**2. Preliminaries**

#### **2.1 Notations and function spaces**

Let us consider functions defined in the half space **R***<sup>n</sup>* <sup>+</sup> <sup>=</sup> **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>***n*−1. We sometimes write the space variable *<sup>x</sup>* = (*x*1, ··· , *xn*) <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* <sup>+</sup> as *x* = (*x*1, *x*� ) with *x*<sup>1</sup> ∈ **R**<sup>+</sup> and *x*� = (*x*2, ··· , *xn*) ∈ **<sup>R</sup>***n*−1. The symbols <sup>∇</sup> = (*∂x*<sup>1</sup> , ··· , *<sup>∂</sup>xn* ) and <sup>Δ</sup> <sup>=</sup> <sup>∑</sup>*<sup>n</sup> <sup>j</sup>*=<sup>1</sup> *<sup>∂</sup>*<sup>2</sup> *xj* denote the standard gradient and Laplacian with respect to *x* = (*x*1,..., *xn*), respectively. The symbol ∇*x*� = (*∂x*<sup>2</sup> , ··· , *∂xn* ) denotes the gradient of the tangential direction with respect to *x* = (*x*2,..., *xn*). Thus we have ∇ · *<sup>g</sup>* <sup>=</sup> <sup>∑</sup>*<sup>n</sup> <sup>j</sup>*=<sup>1</sup> *<sup>∂</sup>xj gj* for *<sup>g</sup>* = (*g*1, ··· , *gn*), and <sup>∇</sup>*x*� · *<sup>g</sup>*<sup>∗</sup> <sup>=</sup> <sup>∑</sup>*<sup>n</sup> <sup>j</sup>*=<sup>2</sup> *∂xj gj* for *g*<sup>∗</sup> = (*g*2, ··· , *gn*). Let *v*ˆ(*x*1, *ξ*) be the Fourier transform of *v*(*x*1, *x*� ) with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1:

$$
\hat{v}(\mathbf{x}\_{1},\boldsymbol{\xi}) = \mathcal{F}[v(\mathbf{x}\_{1},\cdot)](\boldsymbol{\xi}) = (2\pi)^{-(n-1)/2} \int\_{\mathbb{R}^{n-1}} v(\mathbf{x}\_{1},\mathbf{x}') e^{-i\mathbf{x}' \cdot \boldsymbol{\xi}} d\mathbf{x}' \,\mathrm{d}\mathbf{x}' \,\mathrm{d}\mathbf{x}' \,\tag{2.1}
$$

where *<sup>ξ</sup>* = (*ξ*2, ··· , *<sup>ξ</sup>n*) <sup>∈</sup> **<sup>R</sup>***n*−<sup>1</sup> *<sup>ξ</sup>* is the Fourier variable corresponding to *x*� = (*x*2, ··· , *xn*) ∈ **<sup>R</sup>***n*−<sup>1</sup> and *<sup>x</sup>*� · *<sup>ξ</sup>* <sup>=</sup> <sup>∑</sup>*<sup>n</sup> <sup>j</sup>*=<sup>2</sup> *xjξj*.

Let 1 <sup>≤</sup> *<sup>p</sup>* <sup>≤</sup> <sup>∞</sup>. We denote by *<sup>L</sup><sup>p</sup> <sup>x</sup>*� <sup>=</sup> *<sup>L</sup>p*(**R***n*−1) the *<sup>L</sup><sup>p</sup>* space with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1, with the norm �·�*L<sup>p</sup> x*� . For a nonnegative integer *s*, we denote by *H<sup>s</sup> <sup>x</sup>*� <sup>=</sup> *<sup>H</sup>s*(**R***n*−1) the Sobolev space over **R***n*−<sup>1</sup> with the norm

$$||v||\_{H\_{\mathbf{x'}}^s} = \left(\sum\_{k=0}^s ||\partial\_{\mathbf{x'}}^k v||\_{L\_{\mathbf{x'}}^2}^2\right)^{1/2}.$$

where *∂<sup>k</sup> <sup>x</sup>*� denotes the totality of all the *<sup>k</sup>*-th order derivatives with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1. Also, we denote by *<sup>L</sup>p*(**R**+) the *<sup>L</sup><sup>p</sup>* space with respect to *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**+, with the norm |·|*Lp* . For a nonnegative integer *<sup>s</sup>*, we denote by *<sup>H</sup>s*(**R**+) the Sobolev space over **<sup>R</sup>**+, with the norm |·|*Hs* . For *<sup>α</sup>* <sup>∈</sup> **<sup>R</sup>**, we denote by *<sup>L</sup>*<sup>2</sup> *<sup>α</sup>*(**R**+) the weighted *<sup>L</sup>*<sup>2</sup> space over **<sup>R</sup>**<sup>+</sup> with the norm

$$\|v\|\_{L^2\_{\mathfrak{a}}} = \left(\int\_0^\infty (1+\mathfrak{x}\_1)^{\mathfrak{a}} |v(\mathfrak{x}\_1)|^2 d\mathfrak{x}\_1\right)^{1/2}.$$

Now we introduce function spaces over the half space **R***<sup>n</sup>* <sup>+</sup> <sup>=</sup> **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>***n*−1. Let 1 <sup>≤</sup> *<sup>p</sup>*, *<sup>q</sup>* <sup>≤</sup> <sup>∞</sup>, *<sup>s</sup>* be a nonnegative integer, and *<sup>α</sup>* <sup>∈</sup> **<sup>R</sup>**. The space *<sup>L</sup>q*(*Lp*) = *<sup>L</sup>q*(**R**+; *<sup>L</sup><sup>p</sup> <sup>x</sup>*�) consists of *<sup>L</sup><sup>q</sup>* functions of *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> with values in *<sup>L</sup><sup>p</sup> <sup>x</sup>*� with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1. The norm is denoted by �·�*Lq* (*Lp* ). When *q* = *p*, we simply write as

$$\mathcal{L}^p = L^p(L^p), \qquad \|\cdot\|\_{\mathcal{L}^p} = \|\cdot\|\_{L^p(L^p)}.$$

The space *<sup>H</sup>s*(*Lp*) = *<sup>H</sup>s*(**R**+; *<sup>L</sup><sup>p</sup> <sup>x</sup>*�) consists of *<sup>H</sup><sup>s</sup>* functions of *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> with values in *<sup>L</sup><sup>p</sup> <sup>x</sup>*� with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1. The norm is denoted by �·�*Hs*(*Lp* ). Also, *<sup>L</sup>*<sup>2</sup> *<sup>α</sup>*(*Lp*) = *<sup>L</sup>*<sup>2</sup> *<sup>α</sup>*(**R**+; *<sup>L</sup><sup>p</sup> <sup>x</sup>*�) denotes the space of *L*<sup>2</sup> *<sup>α</sup>* functions of *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> with values in *<sup>L</sup><sup>p</sup> <sup>x</sup>*� with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1. The norm is denoted by

$$\|v\|\_{L^2\_a(L^p)} = \left(\int\_0^\infty (1+\mathfrak{x}\_1)^a \|v(\mathfrak{x}\_1\cdot)\|\_{L^p\_{\mathfrak{x}'}}^2 d\mathfrak{x}\_1\right)^{1/2}.$$

We sometimes use

4 Will-be-set-by-IN-TECH

with (1.9), (1.10). To overcome the difficulty occured by the non-convex condition, we make a good combination of the weighted energy method in partial Fourier space employed in [14] and the anti-derivative method employed in [2, 9], and get the desired results. Once we obtain the linear stability results for the problem (1.15), (1.9), (1.10), we may apply this results to the

The remainder of this chapter is organized as follows. In Section 2, we introduce function spaces and some preliminaries used in this chapter. Especially, we reformulate our problem (1.15), (1.9), (1.10) by using the anti-derivative method in Section 2.3. In the final section, we treat the half space problem for the reformulated viscous conservation laws and (1.15), (1.9), (1.10), and develop the weighted energy method in the partial Fourier space with the anti-derivative method. In this section, we derive pointwise estimates of solutions and prove

<sup>+</sup> as *x* = (*x*1, *x*�

*<sup>j</sup>*=<sup>1</sup> *<sup>∂</sup>xj gj* for *<sup>g</sup>* = (*g*1, ··· , *gn*), and <sup>∇</sup>*x*� · *<sup>g</sup>*<sup>∗</sup> <sup>=</sup> <sup>∑</sup>*<sup>n</sup>*

. For a nonnegative integer *s*, we denote by *H<sup>s</sup>*

 *<sup>s</sup>* ∑ *k*=0 �*∂k <sup>x</sup>*� *<sup>v</sup>*�<sup>2</sup> *L*2 *x*� 1/2 ,

*<sup>x</sup>*� denotes the totality of all the *<sup>k</sup>*-th order derivatives with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1. Also,

*<sup>α</sup>*(**R**+) the weighted *<sup>L</sup>*<sup>2</sup> space over **<sup>R</sup>**<sup>+</sup> with the norm

<sup>2</sup>*dx*<sup>1</sup> 1/2 .

we denote by *<sup>L</sup>p*(**R**+) the *<sup>L</sup><sup>p</sup>* space with respect to *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**+, with the norm |·|*Lp* . For a nonnegative integer *<sup>s</sup>*, we denote by *<sup>H</sup>s*(**R**+) the Sobolev space over **<sup>R</sup>**+, with the norm |·|*Hs* .

(<sup>1</sup> <sup>+</sup> *<sup>x</sup>*1)*α*|*v*(*x*1)<sup>|</sup>

*<sup>v</sup>*ˆ(*x*1, *<sup>ξ</sup>*) = <sup>F</sup>[*v*(*x*1, ·)](*ξ*)=(2*π*)−(*n*−1)/2

�*v*�*H<sup>s</sup> <sup>x</sup>*� =

> <sup>∞</sup> 0


Now we introduce function spaces over the half space **R***<sup>n</sup>*

be a nonnegative integer, and *<sup>α</sup>* <sup>∈</sup> **<sup>R</sup>**. The space *<sup>L</sup>q*(*Lp*) = *<sup>L</sup>q*(**R**+; *<sup>L</sup><sup>p</sup>*

Laplacian with respect to *x* = (*x*1,..., *xn*), respectively. The symbol ∇*x*� = (*∂x*<sup>2</sup> , ··· , *∂xn* ) denotes the gradient of the tangential direction with respect to *x* = (*x*2,..., *xn*). Thus we

*<sup>j</sup>*=<sup>1</sup> *<sup>∂</sup>*<sup>2</sup>

 **R***n*−<sup>1</sup>

) with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1:

*<sup>ξ</sup>* is the Fourier variable corresponding to *x*� = (*x*2, ··· , *xn*) ∈

*<sup>x</sup>*� <sup>=</sup> *<sup>L</sup>p*(**R***n*−1) the *<sup>L</sup><sup>p</sup>* space with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1, with

*v*(*x*1, *x*� )*e* −*ix*� ·*ξdx*�

<sup>+</sup> <sup>=</sup> **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>***n*−1. We sometimes write the

) with *x*<sup>1</sup> ∈ **R**<sup>+</sup> and *x*� = (*x*2, ··· , *xn*) ∈

*xj* denote the standard gradient and

*<sup>j</sup>*=<sup>2</sup> *∂xj gj* for *g*<sup>∗</sup> = (*g*2, ··· , *gn*).

*<sup>x</sup>*� <sup>=</sup> *<sup>H</sup>s*(**R***n*−1) the Sobolev

<sup>+</sup> <sup>=</sup> **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>***n*−1. Let 1 <sup>≤</sup> *<sup>p</sup>*, *<sup>q</sup>* <sup>≤</sup> <sup>∞</sup>, *<sup>s</sup>*

*<sup>x</sup>*�) consists of *<sup>L</sup><sup>q</sup>* functions

, (2.1)

asymptotic stability for the nonlinear problem (1.8)–(1.10).

Let us consider functions defined in the half space **R***<sup>n</sup>*

**<sup>R</sup>***n*−1. The symbols <sup>∇</sup> = (*∂x*<sup>1</sup> , ··· , *<sup>∂</sup>xn* ) and <sup>Δ</sup> <sup>=</sup> <sup>∑</sup>*<sup>n</sup>*

Let *v*ˆ(*x*1, *ξ*) be the Fourier transform of *v*(*x*1, *x*�

*<sup>j</sup>*=<sup>2</sup> *xjξj*.

the corresponding decay estimates.

**2.1 Notations and function spaces**

space variable *<sup>x</sup>* = (*x*1, ··· , *xn*) <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>*

where *<sup>ξ</sup>* = (*ξ*2, ··· , *<sup>ξ</sup>n*) <sup>∈</sup> **<sup>R</sup>***n*−<sup>1</sup>

Let 1 <sup>≤</sup> *<sup>p</sup>* <sup>≤</sup> <sup>∞</sup>. We denote by *<sup>L</sup><sup>p</sup>*

*x*�

space over **R***n*−<sup>1</sup> with the norm

For *<sup>α</sup>* <sup>∈</sup> **<sup>R</sup>**, we denote by *<sup>L</sup>*<sup>2</sup>

**<sup>R</sup>***n*−<sup>1</sup> and *<sup>x</sup>*� · *<sup>ξ</sup>* <sup>=</sup> <sup>∑</sup>*<sup>n</sup>*

the norm �·�*L<sup>p</sup>*

where *∂<sup>k</sup>*

**2. Preliminaries**

have ∇ · *<sup>g</sup>* <sup>=</sup> <sup>∑</sup>*<sup>n</sup>*

$$
\mathcal{L}\_{\mathfrak{a}}^2 = L\_{\mathfrak{a}}^2(L^2), \qquad \|\cdot\|\_{\mathcal{L}\_{\mathfrak{a}}^2} = \|\cdot\|\_{L\_{\mathfrak{a}}^2(L^2)}.
$$

*L*2(*Hs*) = *L*2(**R**+; *H<sup>s</sup> <sup>x</sup>*�) denotes the space of *<sup>L</sup>*<sup>2</sup> functions of *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> with values in *<sup>H</sup><sup>s</sup> <sup>x</sup>*� with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1, whose norm is given by

$$\begin{split} \|\boldsymbol{\upsilon}\|\_{L^{2}(H^{s})} &= \left(\int\_{0}^{\infty} \|\boldsymbol{\upsilon}(\mathbf{x}\_{1\prime} \cdot \boldsymbol{\rangle})\|\_{H^{s}\_{\boldsymbol{\mathsf{x}}\boldsymbol{\mathsf{e}}}}^{2} d\boldsymbol{x}\_{1}\right)^{1/2} \\ &= \left(\sum\_{k=0}^{s} \int\_{0}^{\infty} \|\partial\_{\mathbf{x}^{\prime}}^{k} \boldsymbol{\upsilon}(\mathbf{x}\_{1\prime} \cdot \boldsymbol{\mathsf{e}})\|\_{L^{2}\_{\boldsymbol{\mathsf{x}}\boldsymbol{\mathsf{e}}}}^{2} d\boldsymbol{x}\_{1}\right)^{1/2} = \left(\sum\_{k=0}^{s} \|\partial\_{\mathbf{x}^{\prime}}^{k} \boldsymbol{\upsilon}\|\_{\mathcal{L}^{2}}^{2}\right)^{1/2}. \end{split}$$

By the definition (2.1) of the Fourier transform, we see that

$$\sup\_{\substack{\mathfrak{F}\in\mathbb{R}\_{\xi}^{n-1}}} |\mathfrak{f}(\cdot,\xi)|\_{L^{2}} \leq C \|v\|\_{L^{2}(L^{1})} \tag{2.2}$$

with *C* = (2*π*)−(*n*−1)/2. Also, it follows from the Plancherel theorem that

$$\|\left|\partial\_{\mathbf{x}'}^k v\right|\|\_{\mathcal{L}\_a^2} = \left(\int\_{\mathbb{R}\_\xi^{n-1}} |\tilde{\xi}|^{2k} |\hat{v}(\cdot, \tilde{\xi})|\_{L\_a^2}^2 d\tilde{\xi}\right)^{1/2}.\tag{2.3}$$

Let *T* > 0 and let *X* be a Banach space defined on the half space **R***<sup>n</sup>* <sup>+</sup>. Then *C*([0, *T*]; *X*) denotes the space of continuous functions of *t* ∈ [0, *T*] with values in *X*.

In this paper, positive constants will be denoted by *C* or *c*.

#### **2.2 Stationary solution**

We review the results on the stationary problem (1.6)–(1.7). For the details, we refer the reader to [2, 8, 11–13].

**Proposition 2.1** ([8])**.** *Assume the condition* (1.4)*. Then f* � <sup>1</sup>(0) ≤ 0 *is necessary for the existence of solutions to the stationary problem* (1.6)*–*(1.7)*. Conversely, under the condition f* � <sup>1</sup>(0) ≤ 0*, we have the following existence result:*

(i) *Non-degenerate case where f* � <sup>1</sup>(0) < 0*: In this case the stationary problem* (1.6)*–*(1.7) *admits a unique smooth solution φ*(*x*1) *with φx*<sup>1</sup> > 0 *(resp. φx*<sup>1</sup> < 0*), provided that ub* < 0 *(resp.* 0 < *ub and f*1(*u*) < *f*1(0) *for* 0 < *u* < *ub). The solution verifies*

$$|\phi(\mathfrak{x}\_1)| \le C e^{-c\mathfrak{x}\_1}, \qquad \mathfrak{x}\_1 > 0,$$

where *A* is a positive constant determined in Lemma 2.2. This weight function is very important to derive a priori estimate in the latter section. For this weight function, we obtain

<sup>255</sup> Application of the Weighted Energy Method in the Partial Fourier Space to Linearized Viscous Conservation Laws with Non-Convex Condition

**Lemma 2.2** ([2])**.** *Suppose that f*1(*u*) *satisfies* (1.4)*. Let w*(*u*) *be the weight function defined in* (2.8)*. Then there exists a positive constant δ such that if A* ≥ *δ, then w*(*u*) *satisfies the following conditions:*

*Moreover, let φ be the stationary solution constructed in Proposition 2.1. Then the weight function*

*c* < *w*(*φ*) < *C for φ* ∈ [*ub*, 0].

<sup>1</sup>(0) = 0*: The weight function w*(*φ*) *satisfies*

*c*(1 + *x*1) < *w*(*φ*) < *C*(1 + *x*1) *for φ* ∈ [*ub*, 0].

In the final section, we apply our weighted energy method in the partial Fourier space to the linearized problem. We consider the linearized problem corresponding to the half space problem (2.5)–(2.7). Namely, we consider (2.5) with *gj* = 0 for *j* = 1, ··· , *n*. For this linearized

"*fj*(*u*) are linear in *u* ∈ [*ub*, 0] for *j* = 2, ··· , *n*."

together with (2.6) and (2.7). Taking the Fourier transform with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−<sup>1</sup> for the

<sup>∗</sup>(*φ*) = <sup>∑</sup>*<sup>n</sup>*

<sup>1</sup>(*φ*)*z*ˆ*x*<sup>1</sup> + *iξ* · *f* �

The detail of the proof is omitted here. For the details, we refer the reader to [2].

Then our initial value problem of the linearized equation is written as

*z*ˆ*<sup>t</sup>* − *z*ˆ*x*<sup>1</sup> *<sup>x</sup>*<sup>1</sup> + |*ξ*|

*z*ˆ*x*<sup>1</sup> (0, *ξ*, *t*) = 0,

*z*ˆ(*x*1, *ξ*, 0) = *z*ˆ0(*x*1, *ξ*),

*<sup>n</sup>*(*φ*)), and *ξ* · *f* �

*zt* − Δ*z* + *f* �

<sup>2</sup>*z*ˆ + *f* �

<sup>1</sup>(0) < 0*: The weight function w*(*φ*) *satisfies*

(*u*) < 0 *for u* ∈ [*ub*, 0],

(ii) (*w f*1)��(*u*) <sup>&</sup>lt; <sup>0</sup> *for u* <sup>∈</sup> [*ub*, 0]. (2.9)

(*φ*) · ∇*z* = 0 (3.1)

*<sup>j</sup>*(*φ*). This is the formulation of our

(3.2)

<sup>∗</sup>(*φ*)*z*<sup>ˆ</sup> = 0,

*<sup>ξ</sup>* is the Fourier variable corresponding to *x*� = (*x*2, ··· , *xn*) ∈

*<sup>ξ</sup>* .

*<sup>j</sup>*=<sup>2</sup> *ξ<sup>j</sup> f* �

<sup>+</sup> <sup>=</sup> **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>***n*−<sup>1</sup>

(i) (*w f*1)�

*Here, C and c are some positive constants which independent of x*1*.*

**3. Asymptotic stability with convergence rates**

equation, we treat the special situation that

linearized problem (3.1), (2.6), (2.7), we obtain

<sup>2</sup>(*φ*), ··· , *f* �

linearized problem in the partial Fourier space **R**ˆ *<sup>n</sup>*

where *<sup>ξ</sup>* = (*ξ*2, ··· , *<sup>ξ</sup>n*) <sup>∈</sup> **<sup>R</sup>***n*−<sup>1</sup>

<sup>∗</sup>(*φ*)=(*<sup>f</sup>* �

**R***n*−1, *f* �

the following lemma.

*satisfies the following properties.* (i) *Non-degenerate case where f* �

(ii) *Degenerate case where f* �

*where C and c are positive constants.*

(ii) *Degenerate case where f* � <sup>1</sup>(0) = 0*: In this case the problem* (1.6)*–*(1.7) *admits a unique smooth solution φ*(*x*1) *if and only if ub* < 0*. The solution verifies φx*<sup>1</sup> > 0 *and*

$$|\phi(\mathfrak{x}\_1)| \le C(1+\mathfrak{x}\_1)^{-1/q} \varkappa \qquad \mathfrak{x}\_1 > 0,$$

*where q is the degeneracy exponent of f*<sup>1</sup> *and C is a positive constant.*

In this chapter we only treat the stationary solutions *φ*(*x*1) with *φx*<sup>1</sup> > 0 and discuss their stability; however, we must get the similar stability result of the monotone decreasing stationary solutions by using the same argument introduced in this chapter. (We refer the reader to [2].)

#### **2.3 Reformulated problem**

In this subsection we reformulate our problem by the anti-derivative method. To this end we introduce a new function *z*(*x*, *t*) as

$$z(\mathbf{x},t) = -\int\_{\mathcal{X}\_1}^{\infty} v(y, \mathbf{x}', t) \, dy. \tag{2.4}$$

Here, we assume the integrability of *v*(*x*, *t*) over **R**+. This transformation is motivated by the argument in Liu-Nishihara [9]. By using (2.4), we can reformulate (1.8)–(1.10) in terms of *z*(*x*, *t*) as

$$z\_t - \Delta z + f'(\phi) \cdot \nabla z + \int\_{\mathcal{X}\_1}^{\infty} \phi\_{\mathcal{X}\_1} f\_\*^{\prime\prime}(\phi) \cdot \nabla\_{\mathcal{X}^{\prime}} z \, d\mathcal{y} = -g\_1 + \nabla\_{\mathcal{X}^{\prime}} \cdot h\_\*,\tag{2.5}$$

$$z\_{\mathbf{x}\_1}(0, \mathbf{x}', t) = 0,\tag{2.6}$$

$$z(\mathbf{x}, \mathbf{0}) = z\_0(\mathbf{x}),\tag{2.7}$$

where *<sup>z</sup>*0(*x*) = <sup>−</sup> <sup>∞</sup> *<sup>x</sup>*<sup>1</sup> (*u*0(*y*, *x*� ) − *φ*(*y*))*dy*, *f* �� <sup>∗</sup> (*φ*)=(*<sup>f</sup>* �� <sup>2</sup> (*φ*), ··· , *f* �� *<sup>n</sup>* (*φ*)), and *g*1, ∇*x*� · *h*<sup>∗</sup> are nonlinear terms defined by *h*<sup>∗</sup> = (*h*2, ··· , *hn*) and

$$g\_j = f\_j(\phi + z\_{\ge 1}) - f\_j(\phi) - f\_j'(\phi)z\_{\ge 1}, \qquad h\_j = \int\_{\chi\_1}^{\infty} g\_j \, dy.$$

Once we obtain the solution for the problem (2.5)–(2.7), the differentiation *v* = *zx*<sup>1</sup> is the solution for (1.8)–(1.10). Namely, we will apply the weighted energy method in the partial Fourier space and try ot derive the global solution in time to the reformulated problem (2.5)–(2.7). We will discuss this reformulated problem in Section 3 to prove our main theorems.

#### **2.4 Weight function**

We introduce the weight function employed in the weighted energy method. Our weight function is defined as

$$w(u) = (-e^{Au} + 1) / f\_1(u) \qquad \text{for} \qquad u \in [u\_b, 0]. \tag{2.8}$$

where *A* is a positive constant determined in Lemma 2.2. This weight function is very important to derive a priori estimate in the latter section. For this weight function, we obtain the following lemma.

**Lemma 2.2** ([2])**.** *Suppose that f*1(*u*) *satisfies* (1.4)*. Let w*(*u*) *be the weight function defined in* (2.8)*. Then there exists a positive constant δ such that if A* ≥ *δ, then w*(*u*) *satisfies the following conditions:*

$$\begin{array}{ll} \text{(i) } (wf\_1)'(u) < 0 & \text{for} \quad u \in [u\_{b^\*}, 0]. \\\\ \text{(ii) } (wf\_1)''(u) < 0 & \text{for} \quad u \in [u\_{b^\*}, 0]. \end{array} \tag{2.9}$$

*Moreover, let φ be the stationary solution constructed in Proposition 2.1. Then the weight function satisfies the following properties.*

(i) *Non-degenerate case where f* � <sup>1</sup>(0) < 0*: The weight function w*(*φ*) *satisfies*

*c* < *w*(*φ*) < *C for φ* ∈ [*ub*, 0].

(ii) *Degenerate case where f* � <sup>1</sup>(0) = 0*: The weight function w*(*φ*) *satisfies*

$$c(1+\mathbf{x}\_1) < w(\phi) < \mathcal{C}(1+\mathbf{x}\_1) \qquad \text{for} \qquad \phi \in [\mu\_{b\prime}, 0].$$

*Here, C and c are some positive constants which independent of x*1*.*

The detail of the proof is omitted here. For the details, we refer the reader to [2].
