_{1} |^{\hat{q}} d ξ \\ ≲ & {‖ | \cdot |^{\hat{q} | β |} ‖}_{L^{r} (B_{τ})} ‖ {\hat{u}}_{0} ‖_{L^{s}}^{\hat{q}} + {‖ | \cdot |^{\hat{q} | β | - \hat{q} σ} ‖}_{L^{r} (B_{τ})} ‖ {\hat{u}}_{1} ‖_{L^{s}}^{\hat{q}} . \end{array}$

Using Hausdorff–Young inequality and Lemma A.3 we have for $t ⩾ t_{0}$ : $\begin{array}{rcl} \int_{Z_{low}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ ≲ φ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} . \end{array}$ □

Proposition 3.2.

Under the conditions of Proposition 3.1 , there exists $t_{0}^{*} ⩾ t_{0}$ such that the following estimates hold for $t ⩾ t_{0}^{*}$ : $\begin{array}{l} \int_{Z_{hyp}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ ≲ φ {(t)}^{- \frac{\hat{q} | μ | + \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | μ | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}}; \\ \int_{Z_{ell}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ ≲ φ {(t)}^{- \frac{\hat{q} | μ | + \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | μ | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} . \end{array}$

Let $| β | - σ + \frac{n}{p \hat{q}} (p + \hat{q} - p \hat{q}) > 0$ if $u_{1} \neq 0$ and any $β \in N^{n}$ if $u_{1} = 0$ , then: $\begin{array}{l} \int_{Z_{hyp}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ ≲ φ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}}; \\ \int_{Z_{ell}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ ≲ φ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} . \end{array}$

Proof.

Let $ϕ (t) : = {(1 + t)}^{1 - α} {ln}^{- γ} (1 + t)$ and φ as before. Taking in account the Hypothesis A, we have for $θ \in (0, σ)$ and for all $η > 0$ : $ϕ {(t)}^{- \frac{η}{2 σ - 2 θ}} ≲ φ {(t)}^{- \frac{η}{2 θ}}$ for all $t ⩾ t_{0}$ . The last inequality will be useful for the sake of simplicity.

In $Z_{ell}$ we initially consider $σ \neq θ$ (in the hyperbolic zone this restriction is not necessary). Using Proposition 2.5 and Corollary A.1: $\begin{array}{rcl} \int_{Z_{ell}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ & ≲ & \int_{Z_{ell}} | ξ |^{\hat{q} | μ | + \hat{q} σ} e^{- \frac{\hat{q}}{2 C} | ξ |^{2 σ - 2 θ} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} | {\hat{u}}_{0} |^{\hat{q}} d ξ \\ + \int_{Z_{ell}} | ξ |^{\hat{q} | μ |} e^{- \frac{\hat{q}}{2 C} | ξ |^{2 σ - 2 θ} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} | {\hat{u}}_{1} |^{\hat{q}} d ξ \\ ≲ & {‖ | \cdot |^{\hat{q} | μ | + \hat{q} σ} e^{- c ϕ (t) | \cdot |^{2 σ - 2 θ}} ‖}_{L^{r} (R^{n})} ‖ {\hat{u}}_{0} ‖_{L^{s}}^{\hat{q}} \\ (3.1) & + {‖ | \cdot |^{\hat{q} | μ |} e^{- c ϕ (t) | \cdot |^{2 σ - 2 θ}} ‖}_{L^{r} (R^{n})} ‖ {\hat{u}}_{1} ‖_{L^{s}}^{\hat{q}} \end{array}$ and $\begin{array}{rcl} \int_{Z_{hyp}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ & ≲ & \int_{Z_{hyp}} | ξ |^{\hat{q} | μ | + \hat{q} σ} e^{- \frac{\hat{q}}{2 C} | ξ |^{2 θ} \int_{t_{0}}^{t} g (s) d s} | {\hat{u}}_{0} |^{\hat{q}} d ξ \\ + \int_{Z_{hyp}} | ξ |^{\hat{q} | μ |} e^{- \frac{\hat{q}}{2 C} | ξ |^{2 θ} \int_{t_{0}}^{t} g (s) d s} | {\hat{u}}_{1} |^{\hat{q}} d ξ \\ ≲ & {‖ | \cdot |^{\hat{q} | μ | + \hat{q} σ} e^{- c φ (t) | \cdot |^{2 θ}} ‖}_{L^{r} (R^{n})} ‖ {\hat{u}}_{0} ‖_{L^{s}}^{\hat{q}} \\ (3.2) & + {‖ | \cdot |^{\hat{q} | μ |} e^{- c φ (t) | \cdot |^{2 θ}} ‖}_{L^{r} (R^{n})} ‖ {\hat{u}}_{1} ‖_{L^{s}}^{\hat{q}} . \end{array}$

Using Hausdorff–Young inequality and Lemma A.4 with $k_{1} = \hat{q} | μ | + \hat{q} σ$ or $k_{1} = \hat{q} | μ |$ , $k_{2} = 2 σ - 2 θ$ and $τ = c ϕ (t)$ in inequality (3.1) or $k_{2} = 2 θ$ and $τ = c φ (t)$ in inequality (3.2), we have $\begin{array}{rcl} \int_{Z_{ell}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ & ≲ & ϕ {(t)}^{- \frac{\hat{q} | μ | + \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 σ - 2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + ϕ {(t)}^{- \frac{\hat{q} | μ | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 σ - 2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} \\ ≲ & φ {(t)}^{- \frac{\hat{q} | μ | + \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | μ | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} \end{array}$ and $\begin{array}{rcl} \int_{Z_{hyp}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ & ≲ & φ {(t)}^{- \frac{\hat{q} | μ | + \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | μ | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}}, \end{array}$ for $t ⩾ t_{0}$ .

Let $| β | - σ + \frac{n}{p \hat{q}} (p + \hat{q} - p \hat{q}) > 0$ if $u_{1} \neq 0$ and any $β \in N^{n}$ if $u_{1} = 0$ . Thus, using Proposition 2.5 and Corollary A.1: $\begin{array}{rcl} \int_{Z_{ell}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ & ≲ & \int_{Z_{ell}} | ξ |^{\hat{q} | β |} e^{- \frac{\hat{q}}{2 C} | ξ |^{2 σ - 2 θ} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} | {\hat{u}}_{0} |^{\hat{q}} d ξ \\ + \int_{Z_{ell}} | ξ |^{\hat{q} | β | - \hat{q} σ} e^{- \frac{\hat{q}}{2 C} | ξ |^{2 σ - 2 θ} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} | {\hat{u}}_{1} |^{\hat{q}} d ξ \\ ≲ & {‖ | \cdot |^{\hat{q} | β |} e^{- c ϕ (t) | \cdot |^{2 σ - 2 θ}} ‖}_{L^{r} (R^{n})} ‖ {\hat{u}}_{0} ‖_{L^{s}}^{\hat{q}} \\ (3.3) & + {‖ | \cdot |^{\hat{q} | β | - \hat{q} σ} e^{- c ϕ (t) | \cdot |^{2 σ - 2 θ}} ‖}_{L^{r} (R^{n})} ‖ {\hat{u}}_{1} ‖_{L^{s}}^{\hat{q}} \end{array}$ and $\begin{array}{rcl} \int_{Z_{hyp}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ & ≲ & \int_{Z_{hyp}} | ξ |^{\hat{q} | β |} e^{- \frac{\hat{q}}{2 C} | ξ |^{2 θ} \int_{t_{0}}^{t} g (s) d s} | {\hat{u}}_{0} |^{\hat{q}} d ξ \\ + \int_{Z_{hyp}} | ξ |^{\hat{q} | β | - \hat{q} σ} e^{- \frac{\hat{q}}{2 C} | ξ |^{2 θ} \int_{t_{0}}^{t} g (s) d s} | {\hat{u}}_{1} |^{\hat{q}} d ξ \\ ≲ & {‖ | \cdot |^{\hat{q} | β |} e^{- c φ (t) | \cdot |^{2 θ}} ‖}_{L^{r} (R^{n})} ‖ {\hat{u}}_{0} ‖_{L^{s}}^{\hat{q}} \\ (3.4) & + {‖ | \cdot |^{\hat{q} | β | - \hat{q} σ} e^{- c φ (t) | \cdot |^{2 θ}} ‖}_{L^{r} (R^{n})} ‖ {\hat{u}}_{1} ‖_{L^{s}}^{\hat{q}} . \end{array}$

Using Hausdorff–Young inequality and Lemma A.4 with $k_{1} = \hat{q} | β |$ or $k_{1} = \hat{q} | β | - \hat{q} σ$ , $k_{2} = 2 σ - 2 θ$ and $τ = c ϕ (t)$ in inequality (3.3) or $k_{2} = 2 θ$ and $τ = c φ (t)$ in inequality (3.4), we have: $\begin{array}{rcl} \int_{Z_{ell}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ & ≲ & ϕ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 σ - 2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + ϕ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 σ - 2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} \\ ≲ & φ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} \end{array}$ and $\begin{array}{rcl} \int_{Z_{hyp}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ & ≲ & φ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} \end{array}$ for $t ⩾ t_{0}$ .

For the case $Z_{ell}$ with $θ = σ$ , we directly apply Proposition 2.5: $\begin{array}{rcl} \int_{Z_{ell}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ & ≲ & \int_{Z_{ell}} | ξ |^{\hat{q} | μ | + \hat{q} σ} e^{- \frac{\hat{q}}{2 C} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} | {\hat{u}}_{0} |^{\hat{q}} d ξ + \int_{Z_{ell}} | ξ |^{\hat{q} | μ |} e^{- \frac{\hat{q}}{2 C} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} | {\hat{u}}_{1} |^{\hat{q}} d ξ \\ ≲ & e^{- \frac{\hat{q}}{2 C} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} ({‖ | \cdot |^{\hat{q} | μ | + \hat{q} σ} ‖}_{L^{r} (B_{1})} ‖ {\hat{u}}_{0} ‖_{L^{s}}^{\hat{q}} + {‖ | \cdot |^{\hat{q} | μ |} ‖}_{L^{r} (B_{1})} ‖ {\hat{u}}_{1} ‖_{L^{s}}^{\hat{q}}) \\ ≲ & e^{- \frac{\hat{q}}{2 C} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} (‖ u_{0} ‖_{L^{p}}^{\hat{q}} + ‖ u_{1} ‖_{L^{p}}^{\hat{q}}) \\ ≲ & φ {(t)}^{- \frac{\hat{q} | μ | + \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | μ | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}}, \end{array}$ in which the penultimate inequality is given by Hausdorff–Young inequality, and the last inequality is ensured by Corollary A.2.

Let $| β | - σ + \frac{n}{p \hat{q}} (p + \hat{q} - p \hat{q}) > 0$ if $u_{1} \neq 0$ and any $β \in N^{n}$ if $u_{1} = 0$ , using again Proposition 2.5 and Hausdorff–Young inequality: $\begin{array}{rcl} \int_{Z_{ell}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ & ≲ & e^{- \frac{\hat{q}}{2 C} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} \int_{Z_{ell}} | ξ |^{\hat{q} | β |} | {\hat{u}}_{0} |^{\hat{q}} d ξ + e^{- \frac{\hat{q}}{2 C} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} \int_{Z_{ell}} | ξ |^{\hat{q} | β | - \hat{q} σ} | {\hat{u}}_{1} |^{\hat{q}} d ξ \\ ≲ & e^{- \frac{\hat{q}}{2 C} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} ({‖ | \cdot |^{\hat{q} | β |} ‖}_{L^{r} (B_{1})} ‖ {\hat{u}}_{0} ‖_{L^{s}}^{\hat{q}} + {‖ | \cdot |^{\hat{q} | β | - \hat{q} σ} ‖}_{L^{r} (B_{1})} ‖ {\hat{u}}_{1} ‖_{L^{s}}^{\hat{q}}) \\ ≲ & e^{- \frac{\hat{q}}{2 C} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} (‖ u_{0} ‖_{L^{p}}^{\hat{q}} + ‖ u_{1} ‖_{L^{p}}^{\hat{q}}) \\ ≲ & φ {(t)}^{- (\frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ})} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- (\frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ})} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} . \end{array}$ □

Proposition 3.3.

Consider the conditions of Proposition 3.1 . Then, there exists $t_{0}^{*} ⩾ t_{0}$ such that the following estimates hold for $t ⩾ t_{0}^{*}$ :

If $θ \neq σ$ , $\begin{array}{rcl} \int_{Z^{high}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ ≲ φ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} \end{array}$ and $\begin{array}{rcl} \int_{Z^{high}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ ≲ φ {(t)}^{- \frac{\hat{q} | μ | + \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | μ | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} . \end{array}$

If $θ = σ$ and $ω > \frac{n}{p \hat{q}} (p + \hat{q} - p \hat{q})$ , $\begin{array}{rcl} \int_{Z^{high}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ & ≲ & φ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{W^{| β | + ω, p}}^{\hat{q}} \\ + φ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{W^{| β | - σ + ω, p}}^{\hat{q}} \end{array}$ and $\begin{array}{rcl} \int_{Z^{high}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ & ≲ & φ {(t)}^{- \frac{\hat{q} | μ | + \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{W^{| μ | + σ + ω, p}}^{\hat{q}} \\ + φ {(t)}^{- \frac{\hat{q} | μ | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{W^{| μ | + ω, p}}^{\hat{q}} . \end{array}$

Proof.

In this proof we fix $f (t) : = g (t)$ is g is non-increasing and $f (t) : = \frac{1}{g (t)}$ if g is increasing. Let $ν : = min {2 σ - 2 θ, 2 θ}$ . Given $η > 0$ , by Corollary A.2 there exists sufficient big $t_{0}^{*}$ , such that for all $t ⩾ t_{0}^{*}$ : $\begin{matrix} (3.5) & e^{- \frac{1}{C_{1}} \int_{t_{0}}^{t} f (s) d s} ≲ φ {(t)}^{- η} . \end{matrix}$

Initially, let $θ \neq σ$ and $t_{0}^{*} ⩾ t_{0}$ such that $\int_{t_{0}}^{t_{0}^{*}} f (s) d s ⩾ 1$ (this number exists because $\frac{1}{g}$ and $g \notin L^{1} (R)$ ) and such that inequality (3.5) is satisfied. For $t ⩾ t_{0}^{*}$ , by Proposition 2.3 we have: $\begin{array}{rcl} \int_{Z^{high}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ & ≲ & \int_{Z^{high}} | ξ |^{\hat{q} | β |} e^{- \frac{\hat{q}}{2 C} | ξ |^{ν} \int_{t_{0}}^{t} f (s) d s} | {\hat{u}}_{0} |^{\hat{q}} d ξ \\ + \int_{Z^{high}} | ξ |^{\hat{q} | β | - \hat{q} σ} e^{- \frac{\hat{q}}{2 C} | ξ |^{ν} \int_{t_{0}}^{t} f (s) d s} | {\hat{u}}_{1} |^{\hat{q}} d ξ \\ ≲ & e^{- \frac{\hat{q} R^{ν}}{2 C} \int_{t_{0}^{*}}^{t} f (s) d s} \int_{Z^{high}} | ξ |^{\hat{q} | β |} e^{- \frac{\hat{q}}{2 C} | ξ |^{ν}} | {\hat{u}}_{0} |^{\hat{q}} d ξ \\ + e^{- \frac{\hat{q} R^{ν}}{2 C} \int_{t_{0}^{*}}^{t} f (s) d s} \int_{Z^{high}} | ξ |^{\hat{q} | β | - \hat{q} σ} e^{- \frac{\hat{q}}{2 C} | ξ |^{ν}} | {\hat{u}}_{1} |^{\hat{q}} d ξ \\ ≲ & φ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} {‖ | \cdot |^{\hat{q} | β |} e^{- \frac{\hat{q}}{2 C} | \cdot |^{ν}} ‖}_{L^{r} (R^{n})} ‖ {\hat{u}}_{0} ‖_{L^{s}}^{\hat{q}} \\ + φ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} {‖ | \cdot |^{\hat{q} | β | - \hat{q} σ} e^{- \frac{\hat{q}}{2 C} | \cdot |^{ν}} ‖}_{L^{r} (R^{n})} ‖ {\hat{u}}_{1} ‖_{L^{s}}^{\hat{q}} . \end{array}$

Let $| β | - σ + \frac{n}{p \hat{q}} (p + \hat{q} - p \hat{q}) > 0$ if $u_{1} \neq 0$ and any $β \in N^{n}$ if $u_{1} = 0$ , using Lemma A.4 with $τ = \frac{\hat{q}}{2 C}$ (in this case $k_{2} = ν \neq 0$ ) and Hausdorff–Young inequality: $\begin{array}{rcl} \int_{Z^{high}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ & ≲ & φ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{L^{p}}^{\hat{q}} + φ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{L^{p}}^{\hat{q}} . \end{array}$

Suppose now $θ = σ$ . In this case, since $ω \hat{q} > \frac{n}{p} (p + \hat{q} - p \hat{q}) = \frac{n}{r}$ , we have $‖ | \cdot |^{- ω \hat{q}} ‖_{L^{r} (R^{n} ∖ B_{R})} ≲ 1$ . Using inequality (3.5), Holder inequality and Hausdorff–Young inequality we have: $\begin{array}{rcl} \int_{Z^{high}} | ξ |^{\hat{q} | β |} | \hat{u} (t, ξ) |^{\hat{q}} d ξ & ≲ & e^{- \frac{\hat{q}}{2 C} \int_{t_{0}}^{t} f (s) d s} \int_{Z^{high}} | ξ |^{\hat{q} | β |} | {\hat{u}}_{0} |^{\hat{q}} d ξ \\ + e^{- \frac{\hat{q}}{2 C} \int_{t_{0}}^{t} f (s) d s} \int_{Z^{high}} | ξ |^{\hat{q} | β | - \hat{q} σ} | {\hat{u}}_{1} |^{\hat{q}} d ξ \\ ≲ & φ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} {‖ | \cdot |^{- ω \hat{q}} ‖}_{L^{r} (R^{n} ∖ B_{R})} {‖ | \cdot |^{| β | + ω} {\hat{u}}_{0} ‖}_{L^{s}}^{\hat{q}} \\ + φ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} {‖ | \cdot |^{- ω \hat{q}} ‖}_{L^{r} (R^{n} ∖ B_{R})} {‖ | \cdot |^{| β | - σ + ω} {\hat{u}}_{1} ‖}_{L^{s}}^{\hat{q}} \\ ≲ & φ {(t)}^{- \frac{\hat{q} | β | + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{0} ‖_{W^{| β | + ω, p}}^{\hat{q}} \\ + φ {(t)}^{- \frac{\hat{q} | β | - \hat{q} σ + \frac{n}{p} (p + \hat{q} - p \hat{q})}{2 θ}} ‖ u_{1} ‖_{W^{| β | - σ + ω, p}}^{\hat{q}} . \end{array}$

The proof of estimate for $\int_{Z^{high}} | ξ |^{\hat{q} | μ |} | {\hat{u}}_{t} (t, ξ) |^{\hat{q}} d ξ$ is analogous. □

Proof of Theorem 1.1.

Let $\hat{q}$ be the conjugate of q, $v \in {u, u_{t}}$ and $η \in {β, μ}$ , by Hausdorff–Young inequality (using the inverse Fourier transform instead of the Fourier transform), we have: $\begin{matrix} (3.6) & {‖ D^{η} v (t, \cdot) ‖}_{L^{q}} ≲ {‖ | \cdot |^{η} \hat{v} (t, \cdot) ‖}_{L^{\hat{q}}} ≲ {(\int_{R^{n}} | ξ |^{\hat{q} | η |} | \hat{v} (t, ξ) |^{\hat{q}} d ξ)}^{\frac{1}{\hat{q}}} . \end{matrix}$

It should be noticed that if $η = β$ and $u_{1} \neq 0$ , we have the restriction $| β | - σ + n (\frac{1}{p} - \frac{1}{q}) = | β | - σ + \frac{n}{p \hat{q}} (p + \hat{q} - p \hat{q}) > 0$ . For each fixed t, we separate the integral in inequality (3.6) in four parts, that is, low zone, elliptic zone, hyperbolic zone and high zone. By applying Propositions 3.1, 3.2 and 3.3 the theorem follows. □

Footnotes

Acknowledgements

The second author has been partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico – CNPq, Proc. 308868/2015-3 and 314398/2018-0. In addition, the authors would like to thank the reviewer for his or her valuable time and useful contributions.

Appendix A.

In the appendix we list some results important for proving results. Lemma A.1.

Let $t_{0}^{*} > t_{0}$ , $I = (t_{0}, \infty)$ or $I = (t_{0}, t_{0}^{*})$ , $f \in C^{1} (I, R) \cap C (\bar{I}, (0, \infty))$ and $ψ \in C^{1} (I, R) \cap C (\bar{I}, R)$ with $\frac{1}{ψ} \in L_{loc}^{1} (\bar{I}, R)$ such that there exists $C_{0} < 1$ satisfying: $\begin{matrix} | 1 - ψ^{'} (t) - \frac{f^{'} (t)}{f (t)} ψ (t) | ⩽ C_{0}, \forall t > t_{0} . \end{matrix}$ Thus, $\begin{matrix} \frac{1}{1 + C_{0}} {f (t) ψ (t) - f (a) ψ (a)} ⩽ \int_{a}^{t} f (s) d s ⩽ \frac{1}{1 - C_{0}} {f (t) ψ (t) - f (a) ψ (a)}, \end{matrix}$ for all $t ⩾ a ⩾ t_{0}$ such that $t \in \bar{I}$ .

Proof.

Let $λ (t) : = e^{\int_{t_{0}}^{t} \frac{1}{ψ (s)} d s}$ . Thus, $\begin{matrix} (A.1) & J : = \int_{a}^{t} f (s) d s = \int_{a}^{t} \frac{f (s)}{λ^{'} (s)} λ^{'} (s) d s = \frac{f (s)}{λ^{'} (s)} λ (s) |_{a}^{t} + \int_{a}^{t} φ (s) f (s) d s, \end{matrix}$ where $φ (t) = \frac{λ (t) λ^{″} (t)}{{(λ^{'} (t))}^{2}} - \frac{f^{'} (t) λ (t)}{f (t) λ^{'} (t)} = 1 - ψ^{'} (t) - \frac{f^{'} (t)}{f (t)} ψ (t)$ . Therefore, $\begin{matrix} J ⩽ f (t) ψ (t) - f (a) ψ (a) + C_{0} J . \end{matrix}$ Solving for J, we conclude that: $\begin{matrix} \int_{a}^{t} f (s) d s ⩽ \frac{1}{1 - C_{0}} {f (t) ψ (t) - f (a) ψ (a)} . \end{matrix}$ On the other hand, by equation (A.1) $\begin{matrix} J ⩾ f (t) ψ (t) - f (a) ψ (a) - C_{0} J . \end{matrix}$ Again, solving J we conclude the result. □

Lemma A.2.

Let $g (t) = {(1 + t)}^{α} {ln}^{γ} (1 + t)$ , defined for $t ⩾ t_{0}$ , with $α \in [- 1, 1]$ and $γ \in R$ .

If $α < 1$ , $\begin{matrix} \frac{2}{3 (1 - α)} {\frac{(1 + t)}{g (t)} - \frac{(1 + a)}{g (a)}} ⩽ \int_{a}^{t} \frac{1}{g (s)} d s ⩽ \frac{2}{(1 - α)} {\frac{(1 + t)}{g (t)} - \frac{(1 + a)}{g (a)}}, \end{matrix}$ for all $t ⩾ a ⩾ t_{0} ⩾ e^{\frac{2 | γ |}{(1 - α)}} - 1$ . On the other hand, if $α > - 1$ , $\begin{matrix} \frac{2}{3 (1 + α)} {(1 + t) g (t) - (1 + a) g (a)} ⩽ \int_{a}^{t} g (s) d s ⩽ \frac{2}{(1 + α)} {(1 + t) g (t) - (1 + a) g (a)}, \end{matrix}$ for all $t ⩾ a ⩾ t_{0} ⩾ e^{\frac{2 | γ |}{(1 + α)}} - 1$ .

Proof.

For the first part, choose $f (t) : = \frac{1}{g (t)}$ and $ψ (t) : = \frac{(1 + t)}{1 - α}$ , for the second part choose $f (t) : = g (t)$ and $ψ (t) : = \frac{(1 + t)}{1 + α}$ , and apply Lemma A.1, for both cases with $C_{0} : = \frac{1}{2}$ . □

Corollary A.1.

Let $g (t) = {(1 + t)}^{α} {ln}^{γ} (1 + t)$ , defined for $t ⩾ t_{0}$ , with $α \in (- 1, 1)$ and $γ \in R$ or $α = - 1$ and $γ ⩾ - 1$ . For $ϕ (t) = {(1 + t)}^{1 - α} {ln}^{- γ} (1 + t)$ , we have: $\begin{matrix} \frac{1}{3 (1 - α)} ϕ (t) ⩽ \int_{t_{0}}^{t} \frac{1}{g (s)} d s ⩽ \frac{2}{(1 - α)} ϕ (t), \end{matrix}$ for all $t ⩾ t_{0}^{*} ⩾ ϕ^{- 1} (2 ϕ (t_{0}))$ . On the other hand, there exist $c_{0} > 0$ and $c_{1} > 0$ such that $\begin{matrix} c_{0} φ (t) ⩽ \int_{t_{0}}^{t} g (s) d s ⩽ c_{1} φ (t), \end{matrix}$ for all $t ⩾ t_{0}^{*} ⩾ φ^{- 1} (2 φ (t_{0}))$ , where φ is defined in ( 1.9 ).

Proof.

For $α \in [- 1, 1)$ , to estimate the right side of the first inequality we just apply Lemma A.2. The left side is obtained also using Lemma A.2 and using that, for $t ⩾ t_{0}^{*}$ , we have $\frac{1}{2} \frac{(1 + t)}{g (t)} - \frac{(1 + t_{0})}{g (t_{0})} = \frac{1}{2} ϕ (t) - ϕ (t_{0}) ⩾ 0$ .

For the right side of the second inequality and $α \in (- 1, 1)$ we again apply Lemma A.2. For the left side, we use that for $t ⩾ t_{0}^{*}$ holds $\frac{1}{2} (1 + t) g (t) - (1 + t_{0}) g (t_{0}) = \frac{1}{2} φ (t) - φ (t_{0}) ⩾ 0$ .

Finally, for the second inequality and $α = - 1$ we observe that $\int_{t_{0}}^{t} g (s) d s = \frac{1}{1 + γ} {ln}^{1 + γ} (1 + t) - \frac{1}{1 + γ} {ln}^{1 + γ} (1 + t_{0})$ if $γ > - 1$ , $\int_{t_{0}}^{t} g (s) d s = ln (ln (1 + t)) - ln (ln (1 + t_{0}))$ if $γ = - 1$ , the left side is proved using again the restriction $t_{0}^{*} ⩾ φ^{- 1} (2 φ (t_{0}))$ . The estimate of the right side is trivial. □

Corollary A.2.

Let g and φ as in Corollary A.1 . Furthermore, consider f defined by $f (t) : = \frac{1}{g (t)}$ if g is increasing and $f (t) : = g (t)$ if g is non-increasing. There exists $t_{0}^{*} > t_{0}$ , such that for all $C_{0} > 0$ and $η > 0$ there exists $C_{1}$ (that depends only in $C_{0}$ ) in which: $\begin{matrix} (A.2) & e^{- \frac{1}{C_{0}} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} ≲ e^{- \frac{1}{C_{1}} \int_{t_{0}}^{t} f (s) d s} ≲ φ {(t)}^{- η} \end{matrix}$ for all $t ⩾ t_{0}^{*}$ .

Proof.

Given $C_{1} > 0$ , by applying Corollary A.1 there exists $t_{0}^{*} > t_{0}$ and $c > 0$ such that: $\begin{matrix} e^{- \frac{1}{C_{1}} \int_{t_{0}}^{t} f (s) d s} ≲ \{\begin{matrix} e^{- c {(1 + t)}^{1 - α} {ln}^{- γ} (1 + t)}, & if 0 < α < 1 or α = 0 and γ > 0, \\ e^{- c {(1 + t)}^{1 + α} {ln}^{γ} (1 + t)}, & if - 1 < α < 0 or α = 0 and γ ⩽ 0, \\ e^{- c {ln}^{1 + γ} (1 + t)}, & if α = - 1 and γ > - 1, \\ e^{- c ln (ln (1 + t))}, & if α = - 1 and γ = - 1, \end{matrix} \end{matrix}$ for $t ⩾ t_{0}^{*}$ . Therefore, given $η > 0$ , we have for all $t ⩾ t_{0}^{*}$ : $\begin{matrix} (A.3) & e^{- \frac{1}{C_{1}} \int_{t_{0}}^{t} f (s) d s} ≲ φ {(t)}^{- η} . \end{matrix}$

On the other hand, given $C_{0} > 0$ , there exist $C_{1}$ such that $\frac{1}{C_{0}} \frac{1}{g (t)} ⩾ \frac{1}{C_{1}} f (t)$ for all $t ⩾ t_{0}^{*}$ . Therefore, $e^{- \frac{1}{C_{0}} \int_{t_{0}}^{t} \frac{1}{g (s)} d s} ≲ e^{- \frac{1}{C_{1}} \int_{t_{0}}^{t} f (s) d s}$ . Finally, by using last inequality in (A.3) the result follows. □

Lemma A.3.

For $τ > 0$ , $r \in [1, \infty]$ and $k + \frac{n}{r} > 0$ holds: $\begin{matrix} {‖ | \cdot |^{k} ‖}_{L^{r} (B_{τ})} ≲ τ^{k + \frac{n}{r}} . \end{matrix}$

Lemma A.4.

For $τ > 0$ , $r \in [1, \infty]$ , $k_{1} + \frac{n}{r} > 0$ and $k_{2} > 0$ , holds: $\begin{matrix} {‖ | \cdot |^{k_{1}} e^{- | \cdot |^{k_{2}} τ} ‖}_{L^{r} (R^{n})} ≲ τ^{- \frac{k_{1} + \frac{n}{r}}{k_{2}}} . \end{matrix}$

Appendix B.

Consider the following functions: $f_{- 1} : (- \infty, - 1) \to (- \frac{1}{e}, 0)$ and $f_{0} : (- 1, \infty) \to (- \frac{1}{e}, \infty)$ , both defined by the rule: $x \mapsto x e^{x}$ . These functions are bijective and therefore admits an inverse. Therefore, we can define $W_{- 1} : = {(f_{- 1})}^{- 1}$ and $W_{0} : = {(f_{0})}^{- 1}$ , both known as W-Lambert’s function. Actually, in a more general sense, are the two real-valued branches of W-Lambert’s function. The W-Lambert’s function plays a fundamental role in this paper: it is used to explicitly calculate $t_{ξ}$ and $ψ (ξ)$ . Since the explicit representation of ψ and $t_{ξ}$ is not sufficient by itself, in the following are some useful results to prove Proposition 2.4. The proof of Lemma B.1 can be found in [3], and the proof of Lemma B.2 can be found in [12].

We immediately conclude that $W_{0}$ behaves like the function ln:

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