数理解析研究所講究録
1495 巻 2006 年 60-70

60

Asymptotic behavior of solutions for a fluid
dynamical model of semiconductor equation
Dedicated to Professor $Tai$ -Ping $Liu$ on $h,is$ 60th birthday

A. Matsumura (松村昭孝) and T. Murakami (村上尊弘)
Department of Pure and Applied Mathematics
Graduate School of Information Science and Technology
Osaka University

1

Introduction

In this short paper, we shall show our recent rcsults on the asymptotic stability of the stationary solution to a simple one-dimensional fluid dynamical
model of semiconductor which describes thc motion of clectrons along a thin
imiporlar transistor likechancl connecting the source and drain in a
MOS-FET (cf.[l]):$r\triangleright \mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}_{h}$$\rho_{t}+(\rho u)_{x}=\mathrm{t})$

(1.1)

$\{$

,

$( \rho u)_{t}+(\rho u^{2}+p(\rho))_{x}=\rho\phi_{x}-\frac{1}{\tau}$$\phi_{xx}=\rho-D$pu,$(x, t)\in(0,1)\mathrm{x}[0, \infty)$,,

where $\rho=\rho(t, x)>0$ is the density of electrons, $u=u(t, x)$ is the velocity,$\phi=\phi(t, x)$ is the electrostatic potential, $D=D(x)$ is a given doping profile(positive charge density of impurities), $p=p(\rho)$ is the pressure, and is theandrclaxation time. We assume the pressure has the form$\tau=1$ for simplicity. As for the initial boundary value problem to (1.1) with
the initial data$\tau$$p(\rho)=\rho^{\gamma}(\gamma\geq 1)$(1.2)$(\rho, u)(x, 0)=(\rho_{0}, u_{0})(x)$,$x\in(\mathrm{O}, 1)$,

61

and the Dirichlet type boundary conditions$\rho(\mathrm{O}, t)=\rho_{1}$,$\rho(1, t)=\rho_{2}$,$t\geq 0$,$\phi(1, t)=\Phi_{1}$,$t\geq()$,(1.3)$\phi(0, t)=()$,

Degond, Markowich $(’ 93,[2])$ studied the existence and uniqueness of the subshowedsonic stationary solution, then H.-L. Li, P. Markowich and M.
thestability
undervarithc
the asymptotic stability. However they showed,that
ous smallness conditions
der the situation cverything are very close to the constant states. In particular, sincc actual data of semiconductor often
read the ratio of the maximuIn of the doping profile to the minimum is ofabove seems almost unreal, a smallness (flatness) condition toorder
in a physical and engineering point of view. We also note that even in all, the stability
recent researches on quantum fluid dynamicalof stationary solutions is studied under similar strict smallncss conditions.
In this paper, \‘although so far we have not had satisfactory results on thewith theproblem (1.1)(1.2)(1.3), we insteacl treat the problem (1.1) inperiodie initial and boundary conditions$\mathrm{M}\mathrm{e},\mathrm{i}(’ 03,[v]\ulcorner)$$(|\rho_{1}-\rho_{2}|, |\rho_{0}-\rho_{1}|, |\Phi_{1}|, |D-\rho_{1}|\ll 1)$$\mathrm{i}\mathrm{s}_{\wedge},\mathrm{u}\mathrm{n}-$$(\hat{\rho}, \text{\^{u}}, \phi)=$$(\overline{\rho}, 0,0)(\overline{\rho}:=\rho_{1}=\rho_{2})$$D$$10^{2}$$D\mathrm{a}‘ \mathrm{s}$$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}1_{\mathrm{b}^{\backslash }}(\mathrm{c}\mathrm{f}.[4][3])$$\mathrm{R}$$(\rho, u)(x, 0)=(\rho_{0}, u_{0})(x)$,$x\in R$

(1.4)$(\rho, u, \phi)(t, x+1)=(\rho, u, \phi)(t, x)$,$x\in R,$$t\geq 0$,

and investigate the asymptotic stability of the stationary solution imder as
as possible.
less smallness conditions in particular on the doping profile
We assume that
$D$

(1.5)

$(\rho_{0}, u_{0})\in H_{period}^{2}$(1.6)$D\in H_{period}^{1}$,,$x,\in R\mathrm{i}\iota\iota \mathrm{f}\rho_{0}(x)>0$$\inf_{x\in R}D(x)>0$,,$\int_{0}^{1}\rho_{0}(x)dx=1$$\int_{0}^{1}D(x)dx=1$,.

is the usual -th order Sobolev space of.the periodic funcHere. First we should
tions with the period 1, and denote the norm
satisfying (1.6) the uniquc sationary sonote that for any doping profile
lution corresponding to the problem (1.1)(1.4) is easily obtained in the form
. Then we have
$H_{p\mathrm{e}riod}^{k}$$k$$\}_{)}\mathrm{y}||\cdot||_{k}$$D$$(\hat{\rho}, \text{\^{u}}, \hat{\phi})=(\hat{\rho}, 0,\hat{\phi})(x)$62

Theorem 1.1. Any srnooth global solution in timc of the initial boundary value proble
satisfies the following:such $t,hat$exist positive constants andIf$rn(\mathit{1}.\mathit{1})(\mathit{1}.\mathit{4})$$i)$$\gamma\geq 2,$$tf\iota,ere$$\nu$$E_{0}$$\int_{0}^{1}\rho u^{2}+(\rho-\hat{\rho})^{2}+|(\phi-\hat{\phi})_{x}|^{2}dx\leq E_{0}\exp\{-\nu t\}$

$ii)$

,

$t\geq 0$

If $2>\gamma\geq 1$ and $M:= \sup\rho(t, x)<+\infty$ , then there.

$ex,it$positive$x,t$constants $\nu(M)$ and $E_{0}(M)$ such that$\int_{0}^{1}\rho u^{2}+(\rho-\hat{\rho})^{2}+|(\phi-\hat{\phi})_{x}|^{2}dx\leq E_{0}\exp\{-\nu t\}$,$t\geq 0$.

Here we should note that the above a priori estimates are not enough toinsure thc cxistence of global smooth solution in time. On the other hand,by the arguments in $[6](’ 98)$ , we can not expect the existence of the global
smooth solution for general large data. However we still have a possibility tohavc the global smooth solution around the stationary solution even if thedoping profile $D$ is very large. An answer is the following.

Theorem 1.2. Suppose (1.5) and (1.6). Then there enist positive con, the initial, and $\beta>0$ such thatstantshas a unique global solution in time satisfying
value problem$\delta,$$C$$if||(\rho_{0}-\hat{\rho}, u_{0})||_{2}\leq\delta$$bounda7^{\mathrm{v}}y$$(\mathit{1}.\mathit{1})(\mathit{1}.\mathit{4})$$(\rho, u)\in C^{0}([0, \infty);H_{p\mathrm{e}rid}^{2})()$’$\phi\in C^{0}([0, \infty);H_{period}^{3},)$,and furthermore$|1$$(\rho-\hat{\rho}, u)(t)||_{2}+||(\phi-\hat{\phi})(t)||_{3}\leq C||(\rho_{0}-\hat{\rho}, u_{0})||_{2}\exp\{-\beta t\}$,$t\geq 0$.

Stationary solution

2

In this section, we study the basic properties of the smooth solution of the
problems (1.1)(1.4), and the corresponding stationary solution. In what follows, let us assume $\gamma>1$ for simplicity, since the case $\gamma=1$ can be treated. First, we easily havc
by properly taking limit$\gammaarrow 1$

Lemma 2.1.
(2.1)

The smooth solution of the problem$\int_{0}^{1}\rho(t, x)dx=\int_{0}^{1}\rho \mathrm{o}(x)dx=1$,$(\mathit{1}.\mathit{1})(\mathit{1}.\mathit{4})$$t\geq 0$,

satisfies

63

(2.2)

$\rho(t, x)>0$ ,(2.3)$x\in R,$$t\geq 0$$. \int_{0}^{1}u(t, x)dx\leq e^{-t}\int_{0}^{1}u_{0}(x)dx$,,$t\geq 0$.

Proof The equality (2.1) is shown by integrating the first equation of(1.1) with respect to . The positivity of the solution (2.2) is proved by aproperty of thc ordinary differential equation for $\rho(t, x(t))$ along the characteristic curve $x’(t)=u(t, x(t))$ . For (2.3), we can rewrite the second equationof (1.1) as
$x$

(2.4)

$u_{t}+( \frac{1}{2}u^{2}+\frac{\gamma}{\gamma-1}\rho^{\gamma-1})_{x}=\phi_{x}-u$

then the integration of (2.4) with rcspect toproof is completed.

casily implies (2.3). Thus the$x$

By the above Lemma 2.1, we can see the natural stationary solution
corresponcling to the problcm (1.1)(1.4) is described the following
system$(\hat{\rho}, \text{\^{u}}, \hat{\phi})(x)$$(\hat{\rho}\hat{u})_{x}=0$

(2.5)

$\{$

,

$(\hat{\rho}\hat{u}^{2}+\hat{\rho}^{\gamma})_{x}=\hat{\rho}\hat{\phi}_{x}-\hat{\rho}\hat{u}$,$x\in R$,$\hat{\phi}_{xx}=\hat{\rho}-D$with the boundary condition(2.6)$(\hat{\rho},\hat{u},\hat{\phi})(x+1)=(\hat{\rho},\hat{u},\hat{\phi})(x)$,$x\in R$,

under the additional conditions(2.7)$\int_{0}^{1}\hat{\rho}dx=1$,$\inf_{x}\hat{\rho}>0$,$\int_{0}^{1}$\^u $dx=0$ .

Here we also assume that the doping profile $D$ satisfies (1.6). By the firstandequation of (2.5) and the conditions (2.7), we can easily have$\text{\^{u}}=0$$(_{\overline{\gamma}-\overline{1}}^{\Delta}\hat{\rho}^{\gamma-1})_{x}=\hat{\phi}_{x}$

(2.8)

,

$\{$

$\hat{\phi}_{xx}=\hat{\rho}-D$.

Then the boundary value problem (2.5)(2.6) is virtually rcduced to the problem only for :$\hat{\rho}$$-(_{\overline{\gamma}-\overline{1}}^{2}\hat{\rho}^{\gamma-1})_{xx}+\hat{\rho}=D$

(2.9)

,

$\{$

$\hat{\rho}(x)=\hat{\rho}(x+1)$,$x\in R$$x\in R$

.

,

64

Thus, by the arguments of the Maximum Principle and the Shauder’s Fixed
Point Theorem as in [2], we can have the following lemma. We omit thedetails.
Lemma 2.2. The boundary value problem
such thatsolution$(\mathit{2}.\mathit{5})(\subset)l.\mathit{6})(Q.7)$(2.10)$\underline{D}\leq\hat{\rho}(x)\leq\overline{D}$,has the uniqueand$\text{\^{u}}()=0,\hat{\phi}\in H_{period}^{3}$$\hat{\rho}\in H_{perid}^{2},$$(\hat{\rho}, \text{\^{u}}, \hat{\phi})$$x\in R$

where$\underline{D}:=\inf_{x\in R}D(x)>0$,$\overline{D}:=\sup_{x\in R)}D(x)$,Asymptotic behavior for large initial data

3

In this section, we show the proof of the Theorem 1.1, that is, show anof the initial })of the smooth solution
a.symptotic }), which holds
value problem (1.1) (1.4) toward the stationary solution
andthe doping
without any smallness conditions both on the initial data
profile. We first note that the equation for is written in the form$(\rho, u, \phi)$$\mathrm{t}^{1},\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{o}\mathrm{r}$$(\mathrm{u}.\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{y}$$(\hat{\rho}, \text{\^{u}}, \phi)$

$u$

(3.1)

$u_{t}+uu_{x}+ \frac{\gamma}{\gamma-1}(\rho^{\gamma-1}-\hat{\rho}^{\gamma}‘ 1)_{x}-(\phi-\hat{\phi})_{x}+u=\mathrm{t})$which follows from (2.4) and (2.8). Now we multiply the equation (3.1) by
and integrate with respect to . Then using the relation$x$$\rho u$(3.2)$\rho_{t}=(\phi-\hat{\emptyset})_{xxt}$

which is derived by the third equations of (1.1), and taking careful integration
by parts, we can have the first basic energy estimate
Lemma 3.1. It holds that
(3.3)$\frac{d}{dt}\{\int_{0}^{1}\frac{1}{2}\rho u^{2}+G(\rho,\hat{\rho})+\frac{1}{2}|(\phi-\hat{\phi})_{x}|^{2}dx\}+.\int_{0}^{1}\rho u^{2}dx=0$

where$G( \rho,\hat{\rho}):=\frac{\gamma}{\gamma-1}.\int_{\rho}^{\rho}(\xi^{\gamma-1}-\hat{\rho}^{\gamma-1})d\xi$,$t\geq 0$.

To obtain thc second basic energy estimate, wc in turn use the equation
for written in the another form
$u$

(3.4)

$(\rho u)_{t}+(\rho u^{2}+\rho^{\gamma}-\hat{\rho}^{\gamma})_{x}-(\rho\phi_{x}-\hat{\rho}\hat{\phi}_{x})+\rho u=0$

65

, We multiply the equation (3.4)
which follows from (1.1) and (2.5) withand integrate with respec, to . Then using the relation (3.2)
carrying careful integration by parts, we can have the estimat,
again$\text{\^{u}}=0$$\mathrm{b}\mathrm{y}-(\phi-\hat{\phi})_{x}$$x$$\mathrm{t}$$\mathrm{a}\mathrm{n}\mathfrak{c}1$(3.5)$\mathrm{e}$$\frac{d}{dt}\{\int_{0}^{1}-(\rho u)(\phi-\hat{\phi})_{x}+\frac{1}{2}|(\phi-\hat{\phi})_{x}|^{2}dx\}-\int_{0}^{1}|(\phi-\hat{\phi})_{xt}|^{2}dx+$.

$+ \int_{0}^{1}\rho u^{2}(\rho-\hat{\rho})+(\rho^{\gamma}-\hat{\rho}^{\gamma})(\rho-\hat{\rho})+\frac{1}{2}(\frac{\rho+\hat{\rho}}{2}+D)(\phi-\hat{\phi})_{x}^{2}dx=0$

Although the sign of the integral
it turns out that the iinportant inequality$/-(\phi-\hat{\phi})_{xt}^{2}dx$(3.6)

in (3.5) seems a problem,$\int_{0}^{1}\rho^{2}u^{2}-|(\phi-\hat{\phi})_{x\downarrow}|^{2}dx=(\int_{0}^{1}\rho udx)^{2}\geq 0$

holds, which plays an essential role in the estimates in this section. III fact,
the inequality (3.6) is proved as follows. By the formula (3.2) and the firstequation of (1.1), we have
(3.7)$(\rho u+(\phi-\hat{\phi})_{xt})_{x}=0$,

which shows that we can set $\rho u+(\phi-\hat{\phi})_{xt}=k(t)$ for a function $k(t)$depending only on . Then we can see$t$$k(t)= \int_{0}^{1}$

pu $dx$ ,and can prove (3.6) by$\int_{0}^{1}\rho^{2}u^{2}-|(\phi-\hat{\phi})_{xt}|^{2}dx$$=$$\int_{0}^{1}(k(t)-(\phi-\hat{\phi})_{xt})^{2}-|(\phi-\hat{\phi})_{xt}|^{2}dx$$=$$k(t)^{2}-2k(t) \int_{0}^{1}(\phi-\hat{\phi})_{xt}dx$$=$$k(t)^{2}$.

Hence, applying (2.10) and (3.6) to (3.5), we have the following second basic

energy estimate.
Lemma 3.2.
(3.8)

It holds that$\frac{d}{dt}\{\int_{0}^{1}-(\rho u)(\phi-\hat{\phi})_{x}+\frac{1}{2}|(\phi-\hat{\phi})_{x}|^{2}dx\}-.\int_{0}^{1}\hat{\rho}\rho u^{2}dx+$

66$+ \int_{()}^{1}(\rho^{\gamma}-\hat{\rho}^{\gamma})(\rho-\hat{\rho})+\frac{3}{4}\underline{D}|(\phi-\hat{\phi})_{x}|^{2}dx\leq 0$,.

$t\geq 0$

provc the desired estimates. For
We are now on the stage wherc we
any positive constant , multiplying (3.3) by A and adding it to (3.8), we
have$(_{C}^{it},)_{d}\mathrm{n}$$\lambda$(3.9)$\frac{d}{dt}E(t)+Q(t)\leq 0$,$t\geq 0$

where

pu

(3.10)$E(t)= \int_{0}^{1}\frac{1}{2}$(3.11)$Q(t)= \int_{0}^{1}(1-\lambda\hat{\rho})\rho u^{2}+\lambda(\rho^{\gamma}-\hat{\rho}^{\gamma})(\rho-\hat{\rho})+\frac{3}{4}\lambda\underline{D}|(\phi-\hat{\phi})_{x}|^{2}dx$$2+G(p, \hat{\rho})+\frac{1+\lambda}{2}|(\phi-\hat{\phi})_{x}|^{2}-\lambda(\rho u)(\phi-\hat{\phi})_{x}dx$.

In order to cstimatcs the$\mathrm{m}\mathrm{a}_{\iota}\mathrm{s}$,$\mathrm{t}\mathrm{e}\mathrm{r}\iota \mathrm{n}\mathrm{s}$in $E(t)$ and $Q(t)$ , we need another two lem-.

Lemma 3.3.
(3.12)

It holds that$| \int_{()}^{1}\rho u(\phi-\hat{\phi})_{x}dx|\leq\int_{0}^{1}(\frac{1}{2}\rho u^{2}+\frac{1}{2}(\rho-\hat{\rho})^{2})dx$.

Proof By the equality (2.1), the Sobolev’s lemma and the Poincare’s
inequality, we have$| \int_{0}^{1}\rho u(\phi-\hat{\phi})_{x}dx|$$\leq$$\int_{0}^{1}\frac{1}{2}\rho u^{2}+\frac{1}{2}\rho(\phi_{x}-\hat{\phi}_{x})^{2}dx$$\leq$$\frac{1}{2}\int_{0}^{1}\rho u^{2}d_{X+\frac{1}{2}\mathrm{S}11,r}.\mathrm{p}|\phi_{x}-\hat{\phi}_{x}|^{2}$$\leq$$\frac{1}{2}\int_{0}^{1}pu^{2}dx+\frac{1}{2}\int_{0}^{1}|\phi_{xx}-\hat{\phi}_{xx}|^{2}dx$$=$$\frac{1}{2}\int_{0}^{1}\rho u^{2}dx+\frac{1}{2}\int_{0}^{\mathrm{I}}|\rho-\hat{\rho}|^{2}dx$.

Thus the proof of the Lemma 3.3 is completed.
$i)$Lemma 3.4. It holds the following:, there exists a constant $c>0$ such that for any $\rho>0$ andIf,$\gamma\geq 1$$\hat{\rho}\in[\underline{D}, \overline{D}]$(3.13)$G(\rho,\hat{\rho})\leq c(p^{\gamma}-\hat{\rho}^{\gamma})(p-\hat{\rho})$,$\frac{1}{2}(\rho-\hat{\rho})^{2}\leq c(\rho^{\gamma}-\hat{\rho}^{\gamma})(\rho-\hat{\rho})$.

67$\iota i)$

If$\gamma\geq 2,$$\hat{\rho}\in[\underline{D}, \overline{D}]$there exists a constant $c>0$ svch th,at for any $\rho>0$ and,(3.14)$iii)$If$G(\rho,\hat{\rho})\geq c^{-1}(\rho-\hat{\rho})^{2}$$\gamma\geq 1$

and$\rho(t, x)=:M<+\infty$ ,$\sup$.

then th, $ere$ exists a constant$x\in R,l\geq 0$,$c(M)>0$ such that for$an\uparrow/\rho>()$(3.15)$G(\rho,\hat{\rho})\geq c(M)^{-1}(\rho-\hat{\rho})^{2}$and$\hat{\rho}\in[\underline{D}, \overline{D}]$.

Tbe proof of the Lemma 3.4 is given by standard arguments of the Tayler’s
formula, so we omit it.

canfor simplicity, because the case where
Now suppose that
(3.10)
andbe treated on the same line. Applying thc Lemma 3.3 and 3.4 to(3.11), and choosing A suitably small, we have$\gamma\geq 1$$\gamma\geq 2$(3.16)$\int_{0}^{1}\rho u^{2}+(\rho-\hat{\rho})^{2}+|(\phi-\hat{\phi})_{x}|^{2}dx\leq CE(t)$,$t\geq 0$

and also(3.17)$E(t)\leq\nu^{-1}Q(t)$for some positive constants(3.18)$C$,$t\geq()$and . So we finally$0\}_{)}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}$$\nu$$\frac{d}{dt}E(t)+\nu E(t)\leq 0$,$t\geq 0$.

Thus, by (3.16) and $(3,18)$ , we comple,te the proof of Theorem 1.1.

4

Asymptotic stability of stationary solution, that is,
In this section, we give a rough sketch of the proof of. First, forshow the asymptotic stability of the stationary solution
2,
constructed in the section we set
any fixed stationary solution$\mathrm{T}\mathrm{h}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}\iota \mathrm{n}_{\wedge}1.2$$(\hat{\rho}, 0, \phi)$$(\hat{\rho}, (),\hat{\emptyset})$

(4.1)

th

$=\rho-\hat{\rho}$,$e=\phi-\hat{\phi}$,and rewrite the initial boundary value problem (1.1)(1.4) in terms of $(\psi, u, e)$

as

(4.2)

68

with the initial data and boundary conditions$(\psi, u)(0, x)=(\psi_{0}, u_{0})(x):=(\rho_{0}-\hat{\rho}, u_{0})(x)$(4.3)$(\psi, u, e)(t, x+1)=(\prime \mathrm{v}\text{ノ}, u, e)(t, x)$Then, for suitably small$(\psi_{0}, u_{0})\in H_{period}^{2},$in timeneighborhood of’,$x\in R,$,$x\in R$$t\geq()$,.

we look for the global solution
which stay.s in a$(\psi, u)\in C^{0}([0, \infty);H_{p\epsilon riod}^{2}),$ $e\in C^{0}([0, \infty);H_{p\iota jrtod}^{3}\backslash )$

1

(4.4)

$(\psi, u)||_{2}\leq\overline{\epsilon}_{0}:=\underline{D}/4$

that, we use
and exponentially decays as the time goes to infinity. To
combiningbythe losolution
globalthethe typical arguments to construct
cal existence theorem togethor with the a priori estimate. As for the local
existence, we omit the arguments, because the principal part of the system
is a symmetrizablc hyperbolic one and so standard arguments(4.2) foron the Cauchy problem for qua.silinear symmctric hyperbolic system arc applicable to our case. The a priori estimate we need is as follows,, andProposition 4.1. There exist positive constants$(\psi, u, e)\in C^{0}([0, T];H_{period}^{2})^{2}\cross C^{0}([(), T];H_{pe,riod}^{3})$for some $T>0$
such that if
andis the solution of $thc$ initial boundary value problem$\mathrm{c}\mathrm{l}\mathrm{o}$$(\psi, u)$$\epsilon_{0}(<\overline{\epsilon}_{0})fC$$\beta$$(\mathit{4}\cdot \mathit{2})(\mathit{4}\cdot \mathit{3})$$(4_{\iota)}^{r}.)$$\sup_{0\leq t\leq T}||(\psi, u)(t)||_{2}\leq\epsilon_{0}$,then it holds(4.6)$||(\psi, u)(t)||_{2}+||e(t)||_{3}\leq C||(\psi_{0}, u_{0})||_{2}\in^{\backslash },\mathrm{x}\mathrm{p}\{-\beta t\}$,$0\leq t\leq T$.

To show the Proposition 4.1, in what follows, we assume that the initialboundary value problem (4.2)(4.3) has thc solution$(\psi, u, e)\in C^{0}([0, T];H_{p\mathrm{e}riod}^{2})^{2}\mathrm{x}C^{0}([0, T];H_{period}^{3})$for some $T>0$ and also assume$\sup_{0\leq t\leq T}||(\psi, u)(t)||_{2}\leq\overline{\epsilon_{0}}$.

Then, using the equation (4.2) and (4.4), we can first show
andLemma 4.2. There $ex,ist$ positive constants$\epsilon(<\overline{\epsilon}_{0})$$\sup_{0\leq t\leq T}||(\psi, u)(t)||_{2}\leq\epsilon$,$C$such that if69then it holds(4.7)$||(\psi_{t}, u_{t})||_{1}\leq C||(\psi, u)||_{2}$,$0\leq t\leq T$

and
(4.8)

$||(\psi, u)||_{2}+||e||_{3}\leq C(||(\psi, u)||+||(\uparrow l;_{l}, u_{t})||+||(\uparrow[J_{bi}, u_{ll})||),$$0\leq t\leq T$.

also as small as we want in theBy (4.7), we may assume
estimates, and by (4.8), it turns out that it is enough toprocess ofobtain the exponential decay estimat es forto show thc Proposition 4.1.
$||(\psi_{t}, u_{t})||_{1}$$a$$pr\dot{\tau}ori$$||(\psi, u)||+||(\psi_{t}, u_{t})||+||(\psi_{tt}, u_{\iota\iota})||$

Next, by the arguments in the proof ofthat there exist positivc constants and
$C$

(4.9)

$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\iota \mathrm{n}1.1$$\beta \mathrm{s}\mathrm{u}\mathrm{c},1\iota$, we can easily showthat$||(\psi, u, e_{x})(t)||^{2}\leq C||(\psi_{0}, u_{0})||^{2_{\mathrm{f}_{J}^{1}}}\mathrm{x}\mathrm{p}\{-\beta t\}$,$0\leq t\leq T$.

In order to obtain the estimates for the higher derivatives, we differentiate
the system (4.1) with respect to to have$t$,$\psi_{tt}+((\hat{\rho}+\psi)u_{t})_{x}+(\tau/J_{t}u)_{x}=0$(4.10),$u_{tt}+u_{t}u_{x}+uu_{tx}+\gamma((\hat{p}+\psi)^{\gamma-2}\psi_{t})_{x}-e_{xt}+u_{t}=0$$\{$$e_{txx}=\psi_{t}$.

As in the arguments in the proof of Theorem 1.1, we multiply the second
and integrate it with respect to .
cquation of (4.10) by
Noting the relation$x$$((\hat{\rho}+\psi)u_{t}-\lambda e_{xt})$,$\int_{0}^{1}(\hat{\rho}+\uparrow/J)u_{l}((\hat{p}+\psi)^{\gamma-2}\psi_{l})_{\tau,}dx=\int_{0}^{1}(\psi_{tt}+(\psi_{t}u)_{x})(\hat{\rho}+\psi)^{\gamma-2}\psi_{l}dx$

we may carefully carry the integration by parts. Then, choosing the A andproperly small, we can prove(4.11)$||(\psi_{t}, u_{t})(t)||^{2}\leq C||(\psi_{0}, u_{0})||_{1}^{2}\exp\{-\beta t\}$,$\epsilon$$0\leq t\leq T$

in the samc way as in the section 3.
for some positive constants andSimilarly, after differentiating the system (4.10) with respect to again and, we also can prove
multiplying it by$C$$\beta$$t$$((\hat{\rho}+\psi)u_{ll}-\lambda e_{xll})$(4.12)

1$(1/J\downarrow" u_{tl})(t)||^{2}\leq C||(\psi_{0}, u\mathrm{o})||_{2}^{2}\mathrm{e}\mathrm{x}\iota)\{-\beta t\}$,$0\leq t\leq T$

for some positive constants and . Thus, combining thc estimates (4.8)(4.9)(4.11)(4.12), we can complete the proof of the Proposition 4.1, and then Theorem 1.2.
$\beta$$C$

70

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