Existence of a Stationary Navier–Stokes Flow Past a Rigid Body, with Application to Starting Problem in Higher Dimensions

Abstract

We consider the large time behavior of the Navier–Stokes flow past a rigid body in $\mathbb {R}^n$ with $n\ge 3$. We first construct a small stationary solution possessing the optimal summability at spatial infinity, which is the same as that of the Oseen fundamental solution. When the translational velocity of the body gradually increases and is maintained after a certain finite time, we then show that the nonstationary fluid motion converges to the stationary solution corresponding to a small terminal velocity of the body as time $t\rightarrow \infty$ in $L^q$ with $q\in [n,\infty ]$. This is called Finn’s starting problem and the three-dimensional case was affirmatively solved by Galdi et al. (Arch Ration Mech Anal 138: 307–318, 1997). The present paper extends Galdi et al. (1997) to the case of higher dimensions. Even for the three-dimensional case, our theorem provides new convergence rate, that is determined by the summability of the stationary solution at infinity and seems to be sharp.

1 Introduction and Main Results

We consider a viscous incompressible flow past a rigid body $\mathscr {O}\subset \mathbb {R}^n\,(n\ge 3)$. We suppose that $\mathscr {O}$ is translating with a velocity $-\psi (t)ae_1$, where $a>0$, $e_1=(1,0,\cdots ,0)^\top$ and $\psi$ is a function on $\mathbb {R}$ describing the transition of the translational velocity in such a way that

$$\begin{aligned} \psi \in C^1(\mathbb {R};\mathbb {R}),\quad |\psi (t)|\le 1\quad {\mathrm{for}}~\,t\in \mathbb {R},\quad \psi (t)=0\quad {\mathrm{for}}\,~t\le 0,\quad \psi (t)=1\quad {\mathrm{for}} \,~t\ge 1. \end{aligned}$$

(1.1)

Here and hereafter, $(\cdot )^\top$ denotes the transpose. We take the frame attached to the body, then the fluid motion which occupies the exterior domain $D=\mathbb {R}^n{\setminus }\mathscr {O}$ with $C^2$ boundary $\partial D$ and is started from rest obeys

$$\begin{aligned} \left\{ \begin{array}{r@{}c@{}ll} \displaystyle \frac{\partial u}{\partial t}+u\cdot \nabla u&{}{}={}&{} \Delta u-\psi (t)a\,\displaystyle \frac{\partial u}{\partial x_1} -\nabla p,&{}\quad x\in D,~t>0,\\ \nabla \cdot u&{}{}={}&{}0,&{}\quad x\in D,t\ge 0,\\ u|_{\partial D}&{}{}={}&{}-\psi (t)ae_1,&{}\quad t>0,\\ u&{}{}\rightarrow {}&{}0&{}\quad {\mathrm{as}}~|x|\rightarrow \infty ,\\ u(x,0)&{}{}={}&{}0,&{}\quad x\in D. \end{array}\right. \end{aligned}$$

(1.2)

Here, $u=(u_1(x,t),\cdots ,u_n(x,t))^\top$ and $p=p(x,t)$ denote unknown velocity and pressure of the fluid, respectively. Since $\psi (t)=1$ for $t\ge 1$, the large time behavior of solutions is related to the stationary problem

$$\begin{aligned} \left\{ \begin{array}{r@{}c@{}ll} u_s\cdot \nabla u_s&{}{}={}&{}\Delta u_s-a\, \displaystyle \frac{\partial u_s}{\partial x_1}-\nabla p_s, &{}\quad x\in D,\\ \nabla \cdot u_s&{}{}={}&{}0,&{}\quad x\in D,\\ u_s|_{\partial D}&{}{}={}&{}-ae_1,\\ u_s&{}{}\rightarrow {}&{} 0&{}\quad {\mathrm{as}}~|x|\rightarrow \infty . \end{array}\right. \end{aligned}$$

(1.3)

When $n=3$, the pioneering work due to Leray [27] provided the existence theorem for weak solution to problem (1.3), what is called D-solution, having finite Dirichlet integral without smallness assumption on data. From the physical point of view, solutions of (1.3) should reflect the anisotropic decay structure caused by the translation, but his solution had little information about the behavior at large distances. To fill this gap, Finn [11,12,13,14] introduced the class of solutions with pointwise decay property

$$\begin{aligned} u_s(x)=O(|x|^{-\frac{1}{2}-\varepsilon }) \quad \quad \mathrm{as~}|x|\rightarrow \infty \end{aligned}$$

(1.4)

for some $\varepsilon >0$ and proved that if a is small enough, (1.3) admits a unique solution satisfying (1.4) and exhibiting paraboloidal wake region behind the body $\mathscr {O}$. He called the Navier–Stokes flows satisfying (1.4) physically reasonable solutions. It is remarkable that D-solutions become physically reasonable solutions no matter how large a would be, see Babenko [2], Galdi [18] and Farwig and Sohr [10]. Galdi developed the $L^q$-theory of the linearized system, that we call the Oseen system, to prove that every D-solution has the same summability as the one of the Oseen fundamental solution without any smallness assumption, see [19, Theorem X.6.4]. It is not straightforward to generalize his result to the case of higher dimensions and it remains open whether the same result holds true for $n\ge 4$. We also refer to Farwig [9] who gave another outlook on Finn’s results by using anisotropically weighted Sobolev spaces, and to Shibata [31] who developed the estimates of physically reasonable solutions and then proved their stability in the $L^3$ framework when a is small. There is less literature concerning the problem (1.3) for the case $n\ge 4$. When $n\ge 3$, Shibata and Yamazaki [32] constructed a solution $u_s$, which is uniformly bounded with respect to small $a\ge 0$ in the Lorentz space $L^{n,\infty }$, and investigated the behavior of $u_s$ as $a\rightarrow 0$. If, in particular, $n\ge 4$ and if $a\ge 0$ is sufficiently small, they also derived

$$\begin{aligned} u_s\in L^{\frac{n}{1+\rho _1}}(D)\cap L^{\frac{n}{1-\rho _2}}(D),\quad \nabla u_s\in L^{\frac{n}{2+\rho _1}}(D)\cap L^{\frac{n}{2-\rho _2}}(D) \end{aligned}$$

(1.5)

for some $0<\rho _1,\rho _2<1$, see [32, Remark 4.2].

Let us turn to the initial value problem. Finn [12] conjectured that (1.2) admits a solution which tends to a physically reasonable solution as $t\rightarrow \infty$ if a is small enough. This is called Finn’s starting problem. It was first studied by Heywood [21], in which a stationary solution is said to be attainable if the fluid motion converges to this solution as $t\rightarrow \infty .$ Later on, by using Kato’s approach [24] (see also Fujita and Kato [15]) together with the $L^q$-$L^r$ estimates for the Oseen initial value problem established by Kobayashi and Shibata [26], Finn’s starting problem was affirmatively solved by Galdi et al. [20]. After that, Hishida and Maremonti [23] constructed a sort of weak solution u that enjoys

$$\begin{aligned} \Vert u(t)-u_s\Vert _{\infty }=O(t^{-\frac{1}{2}}) \quad \mathrm{as}~ t\rightarrow \infty \end{aligned}$$

(1.6)

if a is small, but $u(\cdot ,0)\in L^{3}(D)$ can be large. Here and hereafter, $\Vert \cdot \Vert _q$ denotes the norm of $L^q(D)$. Although we concentrate ourselves on attainability in this paper, stability of stationary solutions was also studied by Shibata [31], Enomoto and Shibata [8] and Koba [25] in the $L^q$ framework. Those work except [8] studied the three-dimensional exterior problem, while [8] showed the stability of a stationary solution satisfying (1.5) for some $0<\rho _1,\rho _2<1$ in n-dimensional exterior domains with $n\ge 3$. Stability of physically reasonable solutions in 2D is much more involved for several reasons and it has been recently proved by Maekawa [29].

The aim of this paper is two-fold. The first one is to construct a small stationary solution possessing the optimal summability at spatial infinity, which is the same as that of the Oseen fundamental solution $\mathbf{E}$:

$$\begin{aligned} \mathbf{E}\in L^q(\{x\in \mathbb {R}^n\mid |x|>1\}), \quad q>\frac{n+1}{n-1},\quad \quad \nabla \mathbf{E}\in L^r(\{x\in \mathbb {R}^n\mid |x|>1\}),\quad r>\frac{n+1}{n}, \end{aligned}$$

(1.7)

see Galdi [19, Section VII]. As already mentioned above, this result is well known in three-dimensional case even for large $a>0$, but it is not found in the literature for higher dimensional case $n\ge 4$. Our theorem covers the three-dimensional case as well and the proof is considerably shorter than the one given by authors mentioned above since we focus our interest only on summability at infinity rather than anisotropic pointwise estimates. The second aim is to give an affirmative answer to the starting problem as long as a is small enough, that is, to show the attainability of the stationary solution obtained above. The result extends Galdi, Heywood and Shibata [20] to the case of higher dimensions. Even for the three-dimensional case, our theorem not only recovers [20] but also provides better decay properties, for instance,

$$\begin{aligned} \Vert u(t)-u_s\Vert _{\infty }=O(t^{-\frac{1}{2}-\frac{\rho }{2}}) \quad \mathrm{as}~ t\rightarrow \infty \end{aligned}$$

(1.8)

for some $\rho >0$, that should be compared with (1.6). This is because the fluid is initially at rest and because the three-dimensional stationary solution $u_s$ belongs to $L^q(D)$ with $q<3$; to be precise, since q can be close to 2, one can take $\rho$ close to 1/2 in (1.8). Due to the $L^q$-$L^r$ estimates of the Oseen semigroup established by Kobayashi and Shibata [26], Enomoto and Shibata [7, 8], see Proposition 3.1, this decay rate is sharp in view of presence of $u_s$, see (1.19), in forcing terms of the Eq. (1.18) for the perturbation. Our result can be also compared with [34] by the present author on the starting problem in which translation is replaced by rotation of the body $\mathscr {O}\subset \mathbb {R}^3$. Under the circumstance of [34], the optimal spatial decay of stationary solutions observed in general is the scale-critical rate $O(|x|^{-1})$, so that they cannot belong to $L^q(D)$ with $q\le 3=n$, and therefore, we have no chance to deduce (1.8). Another remark is that, in comparison with stability theorem due to [8] for $n\ge 3$, more properties of stationary solutions are needed to establish the attainability theorem. Therefore, those properties must be deduced in constructing a solution of (1.3).

Let us state the first main theorem on the existence and summability of stationary solutions.

Theorem 1.1

Let $n\ge 3$. For every $(\alpha _1,\alpha _2,\beta _1,\beta _2)$ satisfying

$$\begin{aligned} \frac{n+1}{n-1}<\alpha _1\le n+1\le \alpha _2<\frac{n(n+1)}{2},\quad \frac{n+1}{n}<\beta _1\le \frac{n+1}{2}\le \beta _2<\frac{n(n+1)}{n+2}, \end{aligned}$$

(1.9)

there exists a constant $\delta =\delta (\alpha _1,\alpha _2,\beta _1,\beta _2,n,D)\in (0,1)$ such that if

$$\begin{aligned} 0<a^{\frac{n-2}{n+1}}<\delta , \end{aligned}$$

problem (1.3) admits a unique solution $u_s$ along with

$$\begin{aligned} \Vert u_s\Vert _{\alpha _1}+\Vert u_s\Vert _{\alpha _2}\le Ca^{\frac{n-1}{n+1}},\quad \Vert \nabla u_s\Vert _{\beta _1}+\Vert \nabla u_s\Vert _{\beta _2} \le Ca^{\frac{n}{n+1}}, \end{aligned}$$

(1.10)

where $C>0$ is independent of a.

The upper bounds of $\alpha _2$ and $\beta _2$ come from (2.2) with $q<n/2$ in Proposition 2.1 on the $L^q$-theory of the Oseen system, whereas the lower bounds of $\alpha _1$ and $\beta _1$ are just (1.7).

For the proof of Theorem 1.1, we define a certain closed ball N and a contraction map $\Psi :N\ni v\mapsto u\in N$ which provides the solution to the problem

$$\begin{aligned} \left\{ \begin{array}{r@{}c@{}ll} \Delta u-a\,\displaystyle \frac{\partial u}{\partial x_1} &{}{}={}&{}\nabla p+v\cdot \nabla v, &{}\quad x\in D,\\ \nabla \cdot u&{}{}={}&{}0,&{}\quad x\in D,\\ u|_{\partial D}&{}{}={}&{}-ae_1,\\ u&{}{}\rightarrow {}&{} 0&{}\quad {\mathrm{as}}~|x|\rightarrow \infty . \end{array}\right. \end{aligned}$$

(1.11)

In doing so, we rely on the $L^q$-theory of the Oseen system developed by Galdi [19, Theorem VII.7.1], see Proposition 2.1, which gives us sharp summability estimates of solutions at infinity together with explicit dependence on $a>0$. As long as we only use Proposition 2.1, the only space in which estimates of $\Psi$ are closed is

$$\begin{aligned} \{u\in L^{n+1}(D)\mid \nabla u\in L^{\frac{n+1}{2}}(D)\}. \end{aligned}$$

From this, we can capture neither the optimal summability at infinity nor regularity required in the study of the starting problem. We thus use at least two spaces $L^{\alpha _i}(D)~(i=1,2)$ for u and $L^{\beta _i}(D)~(i=1,2)$ for $\nabla u$, and intend to find a solution within a closed ball N of

$$\begin{aligned} \{u\in L^{\alpha _1}(D)\cap L^{\alpha _2}(D)\mid \nabla u\in L^{\beta _1}(D)\cap L^{\beta _2}(D)\}. \end{aligned}$$

(1.12)

However, it is not possible to apply Proposition 2.1 to $f=v\cdot \nabla v$ with

$$\begin{aligned} v\in L^{\alpha _1}(D),\quad \quad \nabla v\in L^{\beta _1}(D) \end{aligned}$$

(1.13)

or

$$\begin{aligned} v\in L^{\alpha _2}(D),\quad \quad \nabla v\in L^{\beta _2}(D) \end{aligned}$$

(1.14)

if $\alpha _1$ and $\beta _1$ are simultaneously close to $(n+1)/(n-1)$ and $(n+1)/n$, or if $\alpha _2$ and $\beta _2$ are simultaneously close to $n(n+1)/2$ and $n(n+1)/(n+2)$, because the relation

$$\begin{aligned} \frac{2}{n}<\frac{1}{\alpha _2}+\frac{1}{\beta _2}< \frac{1}{\alpha _1}+\frac{1}{\beta _1}<1 \end{aligned}$$

required in the linear theory, see Proposition 2.1, is not satisfied. In order to overcome this difficulty, given $(\alpha _1,\alpha _2,\beta _1,\beta _2)$ satisfying (1.9), we choose auxiliary exponents $(q_1,q_2,r_1,r_2)$ fulfilling

$$\begin{aligned} \alpha _1\le q_1\le q_2\le \alpha _2, \quad \beta _1\le r_1\le r_2\le \beta _2, \quad \frac{2}{n}<\frac{1}{q_i}+\frac{1}{r_i}<1,\quad i=1,2 \end{aligned}$$

such that the application of Proposition 2.1 to $f=v\cdot \nabla v$ with $v\in L^{q_1}(D)$ and $\nabla v\in L^{r_1}(D)$ (resp. $v\in L^{q_2}(D)$ and $\nabla v\in L^{r_2}(D)$) recovers (1.13) (resp. (1.14)) with u.

Another possibility to prove Theorem 1.1 is combining Proposition 2.1 with the Sobolev inequality. We then get a solution $(u_s,p_s)\in X_q(n)$ for all $q\in (1,\infty )$ with $n/3\le q\le (n+1)/3$, where $X_q(n)$ is defined in Proposition 2.1. The restriction $n/3\le q\le (n+1)/3$ is removed by applying a bootstrap argument to decrease the lower bound to 1 and to increase the upper bound to n/2. As compared with this way, in our proof, we do not any use a bootstrap argument and directly construct a solution possessing the optimal summability at infinity as well as regularity required in the study of the starting problem.

Let us proceed to the starting problem. To study the attainability of the stationary solution $u_s$ of class (1.12) with $(\alpha _1,\alpha _2,\beta _1,\beta _2)$ satisfying (1.9), it is convenient to set

$$\begin{aligned} \alpha _1=\frac{n}{1+\rho _1},\quad \quad \alpha _2=\frac{n}{1-\rho _2},\quad \quad \beta _1=\frac{n}{2+\rho _3},\quad \quad \beta _2=\frac{n}{2-\rho _4} \end{aligned}$$

(1.15)

with $(\rho _1,\rho _2,\rho _3,\rho _4)$ satisfying

$$\begin{aligned} 0<\rho _1<\frac{n^2-2n-1}{n+1},\quad \frac{1}{n+1}\le \rho _2<\frac{n-1}{n+1},\quad 0<\rho _3<\frac{n^2-2n-2}{n+1},\quad \frac{2}{n+1}\le \rho _4<\frac{n}{n+1} \end{aligned}$$

(1.16)

and we need the additional condition

$$\begin{aligned} \rho _2+\rho _4>1. \end{aligned}$$

(1.17)

We note that the set of those parameters is nonvoid. It is reasonable to look for a solution to (1.2) of the form

$$\begin{aligned} u(x,t)=v(x,t)+\psi (t)u_s,\quad p(x,t)=\phi (x,t)+\psi (t)p_s. \end{aligned}$$

Then the perturbation $(v,\phi )$ satisfies the following initial boundary value problem

$$\begin{aligned} \left\{ \begin{array}{r@{}c@{}l} \displaystyle \frac{\partial v}{\partial t}&{}{}={}&{}\Delta v-a\displaystyle \frac{\partial v}{\partial x_1} -v\cdot \nabla v-\psi (t)v\cdot \nabla u_s -\psi (t)u_s\cdot \nabla v +(1-\psi (t))a\frac{\partial v}{\partial x_1}\\ &{}&{}\quad +h_1(x,t)+h_2(x,t) -\nabla \phi , \quad x\in D,\,t>0,\\ \nabla \cdot v&{}{}={}&{}0,\quad x\in D,\,t\ge 0,\\ v|_{\partial D}&{}{}={}&{}0,\quad t>0,\\ v&{}{}\rightarrow {}&{} 0~\quad {\mathrm{as}}~|x|\rightarrow \infty ,\\ v(x,0)&{}{}={}&{}0,\quad x\in D, \end{array}\right. \end{aligned}$$

(1.18)

where

$$\begin{aligned}&h_1(x,t)=-\psi '(t)u_s, \end{aligned}$$

(1.19)

$$\begin{aligned}&h_2(x,t)=\psi (t)\big (1-\psi (t)\big ) \Big (u_s\cdot \nabla u_s+a\frac{\partial u_s}{\partial x_1}\Big ). \end{aligned}$$

(1.20)

In what follows, we study the problem (1.18) instead of (1.2). In fact, if we obtain the solution v of (1.18) which converges to 0 as $t\rightarrow \infty$, the solution u of (1.2) converges to $u_s$ as $t\rightarrow \infty$. Problem (1.18) is converted into

$$\begin{aligned} v(t)&=\int _0^t e^{-(t-\tau )A_a}P\Big [-v\cdot \nabla v -\psi (\tau )v\cdot \nabla u_s-\psi (\tau )u_s\cdot \nabla v\nonumber \\&\quad +\big (1-\psi (\tau )\big )a\frac{\partial v}{\partial x_1} +h_1(\tau )+h_2(\tau )\Big ]d\tau \end{aligned}$$

(1.21)

by using the Oseen semigroup $e^{-tA_a}$ (see Section 3) as well as the Fujita–Kato projection P from $L^q(D)$ onto $L^q_{\sigma }(D)$ associated with the Helmholtz decomposition (see Fujiwara and Morimoto [16], Miyakawa [30] and Simader and Sohr [33]):

$$\begin{aligned} L^q(D)=L^q_{\sigma }(D)\oplus \{\nabla p\in L^q(D)\mid p\in L^q_{\mathrm{loc}}({\overline{D}})\}\quad (1<q<\infty ). \end{aligned}$$

Here,

$$\begin{aligned} L^q_{\sigma }(D)=\overline{C^\infty _{0,\sigma }(D)}^{\Vert \cdot \Vert _q}, \quad C^\infty _{0,\sigma }(D)=\{u\in C^\infty _0(D)^n\mid \nabla \cdot u=0\}. \end{aligned}$$

We are now in a position to give the second main theorem on attainability of stationary solutions.

Theorem 1.2

Let $n\ge 3$ and let $\psi$ be a function on $\mathbb {R}$ satisfying (1.1). We set $M=\displaystyle \max _{t\in \mathbb {R}}|\psi '(t)|$. Suppose that $\rho _1$, $\rho _2$, $\rho _3$ and $\rho _4$ satisfy (1.16)–(1.17) and let $\delta$ be the constant in Theorem 1.1 with (1.15). Then there exists a constant $\varepsilon =\varepsilon (n,D)\in (0,\delta ]$ such that if

$$\begin{aligned} 0<(M+1)a^{\frac{n-2}{n+1}}<\varepsilon , \end{aligned}$$

Equation (1.21) admits a unique solution v within the class

$$\begin{aligned} Y_{0}&:=\big \{v\in BC([0,\infty );L^n_{\sigma }(D))\mid t^{\frac{1}{2}}v\in BC((0,\infty );L^{\infty }(D)), t^{\frac{1}{2}}\nabla v\in BC((0,\infty );L^n(D)),\nonumber \\&\quad \lim _{t\rightarrow 0}~t^{\frac{1}{2}}\big ( \Vert v(t)\Vert _{\infty }+\Vert \nabla v(t)\Vert _{n}\big )=0\big \}. \end{aligned}$$

(1.22)

Moreover, we have the following.

• 1. (sharp decay)   Let $n=3$. Then there exists a constant $\varepsilon _*=\varepsilon _*(D)\in (0,\varepsilon ]$ such that if  $0<(M+1)a^{1/4}<\varepsilon _*$, the solution v enjoys decay properties

  $$\begin{aligned}&\Vert v(t)\Vert _q=O(t^{-\frac{1}{2}+\frac{3}{2q}-\frac{\rho _1}{2}}),\quad \quad 3\le \forall q\le \infty , \end{aligned}$$

  (1.23)
  $$\begin{aligned}&\Vert \nabla v(t)\Vert _3= O(t^{-\frac{1}{2}-\frac{\rho _1}{2}}) \end{aligned}$$

  (1.24)

  as $t\rightarrow \infty$.

  Let $n\ge 4$ and suppose that $\rho _3>1$ and $1<\rho _1\le 1+\rho _3$ in addition to (1.16) (the set of those parameters is nonvoid when $n\ge 4$ ). Then there exists a constant $\varepsilon _*=\varepsilon _*(n,D)\in (0,\varepsilon ]$ such that if  $0<(M+1)a^{(n-2)/(n+1)}<\varepsilon _*$, the solution v enjoys

  $$\begin{aligned}&\Vert v(t)\Vert _q=O(t^{-\frac{1}{2}+\frac{n}{2q}-\frac{\rho _1}{2}}),\quad \quad n\le \forall q\le \infty , \end{aligned}$$

  (1.25)
  $$\begin{aligned}&\Vert \nabla v(t)\Vert _n= O(t^{-\frac{1}{2}-\frac{\rho _1}{2}}) \end{aligned}$$

  (1.26)

  as $t\rightarrow \infty .$

• 2. (Uniqueness)   There exists a constant ${\hat{\varepsilon }}={\hat{\varepsilon }}(n,D)\in (0,\varepsilon ]$ such that if  $0<(M+1)a^{(n-2)/(n+1)}<{\hat{\varepsilon }}$, the solution v obtained above is unique even within the class

  $$\begin{aligned} Y:=\{v\in BC([0,\infty );L^n_{\sigma }(D))\mid t^{\frac{1}{2}}v\in BC((0,\infty );L^{\infty }(D)), t^{\frac{1}{2}}\nabla v\in BC((0,\infty );L^n(D))\}. \end{aligned}$$

  (1.27)

For the sharp decay properties (1.23)–(1.26), the key step is to prove the $L^n$-decay of the solution, that is,

$$\begin{aligned} \Vert v(t)\Vert _n=O(t^{-\frac{\rho _1}{2}}) \end{aligned}$$

(1.28)

as $t\rightarrow \infty$. Once we have (1.28), the other decay properties can be derived by the similar argument to [8]. Note that the condition $\rho _1\le 1+\rho _3$ is always fulfilled and thus redundant for $n=3$ since $\rho _1<1/2$ and $\rho _3<1/4$. On the other hand, it is enough for $n\ge 4$ to consider the case $\rho _1,\rho _3>1$. To prove (1.28), we first derive slower decay

$$\begin{aligned} \Vert v(t)\Vert _n=O(t^{-\frac{\rho }{2}}) \end{aligned}$$

with some $\rho \in (0,1)$ by making use of $u_s\in L^{n/(1+\rho _1)}(D)$ and $\nabla u_s\in L^{n/(2+\rho _3)}(D)$, see Lemma 3.6 in Section 3. When $n=3$, one can take $\rho :=\min \{\rho _1,\rho _3\}$, yielding better decay properties of the other norms of the solution. With them at hand, we repeat improvement of the estimate of $\Vert v(t)\Vert _n$ step by step to find (1.28). However, this procedure does not work for $n\ge 4$ because of $\rho _1>1$. In order to get around the difficulty, our idea is to deduce the $L^{q_0}$-decay of the solution with some $q_0<n$, that is appropriately chosen, see Lemma 3.8. We are then able to repeat improvement of estimates of several terms to arrive at (1.28), where the argument is more involved than the three-dimensional case above. Finally, to prove the uniqueness within Y, we employ the idea developed by Brezis [5], which shows that the solution $v\in Y$ necessarily satisfies the behavior as $t\rightarrow 0$ in (1.22).

In the next section we introduce the $L^q$-theory of the Oseen system and then prove Theorem 1.1. The final section is devoted to the proof of Theorem 1.2.

2 Proof of Theorem 1.1

In order to prove Theorem 1.1, we first recall the result on the Oseen boundary value problem due to Galdi [19, Theorem VII.7.1], see also Galdi [17] for the first proof of this result.

Proposition 2.1

Let $n\ge 3$ and let $D\subset \mathbb {R}^n$ be an exterior domain with $C^2$ boundary. Suppose $a>0$ and $1<q<(n+1)/2$. Given $f\in L^q(D)$ and $u_*\in W^{2-1/q,q}(\partial D)$, problem

$$\begin{aligned} \left\{ \begin{array}{r@{}c@{}ll} \Delta u-a\,\displaystyle \frac{\partial u}{\partial x_1} &{}{}={}&{}\nabla p+f, &{}\quad x\in D,\\ \nabla \cdot u&{}{}={}&{}0,&{}\quad x\in D,\\ u|_{\partial D}&{}{}={}&{}u_*,\\ u&{}{}\rightarrow {}&{} 0&{}\quad {\mathrm{as}}~|x|\rightarrow \infty \end{array}\right. \end{aligned}$$

(2.1)

admits a unique (up to an additive constant for p) solution (u, p) within the class

$$\begin{aligned} X_q(n)&:=\Big \{(u,p)\in L^1_{\mathrm{loc}}(D)\,\Big |\, u\in L^{s_2}(D),\,\nabla u\in L^{s_1}(D),\,\nabla ^2u\in L^{q}(D),\\&\qquad \displaystyle \frac{\partial u}{\partial x_1}\in L^q(D),\,\nabla p\in L^{q}(D)\Big \}, \end{aligned}$$

where

$$\begin{aligned} \frac{1}{s_1}=\frac{1}{q}-\frac{1}{n+1},\quad \frac{1}{s_2}=\frac{1}{q} -\frac{2}{n+1}. \end{aligned}$$

(2.2)

Here, by $W^{2-1/q,q}(\partial D)$ we denote the trace space on $\partial D$ from the Sobolev space $W^{2,q}(D)$ (see, for instance, [1, 19]).

If, in particular, $a\in (0,1]$ and $q<n/2$, then the solution (u, p) obtained above satisfies

$$\begin{aligned} a^{\frac{2}{n+1}}\Vert u\Vert _{s_2}+ a\left\| \displaystyle \frac{\partial u}{\partial x_1}\right\| _q+ a^{\frac{1}{n+1}}\Vert \nabla u\Vert _{s_1} +\Vert \nabla ^2 u\Vert _{q}+\Vert \nabla p\Vert _{q}\le C\big (\Vert f\Vert _q+\Vert u_*\Vert _{W^{2-\frac{1}{q},q}(\partial D)}\,\big ) \end{aligned}$$

with a constant $C>0$ dependent on q, n and D, however, independent of a.

For later use, we prepare the following lemma. The proof is essentially same as the one of Young’s inequality for convolution, thus we omit it.

Lemma 2.2

Let $R_0,\,d>0$. Assume that $1\le q,s\le \infty$ and $1/q+1/s\ge 1$. Suppose $u\in L^q(\mathbb {R}^n)$ with $\mathrm{supp}\,u\subset B_d:=\{x\in \mathbb {R}^n\mid |x|<d\}$ and $\rho \in L^s(\mathbb {R}^n{\setminus } B_{R_0})$. Then for all $R\ge R_0+d$, $\rho *u$ is well-defined as an element of $\,L^r(\mathbb {R}^n{\setminus } B_R)$ together with

$$\begin{aligned} \Vert \rho *u\Vert _{L^r(\mathbb {R}^n{\setminus } B_{R})}\le \Vert \rho \Vert _{L^s(\mathbb {R}^n{\setminus } B_{R_0})}\Vert u\Vert _{L^q(B_d)}, \end{aligned}$$

where $*$ denotes the convolution and $1/r:=1/q+1/s-1$.

When the external force f is taken from $L^{q_1}(D)\cap L^{q_2}(D)$ with $1<q_1,q_2<(n+1)/2$ and $q_1\ne q_2$, we can apply Proposition 2.1 to $f\in L^{q_i}(D)$ $(i=1,2)$. The following tells us that the corresponding solutions coincide with each other.

Lemma 2.3

Suppose $n\ge 3$, $1<q_1,q_2<(n+1)/2$ and $f\in L^{q_1}(D)\cap L^{q_2}(D)$. Let $(u_i,p_i)$ be a unique solution obtained in Proposition 2.1 with $f\in L^{q_i}(D)$ and $u_*=-ae_1$. Then $u_1=u_2$.

Proof

We first show that $u_1-u_2$ behaves like the Oseen fundamental solution $\mathbf{E}$ at large distances. We fix $R_0>0$ satisfying $\mathbb {R}^n{\setminus } D\subset B_{R_0}$. Let $\zeta \in C^\infty (\mathbb {R}^n)$ be a cut-off function such that $\zeta (x)=0$ for $|x|\le R_0$, $\zeta (x)=1$ for $|x|\ge R_0+1$, and set

$$\begin{aligned} u(x)&:=u_1(x)-u_2(x),&\quad \quad p(x)&:=p_1(x)-p_2(x),\\ v(x)&:=\zeta (x) u(x)-\mathbb {B}[u\cdot \nabla \zeta ],&\pi (x)&:=\zeta (x)p(x). \end{aligned}$$

Here, $\mathbb {B}$ is the Bogovskiĭ operator defined on the domain $B_{R_0+1}{\setminus } B_{R_0}$, see Bogovskiĭ [3], Borchers and Sohr [4] and Galdi [19]. Then we have

$$\begin{aligned} -\Delta v+a\frac{\partial v}{\partial x_1}+\nabla \pi =g(x),\quad \nabla \cdot v=0\quad {\mathrm{in}}~\mathscr {S}'(\mathbb {R}^n), \end{aligned}$$

(2.3)

where $\mathscr {S}'(\mathbb {R}^n)$ is the set of tempered distributions on $\mathbb {R}^n$ and

$$\begin{aligned} g(x)=-(\Delta \zeta )u-2(\nabla \zeta \cdot \nabla )u +a\frac{\partial \zeta }{\partial x_1}u+p\nabla \zeta + \Big (\Delta -a\frac{\partial }{\partial x_1}\Big )\mathbb {B}[u\cdot \nabla \zeta ]. \end{aligned}$$

For (2.3) with $g=0$, we have supp ${\hat{v}}\subset \{0\}$ and supp ${\hat{\pi }}\subset \{0\}$, where $\hat{(\cdot )}$ denotes the Fourier transform. We thus find

$$\begin{aligned} v(x)=\int _{\mathbb {R}^n} \mathbf{{E}}(x-y)g(y)\,dy+P(x),\quad \quad \pi (x)=C(n)\int _{\mathbb {R}^n} \frac{x-y}{|x-y|^n}\cdot g(y)\,dy+Q(x) \end{aligned}$$

with some polynomials P(x), Q(x) and some constant C(n). In view of $v\in L^{(\,1/q_1-2/(n+1)\,)^{-1}}(\mathbb {R}^n)$ $+L^{(\,1/q_2-2/(n+1)\,)^{-1}}(\mathbb {R}^n)$ and $\nabla \pi \in L^{q_1}(\mathbb {R}^n)+L^{q_2}(\mathbb {R}^n)$, we have $P(x)=0$ and $Q(x)={\overline{p}}$. Here, ${\overline{p}}$ is some constant. Then Lemma 2.2 with

$$\begin{aligned} \rho =\mathbf{E},~\nabla \mathbf{E},~\frac{x-y}{|x-y|^n}, \end{aligned}$$

$u=g$, $d=R_0+1$, $q=1$ and $r=s$ leads us to

$$\begin{aligned}&u\in L^q(\mathbb {R}^n{\setminus } B_{2R_0+1}),\quad \quad \nabla u\in L^r(\mathbb {R}^n{\setminus } B_{2R_0+1}),\quad \quad p-{\overline{p}}\in L^s(\mathbb {R}^n{\setminus } B_{2R_0+1}) \end{aligned}$$

(2.4)

for all $q>(n+1)/(n-1)$, $r>(n+1)/n$ and $s>n/(n-1)$, see (1.7).

Let $\varphi \in C^{\infty }[0,\infty )$ be a cut-off function such that $\varphi (t)=1$ for $t\le 1$, $\varphi (t)=0$ for $t\ge 2$, and set $\varphi _R(x)=\varphi (|x|/R)$ for $R\ge 2R_0+1$, $x\in \mathbb {R}^n$. We note that there exists a constant $C>0$ independent of R such that

$$\begin{aligned} \Vert \nabla \varphi _R\Vert _n\le C. \end{aligned}$$

(2.5)

It follows from

$$\begin{aligned} -\Delta u+a\frac{\partial u}{\partial x_1}+\nabla p=0,\quad \nabla \cdot u=0\quad {\mathrm{in}}~D,\quad u|_{\partial D}=0 \end{aligned}$$

that

$$\begin{aligned} 0&=\int _D\Big \{-\Delta u +a\frac{\partial u}{\partial x_1} +\nabla (p-{\overline{p}})\Big \}\cdot (\varphi _R u)\,dx\nonumber \\&=\int _D|\nabla u|^2\varphi _R\,dx +\int _{R\le |x|\le 2R}\Big \{(\nabla u\cdot \nabla \varphi _R)u-\frac{a}{2} \frac{\partial \varphi _R}{\partial x_1}|u|^2- (p-{\overline{p}})\nabla \varphi _R\cdot u\Big \}\,dx. \end{aligned}$$

(2.6)

Since we can see

$$\begin{aligned} |\nabla u||u|,\,|u|^2,\,(p-{\overline{p}})|u|\in L^{n/(n-1)} \big (\mathbb {R}^n{\setminus } B_{2R_0+1}) \end{aligned}$$

from (2.4), letting $R\rightarrow \infty$ in (2.6) yields $\Vert \nabla u\Vert ^2_2=0$ because of (2.5). From this together with $u|_{\partial D}=0$, we conclude $u_1=u_2$. $\square$

Proof of Theorem 1.1

Let $n\ge 3$ and let $(\alpha _1,\alpha _2,\beta _1,\beta _2)$ satisfy (1.9). We first choose parameters $(q_1,q_2,r_1,r_2)$ satisfying

$$\begin{aligned}&\frac{n+1}{n-1}<\alpha _1\le q_1\le n+1 \le q_2\le \alpha _2<\frac{n(n+1)}{2}, \end{aligned}$$

(2.7)

$$\begin{aligned}&\frac{n+1}{n}<\beta _1\le r_1\le \frac{n+1}{2} \le r_2\le \beta _2<\frac{n(n+1)}{n+2}, \end{aligned}$$

(2.8)

$$\begin{aligned}&\max \left\{ \frac{1}{\alpha _1}+\frac{2}{n+1}, \frac{1}{\beta _1}+\frac{1}{n+1}\right\} \le \frac{1}{q_1}+\frac{1}{r_1}<1, \end{aligned}$$

(2.9)

$$\begin{aligned}&\frac{2}{n}<\frac{1}{q_2}+\frac{1}{r_2}\le \min \left\{ \frac{1}{\alpha _2}+\frac{2}{n+1}, \frac{1}{\beta _2}+\frac{1}{n+1}\right\} . \end{aligned}$$

(2.10)

It is actually possible to choose those parameters. In fact, we put

$$\begin{aligned} \alpha _1=\frac{n+1}{n-1-\gamma _1},\quad \alpha _2=\frac{n(n+1)}{2+\gamma _2},\quad \beta _1=\frac{n+1}{n-\eta _1},\quad \beta _2=\frac{n(n+1)}{n+2+\eta _2} \end{aligned}$$

with arbitrarily small $\gamma _i,\eta _i\in (0,n-2]$ and look for $(q_1,q_2,r_1,r_2)$ of the form

$$\begin{aligned} q_1=\frac{n+1}{n-1-{\tilde{\gamma }}_1},\quad q_2=\frac{n(n+1)}{2+{\tilde{\gamma }}_2},\quad r_1=\frac{n+1}{n-{\tilde{\eta }}_1},\quad r_2=\frac{n(n+1)}{n+2+{\tilde{\eta }}_2}. \end{aligned}$$

Then the conditions (2.7)–(2.10) are accomplished by

$$\begin{aligned}&n-2<{\tilde{\gamma }}_1+{\tilde{\eta }}_1\le n-2+ \min \{\gamma _1,\eta _1\},\quad \quad n-2<{\tilde{\gamma }}_2+{\tilde{\eta }}_2\le n-2+\min \{\gamma _2,\eta _2\},\\&\gamma _i\le {\tilde{\gamma }}_i,\quad \quad \eta _i\le {\tilde{\eta }}_i, \quad i=1,2. \end{aligned}$$

For each $i=1,2$, the set of $({\tilde{\gamma }}_i,{\tilde{\eta }}_i)$ with those conditions is nonvoid for given $\gamma _i$ and $\eta _i$; for instance, we may take ${\tilde{\gamma }}_i=\gamma _i$, ${\tilde{\eta }}_i=n-2$ when $\gamma _i\le \eta _i$ and take ${\tilde{\gamma }}_i=n-2$, ${\tilde{\eta }}_i=\eta _i$ when $\gamma _i\ge \eta _i$.

To obtain a small solution, we use the contraction mapping principle. We define

$$\begin{aligned} B:=\{u\in L^{\alpha _1}(D)\cap L^{\alpha _2}(D) \mid \nabla u\in L^{\beta _1}(D)\cap L^{\beta _1}(D)\} \end{aligned}$$

which is a Banach space endowed with the norm

$$\begin{aligned} \Vert u\Vert _B:=\sum ^2_{i=1}(a^\frac{2}{n+1}\Vert u\Vert _{\alpha _i} +a^\frac{1}{n+1}\Vert \nabla u\Vert _{\beta _i}). \end{aligned}$$

Given $v\in B$, which satisfies

$$\begin{aligned} v\cdot \nabla v\in \bigcap _{i=1}^2L^{\kappa _i}(D),\quad \quad \quad \frac{1}{\kappa _i}=\frac{1}{q_i}+\frac{1}{r_i}, \quad 1<\kappa _i<\frac{n}{2} \end{aligned}$$

for $i=1,2,$ we can employ Proposition 2.1 with $f=v\cdot \nabla v$, $q=\kappa _i$ $(i=1,2)$ and $u_*=-ae_1$. Then, due to Lemma 2.3, the problem (1.11) admits a unique solution (u, p) such that

$$\begin{aligned}&a^{\frac{2}{n+1}}\Vert u\Vert _{\mu _i}+ a\left\| \frac{\partial u}{\partial x_1}\right\| _{\kappa _i}+ a^{\frac{1}{n+1}}\Vert \nabla u\Vert _{\lambda _i}+\Vert \nabla ^2 u\Vert _{\kappa _i}+ \Vert \nabla p\Vert _{\kappa _i} \\&\quad \le C'(\Vert v\cdot \nabla v\Vert _{\kappa _i}+a) \le C'(\Vert v\Vert _{q_i}\Vert \nabla v\Vert _{r_i}+a) \le C'(a^{-\frac{3}{n+1}}\Vert v\Vert _B^2+a) \end{aligned}$$

for $i=1,2.$ Here, $1/\lambda _i=1/\kappa _i-1/(n+1),$ $1/\mu _i=1/\kappa _i-2/(n+1)$. Furthermore, because the conditions (2.9) and (2.10) ensure $\mu _1\le \alpha _1\le \alpha _2\le \mu _2$ and $\lambda _1\le \beta _1\le \beta _2\le \lambda _2$, we find $u\in B$ with

$$\begin{aligned} \Vert u\Vert _B\le 4C'(a^{-\frac{3}{n+1}}\Vert v\Vert _B^2+a). \end{aligned}$$

Hence, we assume

$$\begin{aligned} a^{\frac{n-2}{n+1}}<\frac{1}{64C'^2}=:\delta \end{aligned}$$

(2.11)

and set

$$\begin{aligned} N_a:=\{u\in B\mid \Vert u\Vert _B\le 8C'a\} \end{aligned}$$

to see that the map $\Psi :N_a\ni v\mapsto u\in N_a$ is well-defined. Moreover, for $v_i\in N_a~(i=1,2)$, set $u_i=\Psi (v_i)$ and let $p_i$ be the pressure associated with $u_i$. Then we have

$$\begin{aligned} \left\{ \begin{array}{r@{}c@{}ll} \Delta (u_1-u_2)-a\, \displaystyle \frac{\partial }{\partial x_1}(u_1-u_2) &{}{}={}&{}\nabla (p_1-p_2)+(v_1-v_2)\cdot \nabla v_1+ v_2\cdot \nabla (v_2-v_1), &{}\quad x\in D,\\ \nabla \cdot (u_1-u_2)&{}{}={}&{}0,\quad x\in D,\\ (u_1-u_2)|_{\partial D}&{}{}={}&{}0,\\ u_1-u_2&{}{}\rightarrow {}&{} 0\quad {\mathrm{as}}~|x|\rightarrow \infty . \end{array}\right. \end{aligned}$$

By applying Proposition 2.1 again, we find

$$\begin{aligned} \Vert u_1-u_2\Vert _B\le 4C'a^{-\frac{3}{n+1}}(\Vert v_1\Vert _B+\Vert v_2\Vert _B)\Vert v_1-v_2\Vert _B \le 64C'^2a^{\frac{n-2}{n+1}}\Vert v_1-v_2\Vert _B \end{aligned}$$

and the map $\Psi$ is contractive on account of (2.11). The proof is complete. $\square$

3 Proof of Theorem 1.2

In this section, we prove Theorem 1.2. We define the operator $A_a:L^q_{\sigma }(D)\rightarrow L^{q}_{\sigma }(D) ~(a>0,1<q<\infty )$ by

$$\begin{aligned} \mathscr {D}(A_a)=W^{2,q}(D)\cap W^{1,q}_0(D)\cap L^{q}_{\sigma }(D),\quad A_au=-P\left[ \Delta u-a\frac{\partial u}{\partial x_1}\right] . \end{aligned}$$

Here, $W_0^{1,q}(D)$ denotes the completion of $C_0^\infty (D)$ in the Sobolev space $W^{1,q}(D)$. It is well known that $-A_a$ generates an analytic $C_0$-semigroup $e^{-tA_a}$ called the Oseen semigroup in $L^q_{\sigma }(D)$, see Miyakawa [30, Theorem 4.2], Enomoto and Shibata [7, Theorem 4.4]. The following $L^q$-$L^r$ estimates of $e^{-tA_a}$, which play an important role in the proof of Theorem 1.2, were established by Kobayashi and Shibata [26] in the three-dimensional case and further developed by Enomoto and Shibata [7, 8] for $n\ge 3$. We also note that $L^q$-$L^r$ estimates in the two-dimensional case were first established by Hishida [22], and recently Maekawa [28] derived those estimates uniformly in small $a>0$ as a significant improvement of [22].

Proposition 3.1

[7, 8, 26] Let $n\ge 3$, $\sigma _0>0$ and assume $|a|\le \sigma _0$.

• 1. Let $1<q\le r\le \infty ~(q\ne \infty )$. Then we have

  $$\begin{aligned}&\Vert e^{-tA_a}f\Vert _{r}\le Ct^{-\frac{n}{2}(\frac{1}{q}-\frac{1}{r})}\Vert f\Vert _q \end{aligned}$$

  (3.1)

  for $t>0$ and $f\in L^{q}_\sigma (D)$, where $C=C(n,\sigma _0,q,r,D)>0$ is independent of a.

• 2. Let $1<q\le r\le n$. Then we have

  $$\begin{aligned}&\Vert \nabla e^{-tA_a}f\Vert _{r}\le Ct^{-\frac{n}{2} (\frac{1}{q}-\frac{1}{r})-\frac{1}{2}}\Vert f\Vert _q \end{aligned}$$

  (3.2)

  for $t>0$ and $f\in L^{q}_\sigma (D)$, where $C=C(n,\sigma _0,q,r,D)>0$ is independent of a.

• 3. Let $n/(n-1)\le q\le r\le \infty ~(q\ne \infty )$. Then we have

  $$\begin{aligned} \Vert e^{-tA_a}P\nabla \cdot F\Vert _r\le Ct^{-\frac{n}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2}}\Vert F\Vert _q \end{aligned}$$

  (3.3)

  for $t>0$ and $F\in L^q(D)$, where $C=C(n,\sigma _0,q,r,D)>0$ is independent of a.

The proof of the assertion 3 is simply based on duality argument together with semigroup property especially for the case $r=\infty$.

We also prepare the following lemma, which plays a role to prove the uniqueness within Y defined by (1.27).

Lemma 3.2

Let $n\ge 3$ and $a>0$. For each precompact set $K\subset L^n_\sigma (D)$, we have

$$\begin{aligned} \lim _{t\rightarrow 0}\sup _{f\in K} t^{\frac{1}{2}}\big (\Vert e^{-t A_a}f\Vert _\infty + \Vert \nabla e^{-t A_a}f\Vert _n\big )=0. \end{aligned}$$

(3.4)

Proof

By applying Proposition 3.1 and approximating $f\in L^n_{\sigma }(D)$ by a sequence in $C^\infty _{0,\sigma }(D)$, we have

$$\begin{aligned} \lim _{t\rightarrow 0}t^{\frac{1}{2}}\big (\Vert e^{-t A_a}f\Vert _\infty + \Vert \nabla e^{-t A_a}f\Vert _n\big )=0 \end{aligned}$$

(3.5)

for all $f\in L^n_{\sigma }(D)$. Given $\eta >0$, let $f_1,\cdots ,f_m\in K$ fulfill $K\subset \displaystyle \bigcup _{j=1}^mB(f_j;\eta ),$ where $B(f_j;\eta ):=\{g\in L^n_{\sigma }(D)\mid \Vert g-f_j\Vert _{n}<\eta \}$. For each $f\in K$, we choose $f_i\in K$ such that $f\in B(f_i;\eta )$. Then it follows from (3.1) that

$$\begin{aligned} t^{\frac{1}{2}}\Vert e^{-tA_a}f\Vert _{\infty }&\le t^{\frac{1}{2}}\Vert e^{-tA_a}f_i\Vert _{\infty } +t^{\frac{1}{2}}\Vert e^{-tA_a}(f-f_i)\Vert _{\infty }\\&\le t^{\frac{1}{2}}\Vert e^{-tA_a}f_i\Vert _{\infty }+C\Vert f-f_i\Vert _{n} \le \sum _{j=1}^mt^{\frac{1}{2}} \Vert e^{-tA_a}f_j\Vert _{\infty }+C\eta . \end{aligned}$$

Since the right-hand side is independent of $f\in K$ and since $\eta$ is arbitrary, (3.5) yields

$$\begin{aligned} \lim _{t\rightarrow 0}\sup _{f\in K} t^{\frac{1}{2}}\Vert e^{-tA_a}f\Vert _{\infty }=0. \end{aligned}$$

We can discuss the $L^n$ norm of the first derivative similarly and thus conclude (3.4). $\square$

We recall a function space $Y_{0}$ defined by (1.22), which is a Banach space equipped with norm $\Vert \cdot \Vert _{Y}=\Vert \cdot \Vert _{Y,\infty }$, where

$$\begin{aligned}&\Vert v\Vert _{Y,t}:=[v]_{n,t}+[v]_{\infty ,t}+[\nabla v]_{n,t},\\&[v]_{q,t}:= \sup _{0<\tau<t}\tau ^{\frac{1}{2}-\frac{n}{2q}}\Vert v(\tau )\Vert _{q},\quad q=n,\infty ;\quad \quad [\nabla v]_{n,t}:=\sup _{0<\tau <t}\tau ^{\frac{1}{2}}\Vert \nabla v(\tau )\Vert _{n} \end{aligned}$$

for $t\in (0,\infty ]$. Construction of the solution is based on the following.

Lemma 3.3

Suppose $0<a^{(n-2)/(n+1)}<\delta$, where $\delta$ is a constant in Theorem 1.1 with (1.15)–(1.17). Let $\psi$ be a function on $\mathbb {R}$ satisfying (1.1) and set $M=\displaystyle \max _{t\in \mathbb {R}}|\psi '(t)|$. Suppose that $u_s$ is the stationary solution obtained in Theorem 1.1. For $u,v\in Y_{0}$, we set

$$\begin{aligned}&G_1(u,v)(t)=\int _0^t e^{-(t-\tau )A_a} P[u\cdot \nabla v](\tau )\,d\tau ,\quad G_2(v)(t)=\int _0^t e^{-(t-\tau )A_a}P[\psi (\tau )v\cdot \nabla u_s]\,d\tau ,\\&G_3(v)(t)=\int _0^t e^{-(t-\tau )A_a}P[\psi (\tau )u_s\cdot \nabla v]\,d\tau ,\\&G_4(v)(t)=\int _0^t e^{-(t-\tau )A_a}P\left[ (1-\psi (\tau ))a \frac{\partial v}{\partial x_1}(\tau )\right] \,d\tau ,\\&H_1(t)=\int _0^t e^{-(t-\tau )A_a}Ph_1(\tau )\,d\tau ,\quad H_2(t)=\int _0^t e^{-(t-\tau )A_a}Ph_2(\tau )\,d\tau , \end{aligned}$$

where $h_1$ and $h_2$ are defined by (1.19) and (1.20), respectively. Then we have $G_1(u,v),G_i(v),H_j$ $\in Y_{0}$ $(i=2,3,4,j=1,2)$ along with

$$\begin{aligned}&\Vert G_1(u,v)\Vert _{Y,t}\le C[u]^\frac{1}{2}_{n,t}[u]^\frac{1}{2}_{\infty ,t} [\nabla v]_{n,t}, \end{aligned}$$

(3.6)

$$\begin{aligned}&\Vert G_2(v)\Vert _{Y,t}\le C\big (\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}} +\Vert \nabla u_s\Vert _{\frac{n}{2}}+ \Vert \nabla u_s\Vert _{\frac{n}{2-\rho _4}}\big )[v]_{\infty ,t}, \end{aligned}$$

(3.7)

$$\begin{aligned}&\Vert G_3(v)\Vert _{Y,t}\le C\big (\Vert u_s\Vert _{\frac{n}{1+\rho _1}}+\Vert u_s\Vert _{n}+ \Vert u_s\Vert _{\frac{n}{1-\rho _2}}\big )[\nabla v]_{n,t}, \end{aligned}$$

(3.8)

$$\begin{aligned}&\Vert G_4(v)\Vert _{Y,t}\le Ca[\nabla v]_{n,t}, \end{aligned}$$

(3.9)

$$\begin{aligned}&\Vert H_1\Vert _{Y,t}\le CM\Vert u_s\Vert _n, \end{aligned}$$

(3.10)

$$\begin{aligned}&\Vert H_2\Vert _{Y,t}\le C\big (\Vert u_s\Vert _{\frac{n}{1-\rho _2}} \Vert \nabla u_s\Vert _{\frac{n}{2-\rho _4}}+a \Vert \nabla u_s\Vert _{\frac{n}{2-\rho _4}}\big ) \end{aligned}$$

(3.11)

for all $t\in (0,\infty ]$ and

$$\begin{aligned} \lim _{t\rightarrow 0}\Vert H_j(t)\Vert _{Y,t}=0 \end{aligned}$$

(3.12)

for $j=1,2$. Here, C is a positive constant independent of $u,v,\psi ,a$ and t.

Proof

The continuity of those functions in t is deduced by use of properties of analytic semigroups together with Proposition 3.1 in the same way as in Fujita and Kato [15]. Since $L^\infty$ estimate is always the same as $L^n$ estimate of the first derivative, the estimate of $[\cdot ]_{\infty ,t}$ may be omitted. Although (3.6)–(3.8) are discussed in Enomoto and Shibata [8, Lemma 3.1.] we briefly give the proof for completeness. We find that $u\in Y_0$ satisfies $u(t)\in L^{2n}(D)$ and

$$\begin{aligned} \Vert u(t)\Vert _{2n}\le t^{-\frac{1}{4}}[u]^{\frac{1}{2}}_{n,t}[u]^{\frac{1}{2}}_{\infty ,t} \end{aligned}$$

for all $t>0$, which together with Proposition 3.1 implies

$$\begin{aligned} \int _0^t \Vert e^{-(t-\tau )A_a}P[u\cdot \nabla v](\tau )\Vert _n\,d\tau \le C\int _0^t(t-\tau )^{-\frac{1}{4}}\Vert u(\tau )\Vert _{2n} \Vert \nabla v(\tau )\Vert _n\,d\tau \le C[u]^{\frac{1}{2}}_{n,t}[u]^{\frac{1}{2}}_{\infty ,t}[\nabla v]_{n,t} \end{aligned}$$

and

$$\begin{aligned} \int _0^t \Vert \nabla e^{-(t-\tau )A_a}P[u\cdot \nabla v](\tau )\Vert _n\,d\tau&\le C\int _0^t(t-\tau )^{-\frac{3}{4}}\Vert u(\tau )\Vert _{2n} \Vert \nabla v(\tau )\Vert _n\,d\tau \\&\le Ct^{-\frac{1}{2}} [u]^{\frac{1}{2}}_{n,t}[u]^{\frac{1}{2}}_{\infty ,t}[\nabla v]_{n,t}. \end{aligned}$$

We thus conclude (3.6). It follows from Proposition 3.1 that

$$\begin{aligned}&\int _0^t \Vert e^{-(t-\tau )A_a}P[\psi (\tau )v\cdot \nabla u_s]\Vert _n\,d\tau \le C\int _0^t(t-\tau )^{-\frac{1}{2}}\Vert v(\tau )\Vert _\infty \Vert \nabla u_s\Vert _{\frac{n}{2}}\,d\tau \le C[v]_{\infty ,t}\Vert \nabla u_s\Vert _{\frac{n}{2}} \end{aligned}$$

(3.13)

and that

$$\begin{aligned} \int _0^t \Vert \nabla e^{-(t-\tau )A_a}P[\psi (\tau )v\cdot \nabla u_s]\Vert _n\,d\tau&\le C\int _0^t(t-\tau )^{-1+\frac{\rho _4}{2}}\Vert v(\tau )\Vert _\infty \Vert \nabla u_s\Vert _{\frac{n}{2-\rho _4}}\,d\tau \nonumber \\&\le Ct^{-\frac{1}{2}+\frac{\rho _4}{2}} [v]_{\infty ,t}\Vert \nabla u_s\Vert _{\frac{n}{2-\rho _4}} \end{aligned}$$

(3.14)

for $t>0$. Furthermore, for $t\ge 2$, we split the integral into

$$\begin{aligned} \int _0^t \Vert \nabla e^{-(t-\tau )A_a}P[\psi (\tau )v\cdot \nabla u_s]\Vert _n\,d\tau =\int _0^{\frac{t}{2}}+\int _{\frac{t}{2}}^{t-1}+\int _{t-1}^t \end{aligned}$$

(3.15)

as in [6, 8]. By applying (3.2), we have

$$\begin{aligned}&\int _0^{\frac{t}{2}}\le C\int _0^{\frac{t}{2}}(t-\tau )^{-1} \Vert v(\tau )\Vert _\infty \Vert \nabla u_s\Vert _{\frac{n}{2}}\,d\tau \le Ct^{-\frac{1}{2}} [v]_{\infty ,t}\Vert \nabla u_s\Vert _{\frac{n}{2}}, \end{aligned}$$

(3.16)

$$\begin{aligned}&\int _{\frac{t}{2}}^{t-1}\le C\int _{\frac{t}{2}}^{t-1} (t-\tau )^{-1-\frac{\rho _3}{2}}\Vert v(\tau )\Vert _\infty \Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}}\,d\tau \le Ct^{-\frac{1}{2}}[v]_{\infty ,t}\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}}, \end{aligned}$$

(3.17)

$$\begin{aligned}&\int _{t-1}^t\le C\int _{t-1}^t(t-\tau )^{-1+\frac{\rho _4}{2}} \Vert v(\tau )\Vert _\infty \Vert \nabla u_s\Vert _{\frac{n}{2-\rho _4}}\,d\tau \le Ct^{-\frac{1}{2}}[v]_{\infty ,t}\Vert \nabla u_s\Vert _{\frac{n}{2-\rho _4}}. \end{aligned}$$

(3.18)

Combining (3.13)–(3.18) yields (3.7). By the same manner, we obtain (3.8). We use Proposition 3.1 to find

$$\begin{aligned} \int _0^t\left\| \nabla ^ke^{-(t-\tau )A_a}P\left[ \big (1-\psi (\tau )\big )a \frac{\partial v}{\partial x_1}\right] \right\| _nd\tau&\le Ca\int _0^{\min \{1,t\}}(t-\tau )^{-\frac{k}{2}} \Vert \nabla v(\tau )\Vert _n\,d\tau \\&\le Ca[\nabla v]_{n,t} \int _0^{\min \{1,t\}}(t-\tau )^{-\frac{k}{2}} \tau ^{-\frac{1}{2}}\,d\tau \end{aligned}$$

for $k=0,1$, which lead us to (3.9). We see (3.10) from

$$\begin{aligned} \int _0^t\left\| \nabla ^k e^{-(t-\tau )A_a} P[\psi '(\tau )u_s]\right\| _nd\tau \le CM\Vert u_s\Vert _{n} \int _0^{\min \{1,t\}}(t-\tau )^{-\frac{k}{2}}\,d\tau \end{aligned}$$

(3.19)

for $k=0,1$ and (3.11) from

$$\begin{aligned}&\int _0^t\left\| \nabla ^ke^{-(t-\tau )A_a} P\left[ \psi (\tau )(1-\psi (\tau )) \Big (u_s\cdot \nabla u_s +a\frac{\partial u_s}{\partial x_1}\Big )\right] \right\| _n\,d\tau \nonumber \\&\quad \le C\Vert u_s\Vert _{\frac{n}{1-\rho _2}}\Vert \nabla u_s\Vert _{\frac{n}{2-\rho _4}} \int _0^{\min \{1,t\}}(t-\tau )^{\frac{\rho _2+\rho _4}{2}-1-\frac{k}{2}}\,d\tau \nonumber \\&\qquad +Ca\Vert \nabla u_s\Vert _{\frac{n}{2-\rho _4}} \int _0^{\min \{1,t\}}(t-\tau )^{-\frac{1}{2}+\frac{\rho _4}{2}-\frac{k}{2}} \,d\tau \end{aligned}$$

(3.20)

for $k=0,1$, where the condition (1.17) is used. The behavior of $G_1(u,v)(t)$ and $G_i(v)(t)$ as well as the one of $H_j(t)$, see (3.12), as $t\rightarrow 0$ follows from (3.6)–(3.9) and (3.19)–(3.20) with $t<1$, so that $G_1(u,v),G_i(v),H_j\in Y_0$ and $\Vert G_1(u,v)(t)\Vert _n+\Vert G_i(v)(t)\Vert _n+\Vert H_j(t)\Vert _n\rightarrow 0$ as $t\rightarrow 0$. The proof is complete. $\square$

Let us construct a solution of (1.21) by applying Lemma 3.3.

Proposition 3.4

Let $\delta$ be the constant in Theorem 1.1 with (1.15)–(1.17). Let $\psi$ be a function on $\mathbb {R}$ satisfying (1.1) and set $M=\displaystyle \max _{t\in \mathbb {R}}|\psi '(t)|$. Then there exists a constant $\varepsilon =\varepsilon (n,D)\in (0,\delta ]$ such that if  $0<(M+1)a^{(n-2)/(n+1)}<\varepsilon$, (1.21) admits a solution $v\in Y_{0}$ with

$$\begin{aligned} \Vert v\Vert _Y\le C(M+1)a^\frac{n-2}{n+1} \end{aligned}$$

(3.21)

and

$$\begin{aligned} \lim _{t\rightarrow 0}\Vert v(t)\Vert _n=0. \end{aligned}$$

(3.22)

Proof

We set

$$\begin{aligned}&v_{0}(t)=0,\nonumber \\&v_{m+1}(t)=\int _0^t e^{-(t-\tau )A_a}P\Big [-v_m\cdot \nabla v_m -\psi (\tau )v_m\cdot \nabla u_s-\psi (\tau )u_s\cdot \nabla v_m +(1-\psi (\tau ))a\frac{\partial v_m}{\partial x_1}\nonumber \\&\qquad \quad +h_1(\tau )+h_2(\tau )\Big ]d\tau \end{aligned}$$

(3.23)

for $m\ge 0.$ It follows from Theorem 1.1, Lemma 3.3 and $a\in (0,1)$ that $v_m\in Y_0$ together with

$$\begin{aligned}&\Vert v_{m}\Vert _{Y,t}\le \Vert G_1(v_{m-1},v_{m-1})\Vert _{Y,t} +\sum _{i=2}^4\Vert G_i(v_{m-1})\Vert _{Y,t} +\Vert H_1\Vert _{Y,t}+\Vert H_2\Vert _{Y,t}, \end{aligned}$$

(3.24)

$$\begin{aligned}&\Vert v_{m}\Vert _{Y}\le C_1\Vert v_{m-1}\Vert _{Y}^2 +C_2a^\frac{n-2}{n+1}\Vert v_{m-1}\Vert _{Y}+C_3(M+1)a^\frac{n-2}{n+1},\nonumber \\&\Vert v_{m+1}-v_m\Vert _{Y}\le \{C_1(\Vert v_m\Vert _{Y}+\Vert v_{m-1}\Vert _{Y})+C_2a^\frac{n-2}{n+1}\} \Vert v_m-v_{m-1}\Vert _{Y} \end{aligned}$$

(3.25)

for all $m\ge 1$. Hence, if we assume

$$\begin{aligned} (M+1)a^\frac{n-2}{n+1} <\min \left\{ \delta , \frac{1}{2C_2},\frac{1}{16C_1C_3}\right\} =:\varepsilon , \end{aligned}$$

(3.26)

it holds that

$$\begin{aligned}&\Vert v_m\Vert _{Y}\le \frac{1-C_2a^\frac{n-2}{n+1}-\sqrt{\big (1-C_2a^\frac{n-2}{n+1})^2 -4C_1C_3(M+1)a^\frac{n-2}{n+1}}}{2C_1} \le 4C_3(M+1)a^\frac{n-2}{n+1},\nonumber \\&\Vert v_{m+1}-v_m\Vert _{Y}\le \{8C_1C_3(M+1)a^\frac{n-2}{n+1}+C_2a^\frac{n-2}{n+1}\} \Vert v_{m}-v_{m-1}\Vert _{Y} \end{aligned}$$

(3.27)

for all $m\ge 1$ and that

$$\begin{aligned} 8C_1C_3(M+1)a^\frac{n-2}{n+1}+C_2a^\frac{n-2}{n+1}<1. \end{aligned}$$

Therefore, we obtain a solution $v\in Y_{0}$ satisfying (3.21) with $C=4C_3$. Moreover, by letting $m\rightarrow \infty$ in (3.24) and by using (3.6)–(3.9) and (3.12), we have (3.22), which completes the proof. $\square$

Remark 3.5

Let $b\in L_{\sigma }^n(D)$. By the same procedure, we can also construct a solution $T(t)b:=v(t)\in Y_{0}$ for the integral equation

$$\begin{aligned} v(t)&=e^{-tA_a}b+\int _0^t e^{-(t-\tau )A_a}P\Big [-v\cdot \nabla v -\psi (\tau )v\cdot \nabla u_s-\psi (\tau )u_s\cdot \nabla v\nonumber \\&\quad \quad +(1-\psi (\tau ))a\frac{\partial v}{\partial x_1} +h_1(\tau )+h_2(\tau )\Big ]d\tau \end{aligned}$$

(3.28)

whenever

$$\begin{aligned} \Vert b\Vert _n+(M+1)a^\frac{n-2}{n+1}<\min \left\{ \delta , \frac{1}{2C_2},\frac{1}{16C_1C_0},\frac{1}{16C_1C_3}\right\} \end{aligned}$$

is satisfied. Here, the constant $C_0$ is determined by the following three estimates:

$$\begin{aligned} \Vert e^{-tA_a}b\Vert _q\le C_0t^{-\frac{1}{2}+\frac{3}{2q}}\Vert b\Vert _n, \quad \quad q=n,\infty ;\quad \quad \Vert \nabla e^{-tA_a}b\Vert _n\le C_0t^{-\frac{1}{2}}\Vert b\Vert _n. \end{aligned}$$

Moreover, we find that the solution T(t)b is estimated by

$$\begin{aligned} \Vert T(\cdot )b\Vert _{Y}\le 4\big (C_0\Vert b\Vert _n+C_3(M+1)a^\frac{n-2}{n+1}\big ). \end{aligned}$$

This will be used in the proof of uniqueness of solutions within Y, see (1.27).

We further derive sharp decay properties of the solution v(t) obtained above. To this end, the first step is the following. In what follows, for simplicity of notation, we write

$$\begin{aligned} G_1(t)=G_1(v,v)(t),\quad G_i(t)=G_i(v)(t) \end{aligned}$$

for $i=2,3,4$, which are defined in Lemma 3.3.

Lemma 3.6

Let $\varepsilon$ be the constant in Proposition 3.4. Given $\rho \in (0,1)$ satisfying $\rho \le \min \{\rho _1,\rho _3\}$, there exists a constant $\varepsilon '=\varepsilon '(\rho ,n,D)\in (0,\varepsilon ]$ such that if  $0<(M+1)a^{(n-2)/(n+1)}<\varepsilon '$, then the solution v(t) obtained in Proposition 3.4 satisfies

$$\begin{aligned}&\Vert v(t)\Vert _q=O(t^{-\frac{1}{2}+\frac{n}{2q}-\frac{\rho }{2}}), \quad \quad n\le \forall q\le \infty , \end{aligned}$$

(3.29)

$$\begin{aligned}&\Vert \nabla v(t)\Vert _n=O(t^{-\frac{1}{2}-\frac{\rho }{2}}) \end{aligned}$$

(3.30)

as $t\rightarrow \infty$.

Proof

We start with the case $q=n$, that is,

$$\begin{aligned} \Vert v(t)\Vert _n=O(t^{-\frac{\rho }{2}}) \end{aligned}$$

(3.31)

as $t\rightarrow \infty .$ By using (3.1), we have

$$\begin{aligned} \Vert G_1(t)\Vert _n&\le Ct^{-\frac{\rho }{2}} \big (\sup _{0<\tau<t}\tau ^{\frac{1}{2}}\Vert \nabla v(\tau )\Vert _n\big ) \big (\sup _{0<\tau<t}\tau ^{\frac{\rho }{2}}\Vert v(\tau )\Vert _n\big ) \le Ct^{-\frac{\rho }{2}}\Vert v\Vert _Y \sup _{0<\tau <t}\tau ^{\frac{\rho }{2}}\Vert v(\tau )\Vert _n, \end{aligned}$$

(3.32)

$$\begin{aligned} \Vert G_2(t)\Vert _n&\le Ct^{-\frac{\rho _3}{2}}\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}} \sup _{0<\tau <t}\tau ^{\frac{1}{2}}\Vert v(\tau )\Vert _\infty \le Ct^{-\frac{\rho _3}{2}} \Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}}\Vert v\Vert _Y \end{aligned}$$

(3.33)

and

$$\begin{aligned} \Vert G_3(t)\Vert _n\le Ct^{-\frac{\rho _1}{2}} \Vert u_s\Vert _{\frac{n}{1+\rho _1}}\Vert v\Vert _Y \end{aligned}$$

(3.34)

for all $t>0$. Moreover, we obtain

$$\begin{aligned} \Vert G_4(t)\Vert _n\le Ca\int _0^{\min \{1,t\}}(t-\tau )^{-\frac{1}{2}}\Vert v(\tau )\Vert _n\, d\tau \le Cat^{-\frac{1}{2}}\Vert v\Vert _Y \end{aligned}$$

(3.35)

for all $t>0$ by use of (3.3). From (3.1) we see that

$$\begin{aligned} \Vert H_1(t)\Vert _n \le CM t^{-\frac{\rho _1}{2}}\Vert u_s\Vert _{\frac{n}{1+\rho _1}} \end{aligned}$$

(3.36)

and that

$$\begin{aligned} \Vert H_2(t)\Vert _n\le Ct^{-\frac{2-\rho _2}{2}} \Vert u_s\Vert _{\frac{n}{1-\rho _2}} \Vert \nabla u_s\Vert _{\frac{n}{2}}+Cat^{-\frac{1+\rho _3}{2}} \Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}} \end{aligned}$$

(3.37)

for $t>0$. Note that $\rho _2<1$, see (1.16). Collecting (3.32)–(3.37) for $t>1$ and (3.21) with $C=4C_3$ yields

$$\begin{aligned} \sup _{0<\tau<t}\tau ^{\frac{\rho }{2}}\Vert v(\tau )\Vert _n&\le C_4\Vert v\Vert _Y \sup _{0<\tau<t}\tau ^{\frac{\rho }{2}}\Vert v(\tau )\Vert _n+C_5\\&\le 4C_3C_4(M+1)a^{\frac{n-2}{n+1}} \sup _{0<\tau <t}\tau ^{\frac{\rho }{2}}\Vert v(\tau )\Vert _n+C_5 \end{aligned}$$

with some constants $C_4=C_4(\rho )>0$ and $C_5=C_5(\Vert v\Vert _Y,u_s,a,M,\rho _1,\rho _2,\rho _3)>0$ independent of t, where $C_3$ comes from estimates of $H_j(t)$ $(j=1,2)$ in (3.25). Therefore, if we assume

$$\begin{aligned} (M+1)a^{\frac{n-2}{n+1}}<\min \Big \{\varepsilon ,\frac{1}{4C_3C_4} \Big \}=:\varepsilon ', \end{aligned}$$

we have $\Vert v(t)\Vert _n\le Ct^{-\rho /2}$ for all $t>0$, which implies (3.31).

We next show that

$$\begin{aligned} \Vert v(t)\Vert _\infty +\Vert \nabla v(t)\Vert _n=O(t^{-\frac{1}{2}-\frac{\rho }{2}}) \end{aligned}$$

as $t\rightarrow \infty$, which together with (3.31) implies (3.29) and (3.30). It suffices to show that

$$\begin{aligned} t^{\frac{1}{2}}\Vert v(t)\Vert _\infty +t^{\frac{1}{2}}\Vert \nabla v(t)\Vert _n\le C\Big \Vert v\Big (\frac{t}{2}\Big )\Big \Vert _n \end{aligned}$$

(3.38)

for all $t\ge 2$. The following argument is similar to Enomoto and Shibata [8]. When $t\ge T>1$, we have

$$\begin{aligned} v(t)=e^{-(t-T)A_a}v(T)-\int _T^t e^{-(t-\tau )A_a}P\big [v\cdot \nabla v +v\cdot \nabla u_s+u_s\cdot \nabla v\big ]\,d\tau . \end{aligned}$$

(3.39)

By the same argument as in the proof of Lemma 3.3 and by (1.10), (3.26) as well as (3.21) with $C=4C_3$, the integral of (3.39) is estimated as

$$\begin{aligned}&\int _T^t\Vert e^{-(t-\tau )A_a}P[\cdots ]\Vert _\infty \,d\tau + \int _T^t\Vert \nabla e^{-(t-\tau )A_a}P[\cdots ]\Vert _n\,d\tau \\&\quad \le C_1(t-T)^{-\frac{1}{2}}\, \big (\sup _{T\le \tau \le t}\Vert v(\tau )\Vert _n\big )^{\frac{1}{2}}\, \big (\sup _{T\le \tau \le t}(\tau -T)^{\frac{1}{2}}\Vert v(\tau )\Vert _\infty \big )^{\frac{1}{2}}\, \big (\sup _{T\le \tau \le t}(\tau -T)^{\frac{1}{2}}\Vert \nabla v(\tau )\Vert _n \big )\\&\quad \quad ~+C_2a^{\frac{n-2}{n+1}}(t-T)^{-\frac{1}{2}} \big \{\sup _{T\le \tau \le t}(\tau -T)^{\frac{1}{2}}\Vert v(\tau )\Vert _\infty +\sup _{T\le \tau \le t}(\tau -T)^{\frac{1}{2}} \Vert \nabla v(\tau )\Vert _n\big \}\\&\quad \le C_1(t-T)^{-\frac{1}{2}}\Vert v\Vert _Y \sup _{T\le \tau \le t}(\tau -T)^{\frac{1}{2}} \Vert \nabla v(\tau )\Vert _n\\&\quad \quad ~ +\frac{1}{2}(t-T)^{-\frac{1}{2}} \big \{\sup _{T\le \tau \le t}(\tau -T)^{\frac{1}{2}}\Vert v(\tau )\Vert _\infty +\sup _{T\le \tau \le t}(\tau -T)^{\frac{1}{2}} \Vert \nabla v(\tau )\Vert _n\big \}\\&\quad \le \frac{3}{4}(t-T)^{-\frac{1}{2}} \sup _{T\le \tau \le t}(\tau -T)^{\frac{1}{2}}\Vert \nabla v(\tau )\Vert _n +\frac{1}{2}(t-T)^{-\frac{1}{2}} \sup _{T\le \tau \le t}(\tau -T)^{\frac{1}{2}}\Vert v(\tau )\Vert _\infty . \end{aligned}$$

Therefore, we have

$$\begin{aligned} \sup _{T\le \tau \le t}(\tau -T)^{\frac{1}{2}}\Vert \nabla v(\tau )\Vert _n +\sup _{T\le \tau \le t}(\tau -T)^{\frac{1}{2}}\Vert v(\tau )\Vert _\infty \le C\Vert v(T)\Vert _n \end{aligned}$$

for all $t\ge T$. This combined with $t^{1/2}\le \sqrt{2}(t-T)^{1/2}$ for $t\ge 2T$ asserts that

$$\begin{aligned} t^{\frac{1}{2}}\Vert \nabla v(t)\Vert _n +t^{\frac{1}{2}}\Vert v(t)\Vert _\infty \le C\Vert v(T)\Vert _n \end{aligned}$$

for all $t\ge 2T$. We then put $T=t/2~(t\ge 2)$ to conclude (3.38). $\square$

Sharp decay properties (1.23)–(1.24) for the case $n=3$ are established in the following proposition.

Proposition 3.7

Let $n=3$ and set $\varepsilon _*:=\varepsilon '(\rho ,3,D)$ which is the constant in Lemma 3.6 with $\rho :=\min \{\rho _1,\rho _3\}$ (recall that $0<\rho _1<1/2$, $0<\rho _3<1/4$ for $n=3$). If  $0<(M+1)a^{1/4}<\varepsilon _*$, then the solution v(t) obtained in Proposition 3.4 enjoys (1.23) and (1.24).

Proof

The case $\rho _1\le \rho _3$ directly follows from Lemma 3.6. To discuss the other case $\rho _3<\rho _1$, we show by induction that if $0<(M+1)a^{1/4}<\varepsilon _*$, then

$$\begin{aligned} \Vert v(t)\Vert _3=O(t^{-\sigma _k}),\qquad \sigma _k: =\min \Big \{\frac{k}{2}\rho _3,\,\frac{\rho _1}{2}\Big \} \end{aligned}$$

(3.40)

as $t\rightarrow \infty$ for all $k\ge 1$. We already know (3.40) with $k=1$ from Lemma 3.6.

Let $k\ge 2$ and suppose (3.40) with $k-1$. By taking (3.21) (near $t=0$) and (3.38) into account, we have

$$\begin{aligned} J_{k-1}(v):=\sup _{\tau>0}(1+\tau )^{\sigma _{k-1}} \Vert v(\tau )\Vert _3+ \sup _{\tau >0}\tau ^{\frac{1}{2}}(1+\tau )^{\sigma _{k-1}} \big (\Vert v(\tau )\Vert _\infty +\Vert \nabla v(\tau )\Vert _3\big )<\infty . \end{aligned}$$

We use this to see that

$$\begin{aligned} \Vert G_1(t)\Vert _3&\le C\int _0^t(t-\tau )^{-\frac{1}{2}}\tau ^{-\frac{1}{2}} (1+\tau )^{-2\sigma _{k-1}}\,d\tau \\&\qquad \times \big (\sup _{\tau>0}(1+\tau )^{\sigma _{k-1}} \Vert v(\tau )\Vert _3\big ) \big (\sup _{\tau >0}\tau ^{\frac{1}{2}}(1+\tau )^{\sigma _{k-1}} \Vert \nabla v(\tau )\Vert _3\big )\\&\le Ct^{-2\sigma _{k-1}}J_{k-1}(v)^2, \end{aligned}$$

and that

$$\begin{aligned} \Vert G_2(t)\Vert _3&\le C\int _0^t(t-\tau )^{-\frac{1+\rho _3}{2}} \tau ^{-\frac{1}{2}}(1+\tau )^{-\sigma _{k-1}}\,d\tau \, \Vert \nabla u_s\Vert _{\frac{3}{2+\rho _3}} \sup _{\tau >0}\tau ^{\frac{1}{2}}(1+\tau )^{\sigma _{k-1}}\Vert v(\tau )\Vert _\infty \\&\le Ct^{-\frac{\rho _3}{2}-\sigma _{k-1}} \Vert \nabla u_s\Vert _{\frac{3}{2+\rho _3}}J_{k-1}(v) \end{aligned}$$

for $t>0$ due to $\sigma _{k-1}\le \rho _1/2<1/4$. From these and (3.34)–(3.37), we obtain (3.40) with k. We thus conclude (1.23) with $q=3$, which together with (3.38) completes the proof. $\square$

To derive even more rapid decay properties of the solution v(t) for $n\ge 4$, we need the following lemma, which gives the $L^{q_0}$-decay of v(t) with a specific $q_0$, see (3.43).

Lemma 3.8

Let $n\ge 4$. Suppose $1<\rho _1\le 1+\rho _3$ in addition to (1.16) (the set of those parameters is nonvoid when $n\ge 4)$. Let $\varepsilon$ be the constant in Proposition 3.4 and v(t) the solution obtained there. Given $\gamma$ satisfying

$$\begin{aligned} \max \Big \{0,\,\frac{\rho _1+3-n}{2}\Big \}<\gamma <\frac{1}{2} \end{aligned}$$

(3.41)

(note that (1.16) yields $\rho _1<n-2)$, there exists a constant $\varepsilon ''=\varepsilon ''(\gamma ,n,D) \in (0,\varepsilon ]$ such that if  $0<(M+1)a^{(n-2)/(n+1)} <\varepsilon ''$, then $v(t)\in L^{q_0}(D)$ for all $t>0$ and

$$\begin{aligned} \sup _{\tau >0}(1+\tau )^\gamma \Vert v(\tau )\Vert _{q_0}<\infty , \end{aligned}$$

(3.42)

where

$$\begin{aligned} q_0:=\frac{n}{1+\rho _1-2\gamma }\,(<n). \end{aligned}$$

(3.43)

Proof

We show that there exists a constant $\varepsilon ''(\gamma ,n,D)\in (0,\varepsilon ]$ such that if $0<(M+1)a^{(n-2)/(n+1)}<\varepsilon ''$, then $v_m(t)\in L^{q_0}(D)$ for all $t>0$ along with

$$\begin{aligned}&K_m:=\sup _{\tau >0}(1+\tau )^\gamma \Vert v_m(\tau )\Vert _{q_0}<\infty ,&K_m\le \frac{1}{2}K_{m-1}+C(M+1)a^{\frac{n-1}{n+1}} \end{aligned}$$

(3.44)

for all $m\ge 1$, where $v_m(t)$ is the approximate solution defined by (3.23) and C is a positive constant independent of a and m. We use (3.1) to see that

$$\begin{aligned} \int _0^t\Vert e^{-(t-\tau )A_a}Ph_1(\tau )\Vert _{q_0}\,d\tau \le CM\Vert u_s\Vert _{\frac{n}{1+\rho _1}} \int _0^{\min \{1,t\}} (t-\tau )^{-\gamma }\,d\tau \le CM\Vert u_s\Vert _{\frac{n}{1+\rho _1}}(1+t)^{-\gamma } \end{aligned}$$

(3.45)

for $t>0$. Moreover, it holds that

$$\begin{aligned} \int _0^t\Big \Vert e^{-(t-\tau )A_a} P\Big [\psi (\tau )\big (1-\psi (\tau )\big )a \frac{\partial u_s}{\partial x_1}\Big ]\Big \Vert _{q_0}\,d\tau&\le Ca\Vert \nabla u_s\Vert _{r} \end{aligned}$$

for $t\le 2$, where $r:=\min \{n/(2-\rho _4),q_0\}$ and that

$$\begin{aligned} \int _0^t\Big \Vert e^{-(t-\tau )A_a} P\Big [\psi (\tau )\big (1-\psi (\tau )\big )a \frac{\partial u_s}{\partial x_1}\Big ]\Big \Vert _{q_0}\,d\tau&\le Ca\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}} \int _0^1(t-\tau )^{-\gamma -\frac{1+\rho _3-\rho _1}{2}}\,d\tau \\&\le Ca\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}}t^{-\gamma } \end{aligned}$$

for $t>2$ as well as that

$$\begin{aligned} \int _0^t\Vert e^{-(t-\tau )A_a} P[\psi (\tau )\big (1-\psi (\tau )\big )u_s\cdot \nabla u_s]\Vert _{q_0}\,d\tau&\le C\Vert u_s\Vert _{\frac{n}{1+\kappa }} \Vert \nabla u_s\Vert _{\frac{n}{2}}\int _0^{\min \{1,t\}} (t-\tau )^{-1-\gamma +\frac{\rho _1-\kappa }{2}}\,d\tau \\&\le C\Vert u_s\Vert _{\frac{n}{1+\kappa }} \Vert \nabla u_s\Vert _{\frac{n}{2}}(1+t)^{-\gamma } \end{aligned}$$

for $t>0$, where $\max \{0,\rho _1-2\}<\kappa <\min \{n-3,\rho _1-2\gamma \}$ (note that (1.16) yields $\rho _1<n-1$). These estimates imply

$$\begin{aligned} \int _0^t\Vert e^{-(t-\tau )A_a}Ph_2(\tau )\Vert _{q_0}\,d\tau \le C(a\Vert \nabla u_s\Vert _{r}+a\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}}+ \Vert u_s\Vert _{\frac{n}{1+\kappa }} \Vert \nabla u_s\Vert _{\frac{n}{2}})(1+t)^{-\gamma } \end{aligned}$$

for $t>0$, which together with (3.45) and (1.10) leads us to $v_1(t)\in L^{q_0}(D)$ for all $t>0$ with

$$\begin{aligned} K_1\le C(M+1)a^{\frac{n-1}{n+1}}. \end{aligned}$$

(3.46)

This proves (3.44) with $m=1$ since $K_0=0$.

Let $m\ge 2$ and suppose that $v_{m-1}(t)\in L^{q_0}(D)$ for all $t>0$ and (3.44) with $m-1$. Then we have $G_1(v_{m-1},v_{m-1})(t)\in L^{q_0}(D)$ for $t>0$ with

$$\begin{aligned} \sup _{\tau>0}(1+\tau )^{\gamma }\Vert G_1(v_{m-1},v_{m-1})(\tau )\Vert _{q_0} \le CK_{m-1} \sup _{\tau >0}\tau ^{\frac{1}{2}}\Vert \nabla v_{m-1}(\tau )\Vert _n. \end{aligned}$$

(3.47)

Let $t\ge 2$ and split the integral into

$$\begin{aligned} \int _0^t\Vert e^{-(t-\tau )A_a}P[\psi (\tau )u_s\cdot \nabla v_{m-1}] \Vert _{q_0}\,d\tau =\int _0^{\frac{t}{2}}+\int _{\frac{t}{2}}^{t-1}+\int _{t-1}^t. \end{aligned}$$

Let $\lambda \in (0,\rho _1]$ satisfy $\lambda <n-3+2\gamma -\rho _1$; in fact, we can take such $\lambda$ due to (3.41). Then (3.3) with $F=v_{m-1}\otimes u_s$ implies

$$\begin{aligned} \int _0^{\frac{t}{2}}&\le C\int _0^{\frac{t}{2}}(t-\tau )^{-1}\Vert u_s\Vert _n \Vert v_{m-1}(\tau )\Vert _{q_0}\,d\tau \le Ct^{-\gamma }\Vert u_s\Vert _{n}K_{m-1},\\ \int _{\frac{t}{2}}^{t-1}&\le C\int _{\frac{t}{2}}^{t-1} (t-\tau )^{-1-\frac{\lambda }{2}}\Vert u_s\Vert _{\frac{n}{1+\lambda }} \Vert v_{m-1}(\tau )\Vert _{q_0}\,d\tau \le Ct^{-\gamma }\Vert u_s\Vert _{\frac{n}{1+\lambda }}K_{m-1},\\ \int _{t-1}^t&\le C\int _{t-1}^t(t-\tau )^{-1+\frac{\rho _2}{2}} \Vert u_s\Vert _{\frac{n}{1-\rho _2}} \Vert v_{m-1}(\tau )\Vert _{q_0}\,d\tau \le Ct^{-\gamma }\Vert u_s\Vert _{\frac{n}{1-\rho _2}}K_{m-1} \end{aligned}$$

for $t\ge 2$. Moreover, we use (3.3) again to see that

$$\begin{aligned} \int _0^t\Vert e^{-(t-\tau )A_a}P[\psi (\tau )u_s\cdot \nabla v_{m-1}] \Vert _{q_0}\,d\tau&\le C\int _0^t(t-\tau )^{-1+\frac{\rho _2}{2}} \Vert u_s\Vert _{\frac{n}{1-\rho _2}}\Vert v_{m-1}(\tau )\Vert _{q_0}\,d\tau \\&\le C\Vert u_s\Vert _{\frac{n}{1-\rho _2}}K_{m-1} \end{aligned}$$

for $t\le 2$. We thus conclude $G_3(v_{m-1})(t)\in L^{q_0}(D)$ for $t>0$ with

$$\begin{aligned} \sup _{\tau >0}(1+\tau )^\gamma \Vert G_3(v_{m-1})(\tau )\Vert _{q_0}\le C(\Vert u_s\Vert _n+\Vert u_s\Vert _{\frac{n}{1+\lambda }}+\Vert u_s\Vert _{\frac{n}{1-\rho _2}}) K_{m-1}. \end{aligned}$$

(3.48)

By the same calculation, we have $G_2(v_{m-1})(t)\in L^{q_0}(D)$ for $t>0$ with

$$\begin{aligned} \sup _{\tau >0}(1+\tau )^\gamma \Vert G_2(v_{m-1})(\tau )\Vert _{q_0} \le C(\Vert u_s\Vert _n+\Vert u_s\Vert _{\frac{n}{1+\lambda }}+\Vert u_s\Vert _{\frac{n}{1-\rho _2}}) K_{m-1}. \end{aligned}$$

(3.49)

We also have

$$\begin{aligned} \int _0^t\Big \Vert e^{-(t-\tau )A_a}P\Big [\big (1-\psi (\tau )\big )a \frac{\partial v_{m-1}}{\partial x_1}\Big ]\Big \Vert _{q_0}&\le Ca\int _0^{\min \{1,t\}}(t-\tau )^{-\frac{1}{2}}\Vert v_{m-1}(\tau )\Vert _{q_0}\,d\tau \\&\le CaK_{m-1}(1+t)^{-\frac{1}{2}}\le CaK_{m-1}(1+t)^{-\gamma } \end{aligned}$$

for $t>0$ by (3.3). This together with (3.46)–(3.49), (1.10) and (3.27) yields $v_{m}(t)\in L^{q_0}(D)$ for $t>0$ and

$$\begin{aligned} K_m&\le C(M+1)a^{\frac{n-1}{n+1}}+ {\widetilde{C}}_1\Big \{ \big (\sup _{\tau >0}\tau ^{\frac{1}{2}}\Vert \nabla v_{m-1}(\tau )\Vert _n\big )+ \Vert u_s\Vert _n+\Vert u_s\Vert _{\frac{n}{1+\lambda }}+\Vert u_s\Vert _{\frac{n}{1-\rho _2}} +a\Big \}K_{m-1}\\&\le C(M+1)a^{\frac{n-1}{n+1}}+{\widetilde{C}}_1 (4C_3+{\widetilde{C}}_2)(M+1)a^{\frac{n-2}{n+1}}K_{m-1}. \end{aligned}$$

Suppose

$$\begin{aligned} (M+1)a^{\frac{n-2}{n+1}}<\min \left\{ \varepsilon ,\, \frac{1}{2{\widetilde{C}}_1(4C_3+{\widetilde{C}}_2)}\right\} =:\varepsilon '', \end{aligned}$$

then we get (3.44) with m and, thereby, conclude

$$\begin{aligned} K_m\le 2C(M+1)a^{\frac{n-1}{n+1}} \end{aligned}$$

for all $m\ge 1$. Since we know that $\Vert v_m(t)-v(t)\Vert _n\rightarrow 0$ as $m\rightarrow \infty$ for each $t>0$, we obtain $v(t)\in L^{q_0}(D)$ for $t>0$ with

$$\begin{aligned} \sup _{\tau >0}(1+\tau )^\gamma \Vert v(\tau )\Vert _{q_0} \le 2C(M+1)a^{\frac{n-1}{n+1}}<\infty , \end{aligned}$$

which completes the proof. $\square$

In view of Lemmas 3.6 and 3.8, we prove sharp decay properties (1.25)–(1.26) for $n\ge 4$.

Proposition 3.9

Let $n\ge 4$. Suppose $\rho _3>1$ and $1<\rho _1\le 1+\rho _3$ in addition to (1.16) (the set of those parameters is nonvoid when $n\ge 4).$ Let $\varepsilon$ be the constant in Proposition 3.4. There exists a constant $\varepsilon _*=\varepsilon _*(n,D)\in (0,\varepsilon ]$ such that if $0<(M+1)a^{(n-2)/(n+1)}<\varepsilon _*$, then the solution v(t) obtained in Proposition 3.4 enjoys (1.25) and (1.26).

Proof

Fix $1/2<\rho <1$ and $\gamma >0$ such that

$$\begin{aligned} \max \Big \{\frac{1}{2}-\frac{\rho }{2},\,\frac{\rho _1+3-n}{2}\Big \}<\gamma < \frac{1}{2}. \end{aligned}$$

(3.50)

Let $\varepsilon '(\rho ,n,D)$ and $\varepsilon ''(\gamma ,n,D)$ be the constants in Lemmas 3.6 and 3.8, respectively. We show by induction that if

$$\begin{aligned} (M+1)a^{\frac{n-2}{n+1}}< \min \{\varepsilon '(\rho ,n,D),\varepsilon ''(\gamma ,n,D)\} =:\varepsilon _*(n,D), \end{aligned}$$

then v(t) satisfies

$$\begin{aligned} \Vert v(t)\Vert _n=O(t^{-\sigma _k}),\qquad \sigma _k:=\min \Big \{ \frac{k}{2}\rho ,\,\frac{\rho _1}{2}\Big \} \end{aligned}$$

(3.51)

as $t\rightarrow \infty$ for all $k\ge 1$. This implies (1.25) with $q=n$, which together with (3.38) completes the proof. Since $\rho <\rho _1$, (3.51) with $k=1$ follows from Lemma 3.6. We note that $\sigma _1<1/2$ and $\sigma _k>1/2$ for $k\ge 2$.

Let $k\ge 2$ and suppose (3.51) with $k-1$. Then

$$\begin{aligned} L_{k-1}(v):=\sup _{\tau>0}(1+\tau )^{\sigma _{k-1}}\Vert v(\tau )\Vert _n+ \sup _{\tau >0}\tau ^{\frac{1}{2}}(1+\tau )^{\sigma _{k-1}} \big (\Vert v(\tau )\Vert _\infty +\Vert \nabla v(\tau )\Vert _n\big ) <\infty \end{aligned}$$

holds due to (3.21) (near $t=0$) as well as (3.38). In what follows, we always assume $t\ge 2$. From (3.42), it follows that

$$\begin{aligned} \Vert G_1(t)\Vert _n&\le \int _0^{\frac{t}{2}}(t-\tau )^{-\frac{n}{2q_0}} \Vert v(\tau )\Vert _{q_0}\Vert \nabla v(\tau )\Vert _n\,d\tau +\int _{\frac{t}{2}}^t(t-\tau )^{-\frac{1}{2}} \Vert v(\tau )\Vert _n\Vert \nabla v(\tau )\Vert _n\,d\tau =:I+II \end{aligned}$$

(3.52)

with

$$\begin{aligned} I\le Ct^{-\frac{n}{2q_0}} \big (\sup _{\tau>0}(1+\tau )^\gamma \Vert v(\tau )\Vert _{q_0}\big )L_{k-1}(v) \le Ct^{-\frac{\rho _1}{2}} \big (\sup _{\tau >0}(1+\tau )^\gamma \Vert v(\tau )\Vert _{q_0}\big ) L_{k-1}(v), \end{aligned}$$

(3.53)

where (3.43) and (3.50) are taken into account and

$$\begin{aligned} II\le Ct^{-2\sigma _{k-1}}L_{k-1}(v)^2. \end{aligned}$$

(3.54)

For $G_2(t)$, we split the integral into

$$\begin{aligned} \int _0^t\Vert e^{-(t-\tau )A_a} P[\psi (\tau )v\cdot \nabla u_s]\Vert _n\,d\tau =\int _{0}^{\frac{t}{2}}+\int _{\frac{t}{2}}^{t-1}+ \int _{t-1}^{t}. \end{aligned}$$

Then we find

$$\begin{aligned} \int _0^{\frac{t}{2}}&\le C\int _0^{\frac{t}{2}}(t-\tau )^{-\frac{1+\rho _3}{2}}\tau ^{-\frac{1}{2}} (1+\tau )^{-\sigma _{k-1}}\,d\tau \,\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}} \big (\sup _{\tau >0}\tau ^{\frac{1}{2}} (1+\tau )^{\sigma _{k-1}}\Vert v(\tau )\Vert _\infty \big )\nonumber \\&\le {\left\{ \begin{array}{ll} Ct^{-\frac{\rho _3}{2}-\sigma _{k-1}}\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}} L_{k-1}(v) \le Ct^{-\sigma _{k}}\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}} L_{k-1}(v) \quad \text{ if }~k=2,\\ Ct^{-\frac{1+\rho _3}{2}}\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}} L_{k-1}(v)\le Ct^{-\frac{\rho _1}{2}}\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}} L_{k-1}(v) \quad \text{ if }~k\ge 3 \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \int _{\frac{t}{2}}^{t-1}+\int _{t-1}^{t} \le C t^{-\sigma _{k-1}-\frac{1}{2}}\big (\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}} +\Vert \nabla u_s\Vert _{\frac{n}{2}}\big )L_{k-1}(v), \end{aligned}$$

where we have used $\rho _3>1$ and $\rho _1\le 1+\rho _3.$ Estimates above imply that

$$\begin{aligned} \Vert G_2(t)\Vert _n\le Ct^{-\sigma _k} \big (\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}} +\Vert \nabla u_s\Vert _{\frac{n}{2}}\big )L_{k-1}(v). \end{aligned}$$

(3.55)

Similarly, we observe

$$\begin{aligned} \Vert G_3(t)\Vert _n\le Ct^{-\sigma _k} \big (\Vert u_s\Vert _{\frac{n}{1+\rho _1}} +\Vert u_s\Vert _{n}\big )L_{k-1}(v). \end{aligned}$$

(3.56)

Moreover, by the same manner as in the proof of Lemma 3.6, we obtain

$$\begin{aligned} \Vert G_4(t)\Vert _n&\le Ct^{-\frac{n}{2q_0}}\sup _{\tau>0}(1+\tau )^\gamma \Vert v(\tau )\Vert _{q_0} \le Ct^{-\frac{\rho _1}{2}}\sup _{\tau >0}(1+\tau )^\gamma \Vert v(\tau )\Vert _{q_0}, \end{aligned}$$

(3.57)

$$\begin{aligned} \Vert H_1(t)\Vert _n&\le CMt^{-\frac{\rho _1}{2}}\Vert u_s\Vert _{\frac{n}{1+\rho _1}}, \end{aligned}$$

(3.58)

$$\begin{aligned} \Vert H_2(t)\Vert _n&\le Ct^{-\frac{2+\kappa }{2}} \Vert u_s\Vert _{\frac{n}{1+\kappa }}\Vert \nabla u_s\Vert _{\frac{n}{2}} +Ct^{-\frac{1+\rho _3}{2}} \Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}}\nonumber \\&\le Ct^{-\frac{\rho _1}{2}} \big (\Vert u_s\Vert _{\frac{n}{1+\kappa }}\Vert \nabla u_s\Vert _{\frac{n}{2}} +\Vert \nabla u_s\Vert _{\frac{n}{2+\rho _3}}\big ) \end{aligned}$$

(3.59)

for all $t\ge 2$, where $\kappa$ is chosen such that $\max \{0,\rho _1-2\}<\kappa <\min \{n-3,\rho _1\}$. Collecting (3.52)–(3.59), we conclude (3.51) with k. The proof is complete. $\square$

We next consider the uniqueness. We begin with the classical result on the uniqueness of solutions within $Y_0$ as in Fujita and Kato [15].

Lemma 3.10

Let $\psi$ be a function on $\mathbb {R}$ satisfying (1.1) and let $\delta$ be the constant in Theorem 1.1 with (1.15)–(1.17). Then there exists a constant ${\tilde{\varepsilon }}={\tilde{\varepsilon }}(n,D)\in (0,\delta ]$ such that if  $0<a^{(n-2)/(n+1)}<{\tilde{\varepsilon }}$, (1.21) admits at most one solution within $Y_{0}$.

Proof

The following argument is based on [15]. Suppose that $v,{\tilde{v}}\in Y_{0}$ are solutions. Then we have

$$\begin{aligned} \Vert v-{\tilde{v}}\Vert _{Y,t}\le \big \{C_1\big ([\nabla v]_{n,t}+[{\tilde{v}}]_{n,t}^{\frac{1}{2}} [{\tilde{v}}]^{\frac{1}{2}}_{\infty ,t}\big ) +C_2a^{\frac{n-2}{n+1}}\big \}\Vert v-{\tilde{v}}\Vert _{Y,t},\quad t>0 \end{aligned}$$

(3.60)

by applying (1.10) and Lemma 3.3. If we assume

$$\begin{aligned} a^{\frac{n-2}{n+1}}<\min \Big \{\delta ,\frac{1}{2C_2}\Big \} =:{\tilde{\varepsilon }} \end{aligned}$$

(3.61)

and choose $t_0>0$ such that

$$\begin{aligned} C_1\big \{[\nabla v]_{n,t_0}+ \big (\sup _{0<\tau<\infty }\Vert {\tilde{v}}(\tau )\Vert _n\big )^{\frac{1}{2}}\, [{\tilde{v}}]^{\frac{1}{2}}_{\infty ,t_0}\big \}<\frac{1}{2}, \end{aligned}$$

then (3.60) yields $[v-{\tilde{v}}]_{Y,t_0}=0$. Hence, we conclude $v={\tilde{v}}$ on $(0,t_0]$ and obtain

$$\begin{aligned} v(t)-{\tilde{v}}(t)&=\int _{t_0}^t e^{-(t-\tau )A_a}P\Big [-(v-{\tilde{v}})\cdot \nabla v -{\tilde{v}}\cdot \nabla (v-{\tilde{v}})-\psi (\tau )(v-{\tilde{v}})\cdot \nabla u_s \nonumber \\&\quad -\,\psi (\tau )u_s\cdot \nabla (v-{\tilde{v}}) +(1-\psi (\tau ))a\frac{\partial }{\partial x_1}(v-{\tilde{v}})\Big ]\,d\tau . \end{aligned}$$

By the same argument as in the proof of Lemma 3.3 together with (1.10), we see that

$$\begin{aligned} \Vert v-{\tilde{v}}\Vert _{Y,t_0,t}\le C_*\Vert v-{\tilde{v}}\Vert _{Y,t_0,t} \end{aligned}$$

(3.62)

for all $t>t_0$, where

$$\begin{aligned} \Vert v\Vert _{Y,t_0,t}:&= \sup _{t_0\le \tau \le t}\Vert v(\tau )\Vert _n +\sup _{t_0\le \tau \le t}\Vert v(\tau )\Vert _\infty + \sup _{t_0\le \tau \le t}\Vert \nabla v(\tau )\Vert _n,\end{aligned}$$

(3.63)

$$\begin{aligned} C_*&=C\Big [\big (t_0^{-\frac{1}{2}}\Vert v\Vert _Y+t_0^{-\frac{1}{4}} \Vert {\tilde{v}}\Vert _Y\big ) \big \{(t-t_0)^{\frac{3}{4}} +(t-t_0)^{\frac{1}{4}}\big \}\nonumber \\&\quad +a^\frac{n-1}{n+1} \big \{(t-t_0)^{\frac{1}{2}} +(t-t_0)^{\frac{\rho _2}{2}}+(t-t_0)^{\frac{\rho _4}{2}}\big \} +a\big \{(t-t_0)+(t-t_0)^{\frac{1}{2}}\big \} \Big ] \end{aligned}$$

(3.64)

and the constant C is independent of v, ${\tilde{v}}$, t and $t_0$. We choose $\eta >0$ such that

$$\begin{aligned} \xi :=C\Big [\big (t_0^{-\frac{1}{2}}\Vert v\Vert _Y+t_0^{-\frac{1}{4}} \Vert {\tilde{v}}\Vert _Y\big ) \big (\eta ^{\frac{3}{4}} +\eta ^{\frac{1}{4}}\big )+a^\frac{n-1}{n+1} \big (\eta ^{\frac{1}{2}} +\eta ^{\frac{\rho _2}{2}}+\eta ^{\frac{\rho _4}{2}} \big )+a\big (\eta +\eta ^{\frac{1}{2}}\big ) \Big ]<1. \end{aligned}$$

On account of (3.62), we have $\Vert v-{\tilde{v}}\Vert _{Y,t_0,t_0+\eta } \le \xi \Vert v-{\tilde{v}}\Vert _{Y,t_0,t_0+\eta }$, which leads us to $v={\tilde{v}}$ on $[t_0,t_0+\eta ]$. By the same calculation, we can obtain (3.62)–(3.64), in which $t_0$ should be replaced by $t_0+\eta$ and hence

$$\begin{aligned} \Vert v-{\tilde{v}}\Vert _{Y,t_0+\eta ,t_0+2\eta }&\le C\Big [\big \{(t_0+\eta )^{-\frac{1}{2}}\Vert v\Vert _Y+(t_0+\eta )^{-\frac{1}{4}} \Vert {\tilde{v}}\Vert _Y\big \} \big (\eta ^{\frac{3}{4}} +\eta ^{\frac{1}{4}}\big )\\&\quad +a^\frac{n-1}{n+1} \big (\eta ^{\frac{1}{2}} +\eta ^{\frac{\rho _2}{2}}+\eta ^{\frac{\rho _4}{2}}\big ) +a\big (\eta +\eta ^{\frac{1}{2}}\big ) \Big ] \Vert v-{\tilde{v}}\Vert _{Y,t_0+\eta ,t_0+2\eta }\\&<\xi \Vert v-{\tilde{v}}\Vert _{Y,t_0+\eta ,t_0+2\eta } \end{aligned}$$

holds. This implies $v={\tilde{v}}$ on $[t_0+\eta ,t_0+2\eta ].$ Repeating this procedure, we conclude $v={\tilde{v}}$. $\square$

Remark 3.11

It is clear that the Eq. (3.28) admits at most one solution within $Y_{0}$ under the same condition as in Lemma 3.10.

Let us close the paper with completion of the proof of Theorem 1.2.

Proof of Theorem 1.2

Since we know $\varepsilon \le {\tilde{\varepsilon }}$ from (3.26) and (3.61), Proposition 3.4 and Lemma 3.10 yield the unique existence of solutions in $Y_{0}$ when $(M+1)a^{(n-2)/(n+1)}<\varepsilon$. Moreover, Propositions 3.7 and 3.9 give us sharp decay properties of the solution provided a is still smaller. We finally show the uniqueness of the solution constructed above within Y by following the argument due to Brezis [5]. It suffices to show that if $v\in Y$ is a solution, it necessarily satisfies

$$\begin{aligned} \lim _{t\rightarrow 0}\,[v]_t=0, \end{aligned}$$

(3.65)

where

$$\begin{aligned}{}[v]_t:=\sup _{0<\tau <t}\tau ^{\frac{1}{2}}(\Vert v(\tau )\Vert _\infty +\Vert \nabla v(\tau )\Vert _n). \end{aligned}$$

We assume

$$\begin{aligned} (M+1)a^\frac{n-2}{n+1}<\min \left\{ \delta , \frac{1}{2C_2},\frac{1}{16C_1C_0},\frac{1}{16C_1C_3}\right\} =: {\hat{\varepsilon }}(n,D)\,(\le \varepsilon ) \end{aligned}$$

(3.66)

and let $v\in Y$ be a solution. Here, the constants $C_i$ are as in Remark 3.5 as well as in the proof of Proposition 3.4. Since $v\in BC([0,\infty );L_\sigma ^n(D))$ with $v(0)=0$, there exists $s_0>0$ such that

$$\begin{aligned} \Vert v(s)\Vert _n+(M+1)a^\frac{n-2}{n+1}<{\hat{\varepsilon }} \end{aligned}$$

for all $0<s\le s_0$. Hence by Remark 3.5, the integral equation (3.28) with $b=v(s)$ admits a solution $T(t)v(s)\in Y_{0}$ along with

$$\begin{aligned} \Vert T(\cdot )v(s)\Vert _{Y}\le 4\big (C_0\Vert v(s)\Vert _n+C_3(M+1)a^\frac{n-2}{n+1}\big ) <4(C_0+C_3){\hat{\varepsilon }}\le \frac{1}{2C_1}. \end{aligned}$$

(3.67)

On the other hand, given $s\in (0,s_0]$, we can see that $z_s(t):=v(t+s)$ for $t\ge 0$ also satisfies (3.28) with $b=v(s)$ and $z_s\in Y_{0}$. In view of Remark 3.11, we have $z_s(t)=T(t)v(s)$ for $s\in (0,s_0]$, which implies

$$\begin{aligned} t^{\frac{1}{2}}\big (\Vert v(t+s)\Vert _\infty +\Vert \nabla v(t+s)\Vert _n\big ) \le \displaystyle \sup _{f\in K}[T(\cdot )f]_t,\quad K=v((0,s_0]):=\{v(t)\in L^n_\sigma (D)\mid t\in (0,s_0]\} \end{aligned}$$

for all $s\in (0,s_0]$ and $t>0$. Passing to the limit $s\rightarrow 0$, we find

$$\begin{aligned}{}[v(\cdot )]_t\le \displaystyle \sup _{f\in K}[T(\cdot )f]_t. \end{aligned}$$

(3.68)

Furthermore, applying Lemma 3.3 to (3.28) with $b=f\in v((0,s_0])$ as well as Proposition 3.1 and (1.10), we have

$$\begin{aligned}{}[T(\cdot )f]_t&\le C_0[S(\cdot )f]_t +\Big (C_1\sup _{f\in K}\Vert T(\cdot )f\Vert _{Y} +C_2a^\frac{n-2}{n+1}\Big )[T(\cdot )f]_t +\Vert H_1\Vert _{Y,t}+\Vert H_2\Vert _{Y,t}, \end{aligned}$$

where $S(t)f:=e^{-tA_a}f$, and deduce from (3.66) and (3.67) that

$$\begin{aligned}{}[T(\cdot )f]_t\le \frac{C_0[S(\cdot )f]_t+ \Vert H_1\Vert _{Y,t}+\Vert H_2\Vert _{Y,t}}{1-\Big (C_1\displaystyle \sup _{f\in K}\Vert T(\cdot )f\Vert _{Y} +C_2a^\frac{n-2}{n+1}\Big )} \end{aligned}$$

(3.69)

for all $f\in K$ and $t>0$. Collecting (3.69), (3.4), (3.68) and $H_1,H_2\in Y_{0}$ leads to (3.65). The proof is complete. $\square$