数理解析研究所講究録
第 824 巻 1993 年 283-296

283

Neumann Problem
of One-Dimensional Nonlinear Thermoelastic Equations
筑波大・数学系
柴田良弘

Yoshihiro Shibata
Inst. of Math. Univ. of Tsukuba, Tsukuba-shi, 305 JapanI will report here ajoint work [3] with S. Kawashima, Kyushu Univ. about a global
in time solvability and an asymptotic stability of Neumann problem of one-dimensional
nonlinear thermoelastic equations. This problem was first solved by Shibata [5] in thebounded interval case, but the assumption on the regularity of initial data was not
optimal and polynomial decay rate was only shown. And then, Jiang Song [2] improved
the result concerning the solvability, regarding an optimal assumption of the regularity
of initial data. He also proved an existence theorem globally in time in the half line
case and he obtained an exponential decay result in the linear case. In the both works,
it is essential that no external force appears. In this note, I essentially assume that theexternal force depends on the spatial variable only. When the external force depends
on not only the spacial variable but also the time, to solve the problem seems to bedifficult, in particular such difficulty appears when we decide an as ymptotic behaviourof solutions globally in time. Such difficulty sometimes appears in the Neumann casein general and as a technical difficulty, we can not use a Poincar\’e’s inequality.
First, let me formulate the problem. Let $\Omega=(0,1)$ be the unit interval of $R$ identffied. The deformation of the referencewith reference body with natural temperaturebody after time past is described by the deformation map:$\tau_{0}>0$$\Omega$$t$$X$: $x\in\Omega X(t, x)\in R$ .

Let $T(t, x)$ be the absolute temperature of the point $X(t, x)$ . The equation of motion
and the balance of energy are described by the following equations:(1.1)$p(x)X_{tt}-S_{x}=f$(1.2)$(e+ \frac{\rho(x)}{2}X_{t}^{2})_{t}-(SX_{t})_{x}$$=q_{x}+X_{t}f+g$for$x\in\Omega$

and $t>0$ ,for$x\in\Omega$

and $t>0$ .

Here, the subscripts stand for partial derivatives, is the stress function, is the internal
is the mass density, $f$ is an external force andenergy function, is the heat flux,
is an external heat supply. For simplicity, I assume that
$S$

$q$

$e$

$\rho(x)$$g$$\rho(x)=1,$$g=0$ and $q=\kappa T_{x}$ ,being a positive constant. In this lecture note, as a boundary condition, I consider
the traction free and thermally insulated condition which is described by the following
formula:$\kappa$(N.N)$S=q=0$for$x\in\partial\Omega$.

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Note that
formula:$\partial\Omega$(1.3)

consists of only two points$X(O, x)=X_{0}(x),$$0$

and 1. The initial condition is given by the$X_{t}(0, x)=X_{1}(x))T(O, x)=T_{0}(x)$ .

Now, let me explain as sumptions. The first one is that$f=f(x)$ , that is $f$ depends only on

(A.1)

$x$

.

Next one is concerned with the constitutive relation. Let $F$ be a variable correspondingto. Let be a Helmholtz free energy function and be an entropy function. The
second assumption is that
$X_{x}$(A.2)$\psi$$\eta$$S,$$e,$$\psi$

and$\eta$

are functions in $(F, T)$ only, that is$S=S(F, T),$ $e=e(F, T),$and they are in$C^{\infty}(G(\delta))$$\psi=\psi(F, T),$$\eta=\eta(F,T)$

,

,

where
$G(\delta)=\{(F,T)\in R^{2}||(F,T)-(1, \tau_{0})|\leq\delta\}$ .

In this lecture, all the functions are real-valued unless it is mentioned. The 2nd Law ofThermodynamics tells us that the following two relations are equivalent:$de=SdF+Td\eta\Leftrightarrow d\psi=SdF-\eta dT$,from which the following constitutive relations follow:(A.3)$S= \frac{\partial\psi}{\partial F},$

$e= \psi-T\frac{\partial\psi}{\partial T}$$\eta=-\frac{\partial\psi}{\partial T},$.

The next assumption is that
(A.4)$\frac{\partial^{2}\psi}{\partial F^{2}}>0,$$\frac{\partial^{2}\psi}{\partial F\partial T}\neq 0,$$\frac{\partial^{2}\psi}{\partial T^{2}}<0$in$G(\delta)$.

Under the constitutive relation (A.3), (1.2) is equivalent to the following equation:(1.4)$T\eta_{t}=q_{x}$forand $t>0$ .

$x\in\Omega$

In fact,$e_{t}= \frac{\partial\psi}{\partial T}T_{t}+\frac{\partial\psi}{\partial F}X_{xt}-T_{t}\frac{\partial\psi}{\partial T}-T(\frac{\partial\psi}{\partial T}I_{t}=T\eta_{t}+SX_{xt)}$$( \frac{1}{2}X_{t}^{2})_{t}=X_{t}X_{tt}=S_{x}X_{t}+fX_{t}$.

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Combining these two equations implies (1.4). Except for finding a conservative quantity,usually we solve (1.1) and (1.4) instead of (1.1) and (1.2).

EXAMPLE 1.1. As a Helmholtz free energy function, let us choose$\psi(F,T)=\sqrt{1+F^{2}}-T^{2}-\gamma TF,$$\gamma\neq 0$

.

Then,
$S=F/\sqrt{1+F^{2}}-\gamma T,$$e=\sqrt{1+F^{2}}+T^{2},$$\eta=\gamma F+2T$.

The corresponding equations are that,$\{T(2T+\gamma X_{x})_{t}=\kappa T_{xx}X_{tt}-(X_{x}/\sqrt{1+X_{x^{2}}}-\gamma T)_{x}=f$which is one of thermo-damping equations corresponding to$(^{*})$

case.

in one-dimensional

As a class of solutions, let me consider the following space: for $t_{0}>0$ I put$Z(t_{0})=\{(X(t, x),$ $T(x,t))|$(1.5)$X \in\bigcap_{j=0}^{3}C^{j}([0,t_{0});H^{3-j})$(1.6)$T \in C^{2}([0,t_{0});L^{2})\cap\bigcap_{j=0}^{1}C^{j}([0,t_{0});H^{3-j}),$(1.7)$(X_{x}(t, x),$ $T(t, x))\in G(\delta)$,$\partial_{t^{2}}T\in L^{2}((0,t_{0});H^{1})$and $T(t, x)>0$ for any,$(t, x)\in[0,t_{0})\cross\overline{\Omega}$}.

I will look for solutions (X, $T$) $\in Z(\infty)$ . Here and hereafter,
denotes the set of
denotes the set of all
functions whose
all square integrable functions on .
distributional derivatives of order up to also belong to . $C^{k}(I;B)$ denotes the setof all -valued -times continuously differentiable functions on I. $L^{2}(I;B)$ denotes theset of all B-valued square integrable functions on .
Now, let me explain the conditions on initial data
and a right member .
To do this, for a moment I assume that solutions (X, $T$) $\in Z(t_{0})$ exist. Put$L^{2}$$\Omega$$H^{k}$$L^{2}$$L^{2}$

$k$

$B$

$k$

$I$

$X_{0},$$X_{j}(x)=\partial\dot{i}X(0,x)$and$X_{1},$$f$$T_{0}$$T_{j}(x)=\dot{\theta}_{t}T(0, x)$which are successively determined through the equations (1.1) and (1.4) in terms ofand their derivatives. For example,$X_{1},$$T_{0},$$f$$X_{2}(x)=S(X_{0}’(x),T_{0}(x))+f(x)$ ,,$T_{1}(x)=(T_{0}(x) \frac{\partial\eta}{\partial T}(X_{0}’(x), T_{0}(x))^{-1}\{\kappa T_{0}’(x)-T_{0}(x)\frac{\partial\eta}{\partial F}(X_{0}’(x))T_{0}(x))X_{1}’(x)\}$$X_{0}$,

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and so on. According to (1.5) and (1.6), I as sume that(A.5)
$T_{j}(x)\in H^{3-j},$ $j=0,1$ ;$X_{j}(x)\in H^{3-j},$ $0\leq j\leq 3$ ;$T_{2}(x)\in L^{2}$;$f(x)\in H^{1}$

.

Since
$T_{x}(t,x),$$S(X_{x}(t,x),T(t, x)) \in\bigcap_{j=0}^{1}C^{j}([0,t_{0});H^{2-j})$,in view of the trace theorem to the boundary, the boundary condition (N.N) requires
the following conditions:(A.6)$S(X_{0}’(x), T_{0}(x))=0$,$\frac{\partial S}{\partial F}(X_{0}’(x), T_{0}(x))X_{1}’(x)+\frac{\partial S}{\partial T}(X_{0}’(x),T_{0}(x))T_{1}(x)=0$,$T_{0}’(x)=T_{1}’(x)=0$

for$\partial\Omega$. In fact, these conditions come from the facts that $S=S_{t}=T_{x}=T_{xt}=0$ on. (A.6) is called the compatibility condition. In addition, I as sume that$x\in\Omega$(A.7)$\int_{0}^{1}f(x)dx=\int_{0}^{1}X_{0}(x)dx=\int_{0}^{1}X_{1}(x)dx=0$.

But, (A.7) does not give us any restrictions. In fact, let me consider the compensating
function $r(t)$ defined by the formula:$r(t)= \int_{0}^{1}X_{0}(x)dx+t\int_{0}^{1}X_{1}(x)dx+\frac{t^{2}}{2}\int_{0}^{1}f(x)dx$.

Put $\tilde{X}(t, x)=X(t, x)-r(t)$ . Then,$\int_{0}^{1}\tilde{X}(0, x)dx=\int_{0}^{1}\tilde{X}_{t}(0, x)dx=0$,$\tilde{X}_{tt}-S(\tilde{X}_{x}, T)_{x}=X_{tt}-S(X_{x}, T)_{x}-r’’(t)=f(x)-\int_{0}^{1}f(x)dx$$\int_{0}^{1}(f(x)-\int_{0}^{1}f(x)dx)dx=0$,.

Rom these observations, you see that (A.7) actually does not give us any restrictions
on initial data and right members.
and, for unknown functions
To find the possible asymptotic behaviour as
let me consider the stationary problem corresponding to (1.1), (1.4) and (N.N):$tarrow\infty$$X_{\infty}$$T_{\infty}$$S(X_{\infty}’(x), T_{\infty}(x))’=-f(x)$and $T_{\infty}’’(x)=0$ in$S(X_{\infty}’(x), T_{\infty}(x))=T_{\infty}’(x)=0$

on$\Omega$,$\partial\Omega$.

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It follows from the boundary condition that
(1.8)

$T_{\infty}$

is a constant and that

$S(X_{\infty}(x),T_{\infty})=-F(x)$

,

where
$F(x)= \int_{0}^{x}f(y)dy$.

satisfying
Note that it follows from (A.7) that $F(O)=F(1)=0$ . A paircomes from the following(1.8) is not unique. Another requirement on
and noting the relation:energy conservation law. Integrating (1.2) over$(X_{\infty}(x), T_{\infty})$$(X_{\infty}(x),T_{\infty})$$(0,t)\cross\Omega$$X_{t}f=(X_{t}F)_{x}- \frac{\partial}{\partial t}(X_{x}F)$,

you see that
(1.9)$\int_{0}^{1}\{e(X_{x}(t, x),$ $T(t, x))+ \frac{1}{2}X_{t}(t,x)^{2}+X_{x}(t, x)F(x)\}dx=e_{0}$

,

where
$e_{0}= \int_{0}^{1}\{e(X_{0}’(x),T_{0}(x))+\frac{1}{2}X_{1}(x)^{2}+X_{0}’(x)F(x)\}dx$.

, and sinceas
Since the motion is expected to stop at $t=\infty$ , that is$(X_{x},T)$
as tends to infinity, (1.9) implies
a pairis expected to converge to$X_{t}arrow 0$$(X_{\infty}’,T_{\infty})$$tarrow\infty$

$t$

that
(1.10)$\int_{0}^{1}\{e(X_{\infty}’(x), T_{\infty})+X_{\infty}’(x)F(x)\}dx=e_{0}$.

Put$M_{g}(F,T)= \frac{\partial S}{\partial F}(F, T)\frac{\partial g}{\partial T}(F, T)-\frac{\partial S}{\partial T}(F,T)\frac{\partial g}{\partial F}(F, T)$.

for $g=e$ and .
$\eta$

Assume that
(A.8)

$S(1, \tau_{0})=0$

which means thatand (A.8) imply that$(1, \tau_{0})$(1.11),

is an equilibrium state with $f=g=0$ . And then, (A.3), (A.4)$M_{e}(1, \tau_{0})=M_{\eta}(1, \tau_{0})=\tau_{0}\{-\frac{\partial^{2}\psi}{\partial F^{2}}(1, \tau_{0})\frac{\partial^{2}\psi}{\partial T^{2}}(1, \tau_{0})+\frac{\partial^{2}\psi}{\partial F}\partial T(1, \tau_{0})^{2}\}>0$.

In view of (1.11), by the implicit function theorem you can prove the following lemma
satisfying (1.8) and (1.10).
concerning the unique existence of a pair$(X_{\infty}(x), T_{\infty})$

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LEMMA 1.2. Suppose that (A.3), (A.4), (A.5), (A.7) and (A.8) hold. Then, for any
there exists a $\kappa>0$ such that if$\sigma>0$$||(X_{0}’,T_{0})-(1,\tau_{0})||_{\infty}+||X_{1}||+||f||_{1}<\kappa$

then th ere exist a $X_{\infty}(x)\in H^{3}$ and a const an $tT_{\infty}>0$ satisfying (1.8) and (1.10) andthe following conditions:$||X_{\infty}’-1||_{2}+|T_{\infty}-\tau_{0}|<\sigma$

In particular, $S(X_{\infty}’(x), T_{\infty})=0$ for
Here and hereafter,$||\cdot||$

an $d(X_{\infty}’(x),T_{\infty})\in G(\delta/2)$$x\in\partial\Omega$for all$x\in\overline{\Omega}$.

.

denotes the usual$||v||_{k}= \{\sum_{j=0}^{k}||\frac{d^{j}v}{dx^{j}}||^{2}\}^{1/2}$$L^{2}$-norm onand$\Omega$and put$||v||_{\infty}= \sup_{x\in\Omega}|v(x)|$.

To state a main result exactly, let me introduce an additional notation. Put$u(t, x)=X(t, x)-X_{\infty}(x),$$\theta(t, x)=T(t, x)-T_{\infty}$$N(t)= \sup_{0<s<t}||\overline{D}^{2}(u_{x}, u_{t}, \theta)(s, \cdot)||$,,$N_{\alpha}(t)= \sup_{0<s<t}e^{\alpha s}\{||\overline{D}^{2}(u_{x}, u_{t}, \theta)(s, \cdot)||+||(\theta_{xxt}, \theta_{xxx})(s, \cdot)||\}$,$M_{\alpha}(t)= \{\int_{0}^{t}e^{2\alpha s}||(D^{2}u, D^{3}u, D^{1}\theta, D^{2}\theta, \theta_{xtt}, \theta_{xxt}, \theta_{xxx})(s, \cdot)||^{2}ds\}^{1/2}$$E_{0}=||X_{0}’-X_{\infty}’||_{2}+||T_{0}-T_{\infty}||_{3}+ \sum_{j=1}^{3}||X_{j}||_{3-j}+||T_{1}||_{2}+||T_{2}||$,.

Here and hereafter, I use the following symbols:$D^{k}u=( \frac{\partial^{k}u}{\partial t^{j}\partial x^{j-k}},j=0,1, \cdots, k)$

and $\overline{D}^{k}u=(u, D^{1}u, \ldots, D^{k}u)$ .

Under these preparations, I can state the main result of this section which was provedby Kawashima and Shibata [3] in the following way.
. Then,THEOREM 1.3. Suppose that (A. ) $-(A.8)$ hold and that $T_{0}(x)>0$ forthere exis ansuch that if $||(X_{0}’, T_{0})-(1, \tau_{0})||_{\infty}+E_{0}+||f||_{1}<\epsilon$ , then the
problem (1.1), (1.2), (N.N) and (1.3) $ad$mi a unique solution (X $(t,$ $x),$ $T(t,$ $x)$ )satisfying the estim ate:$x\in\overline{\Omega}$

$1$

$ts$

$\epsilon>0$$ts$(1.12)$N_{\alpha}(t)^{2}+M_{\alpha}(t)^{2}\leq CE_{0}^{2}$$\in Z(\infty)$

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for $sui$ table posi tive constants$\alpha$and $C$ .

A Sketch of A Proof of Theorem 1.3.

The first step of a proof of Theorem 1.3 is a local in time existence theorem. I quote
the result due to W. Dan [1], which treated the Neumann problem for more general
quasilinear hyperbolic-parabolic coupled systems in any dimensional space.

Local in time existence theorem. Suppose that (A.2), (A.3), (A.4), (A.5)
an (A.6) hold. In addition, suppose that
$d$

$f,$

$(X_{0}’(x), T_{0}(x))\in G(\delta/2)$

and $T_{0}(x)>0$$g \in\bigcap_{j=0}^{1}C^{j}([0,t_{0});H^{1-j})$and$\partial_{t^{2}}f,$for$x\in\overline{\Omega}$,$\partial_{t^{2}}g\in L^{2}((0,t_{0});L^{2})$.

Le $tB$ be a posi tive numb er $such$ that$\sum_{j=0}^{3}||X_{j}||_{3-j}+\sum_{j=0}^{1}||T_{j}||_{3-j}+||T_{2}||$$+ \sum_{j=0}^{1}\sup_{0\leq s\leq t_{0}}||(\dot{\theta}_{t}f,\dot{\theta}_{t}g)(s, \cdot)||_{1-j}+\{\int_{0}^{t_{0}}||(\partial_{t}^{2}f, \partial_{t^{2}}g)(s, \cdot)||^{2}ds\}^{1/2}$$\leq B$.

Then, there exists a time $t_{1}\in(0,t_{0})$ depending only on
essentially such th at the
problem (1.1), (1.2), (N.N) and (1.3) admits a unique solution (X, )satisfying
the condition:
$B$

$T$

for all$(X_{x}(t, x),$ $T(t, x))\in G(2\delta/3)$$\in Z^{3}(t_{1})$$(t, x)\in[0,t_{1}]\cross\overline{\Omega}$.

Combining this local in time existence theorem and a prio estimates of local in time
solutions, we can extend local in time solutions to any time interval, and then Theorem
1.3 is established. Therefore, the main step of the proof in Kawashima and Shibata [3]is to show the following.
estimates of a local in time solution (X, )$\tau\dot{\tau}$

$a$

$T$

$p_{7\dot{\eta}}or\dot{\eta}$

A priori estimates. There exist positive cons tants $C,$
holds provided that

(1.13)(1.14)$N(t)\leq\sigma$

for$0\leq t\leq t_{0}$$\alpha$andsuch that (1.12)$\sigma$,$||(X_{\infty}’, T_{\infty})-(1, \tau_{0})||_{\infty}+||X_{\infty}’’||_{\infty}+||X_{\infty}’’||_{1}+||f||\leq\sigma$$\in Z^{3}(t_{0})$.

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Our derivation of (1.12) is divided into eight steps.
STEP 1. I verify th relations:$e$(1.15)(1.16)$||u_{t}(t, \cdot)||\leq C||u_{tx}(t, \cdot)||$,$||(u_{x)}\theta)(t, \cdot)||\leq C||(u_{t}, u_{x}, \theta)_{x}(t, \cdot)||$,provided that is $sm$ all enough. Here and hereafter, the le $tterC$ denotes various
constants independent of and .
$\sigma$$\alpha$$\sigma$

To prove Step 1, we use the following Poincar\’e’s inequalities:(1.17)$||v|| \leq C\{|\int_{0}^{1}p(x)v(x)dx|+||v’||\}$

for $v\in H^{1}$ , where $p(x)\in L^{2}$ such that
(1.18)$\int_{0^{1}}p(x)dx\neq 0$;$||v||\leq C\{<v>+||v’||\}$

for $v\in H^{1}$ , where $<v>=|v(0)|+|v(1)|$ . (A.7) implies that $\int_{0^{1}}u_{t}(t, x)dx=0$ , which. Since $S-S_{\infty}=0$
combined with (1.17) implies (1.15). Put
on(1.8)(N.N),
as follows fromand the fact: $F(O)=F(1)=0$ , applying (1.18) to$S-S_{\infty}$ implies that
$S_{\infty}=S(X_{\infty}’(x), T_{\infty})$$\partial\Omega$(1.19)$||(S_{F}^{0}u_{x}+S_{T}^{0}\theta)(t,\cdot)||\leq C\{||(u_{x},\theta)_{x}(t,\cdot)||+\sigma||(u_{x},\theta)(t, \cdot)||\}$

where we have used the formula:
expansion, that is$S-S_{\infty}=S_{F}^{0}u_{x}+S_{T}^{0}\theta$,

which is obtained by the Taylor$S_{G}^{0}= \int_{0}^{1}\frac{\partial S}{\partial G}((X_{\infty}’(x), T_{\infty})+\ell(u_{x}(t, x),$ $\theta(t, x)))\mathscr{U}$for $G=F$ and$T$.

In the same manner, we can write: $e(X_{x}, T)=e(X_{\infty}’(x), T_{\infty})+e_{F}^{0}u_{x}+e_{T}^{0}\theta$. Inserting
this formula into (1.9) and using (1.10), we see that
(1.20)$\int_{0}^{1}\{e_{F}^{0}u_{x}+e_{T}^{0}\theta+\frac{1}{2}u_{t}^{2}+u_{x}F\}dx=0$,

which combined with (1.17) and (1.15) implies that
(1.21)$||(e_{F}^{0}u_{x}+e_{T}^{0}\theta)(t, \cdot)||\leq C\{\sigma||(u_{x}, \theta)(t, \cdot)||+||(u_{x}, u_{t}, \theta)_{x}(t, \cdot)||\}$Let us define the matrix

$U$

by:
$U=$

(

$e_{T}^{0}e_{F}^{0}$

).

.

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(1.11) guarantees that the determinant of is positive provided thatso that combining (1.19) and (1.21) implies (1.16).
Note that (1.1) can be rewritten in the form:$U$(1.22)$\sigma$

is small enough,$u_{tt}-(S_{F}^{0}u_{x}+S_{T}^{0}\theta)_{x}=0$

on. Differentiating (1.22) and (1.4) -times ( $k=0,1$ and 2)
with
,
and
with respect to and multiplying the resulting equations by
we can derive the following step.
$S_{F}^{0}u_{x}+S_{T}^{0}\theta=0$$k$$\partial\Omega$$\partial_{t^{k+1}}u$$t$$(T_{\infty}+\theta)^{-1}\partial_{t^{k}}\theta$

STEP 2. We have
(1.23)$e^{2\alpha t}|| \overline{\partial}_{t}^{2}(u_{t}, u_{x}, \theta)(t, \cdot)||^{2}+c_{1}\int_{0}^{1}e^{2\alpha s}||\overline{\partial}_{t}^{2}\theta_{x}(s, \cdot)||^{2}ds\leq CR_{\alpha}(t)$for a suita $ble$ constant $c_{1}>0$ , where$R_{\alpha}(t)=E_{0}^{2}+(\sigma+\alpha)M_{\alpha}(t)^{2}+\sigma N_{\alpha}(t)^{2}$and$\overline{\partial}_{t}^{k}v=(v, \partial_{t}v, \ldots\partial_{t^{k}}v)$.

In the derivation of (1.12), we use the identity:$e^{2\alpha t}g(t)= \int_{0}^{t}e^{2\alpha s}\frac{dg}{ds}(s)ds+g(0)+2\alpha\int_{0}^{t}e^{2\alpha s}g(s)ds$

STEP 3. For$\sigma$.

small enough, I verify the relation:(1.24)$N_{\alpha}(t)^{2}\leq CR_{\alpha}(t)$In view of (1.23), we have to estimate the terms:. Fkom (1.4) it follows that
and.
$\theta_{x},$$\theta_{xt},$$u_{xx},$$\theta_{xx},$$u_{xxx},$$u_{xxt},$$\theta_{xxx}$$\theta_{xxt}$$(Q\partial_{t^{k}}\theta_{x})_{x}=-k(Q_{t}\theta_{x})_{x}+\partial_{t^{k}}((T_{\infty}+\theta)\eta_{t})$for $k=0$ and 1.

Multiplying this by
and integration by parts and using the fact thatwhich follows from Sobolev’s inequality and (1.13), we have$\partial_{t^{k}}\theta$$||Q_{t}||_{\infty}\leq C\sigma$$|(Q\theta_{x}, \theta_{x})|+|(Q\theta_{xt}, \theta_{xt})|\leq C\{\sigma||(\theta_{x}, \theta_{xt})(t, \cdot)||^{2}+||\overline{\partial}_{t}^{2}(u_{t)}u_{x}, \theta)(t, \cdot)||^{2}\}$where(1.25)$(\cdot, \cdot)$

denotes the usual$L^{2}$,-innerproduct, which combined with (1.23) implies that$e^{2\alpha t}||(\theta_{x}, \theta_{xt})(t, \cdot)||^{2}\leq CR_{\alpha}(t)$.

From (1.4) and (1.22) it follows that(1.26)(1.27),$S_{F}^{0}u_{xx}=u_{tt}-\{S_{T}^{0}\theta_{x}+(S_{F}^{0})_{x}u_{x}+(S_{T}^{0})_{x}\theta\}$$Q\theta_{xx}=-Q_{x}\theta_{x}+(T_{\infty}+\theta)(\eta_{T}\theta_{t}+\eta_{F}u_{xt})$

.

,

292
By using these formulae, (1.23) and (1.25) we have

(1.28)$e^{2\alpha t}||(u_{xx}, \theta_{xx})(t, \cdot)||\leq CR_{\alpha}(t)$Differentiating (1.26) and (1.27) with respect to(1.28), we have
(1.29)

$t$

and

$x$

.

and using (1.23), (1.25) and$e^{2\alpha t}||(u_{xxx},u_{xxt}, \theta_{xxx}, \theta_{xxt})(t, \cdot)||^{2}\leq CR_{\alpha}(t)$Combining (1.23), (1.25), (1.28) and (1.29) implies (1.24).
. The estimation forNow, I am going to estimate
obtained.
$M_{\alpha}(t)$.

was already$||\overline{\partial}_{t}^{2}\theta_{x}(t, \cdot)||^{2}$

STEP 4. I verify the relation:;(1.30)$\int_{0}^{t}e^{2\alpha s}||\theta_{xxx}(s, \cdot)||^{2}ds\leq C\{\int_{0}^{t}e^{2\alpha s}||u_{xxt}(s, \cdot)||^{2}ds+R_{\alpha}(t)\}$(1.31)$\int_{0}^{t}e^{2\alpha s}||(\theta_{xx}, \theta_{xxt})(s, \cdot)||^{2}ds\leq\delta\int_{0}^{t}e^{2\alpha s}||u_{xxt}(s, \cdot)||^{2}ds+C\delta^{-1}R_{\alpha}(t)$for $\delta\in(0,1)$ .

Differentiating (1.27) once with respect to , we have (1.30) immediately. Using (1.27)
and the formula of differentiation of (1.27) once with respect to , we have also (1.31)
immediately.
$x$

$t$

STEP 5. I verify the relation:(1.32)$\int_{0}^{t}e^{2\alpha s}||D^{3}u(s, \cdot)||ds\leq C\{\int_{0}^{t}e^{2\alpha s}||u_{xxt}(s, \cdot)||^{2}ds+R_{\alpha}(t)\}$Put
for $K=S$ andrepresentation formula for:$K_{L}=\partial K/\partial L$$\eta$

and $L=F$ and$T$.

. First, I shall use the following$u_{xtt}$(1.33)$Vu_{xtt}=S_{tt}+(((T_{\infty}+\theta)\eta_{T})^{-1}S_{T}(Q\theta_{x})_{x})_{t}-V_{t}u_{xt}$

,

where
$V=\eta^{-1}M_{\eta}(X_{\infty}’+u_{x)}T_{\infty}+\theta)$(cf. (1.11)).

This representation follows from the differentiation with respect to of the formula:$t$(1.34)$S_{t}=Vu_{xt}+((T_{\infty}+\theta)7\Pi)^{-1}S_{T}(Q\theta_{x})_{x}$,293which follows from the combination of the following two formulae:$S_{t}=S_{F}u_{xt}+S_{T}\theta_{t}$Since $V>0$ for$\sigma$and$\theta_{t}=((T_{\infty}+\theta)m^{1})^{-1}\{(Q\theta_{x})_{x}-(T_{\infty}+\theta)\eta_{F}u_{xt}\}$small enough, multiplying (1.33) by$u_{xtt}$.

and noting the identity:$(S_{tt}, u_{xtt})= \frac{d}{dt}(S_{t}, u_{xtt})+(S_{xt}, u_{ttt})=\frac{d}{dt}(S_{t}, u_{xtt})+(S_{xt}, S_{xt})$,

we have.

$\int_{0}^{t}e^{2\alpha s}||u_{xtt}(s, \cdot)||^{2}ds\leq C\{\int_{0}^{t}e^{2\alpha s}||u_{xxt}(s, \cdot)||^{2}ds+R_{\alpha}(t)\}$Differentiating (1.26) once with respect toeasily that(1.35)$x$and using the formula: $u_{ttt}=S_{xt}$ , we have$\int_{0}^{t}e^{2\alpha s}||(u_{xxx}, u_{ttt})(s, \cdot)||^{2}ds\leq C\{\int_{0}^{t}e^{2\alpha s}||u_{xxt}(s, \cdot)||^{2}ds+R_{\alpha}(t)\}$,where we have used the relation: 11 $u_{tt}||\leq C||u_{xtt}||$ , which follows from the fact that$\int_{0^{1}}u_{tt}(t, x)dx=0$ and (1.17).
One of the main points of the proof is the following two steps and the idea goes backto the paper due to Munos Rivera [4].

STEP 6. I verify the relation:(1.36)$\int_{0}^{t}e^{2\alpha s}||(D^{3}u, \theta_{xx}, \theta_{xxx}, \theta_{xxt})(s, \cdot)||^{2}ds$$\leq\mu\int_{0}^{t}e^{2\alpha s}<u_{xxt}(s, \cdot)>^{2}ds+C\mu^{-4}R_{\alpha}(t)$for $\mu>0$ small enough.
In view of (1.30), (1.31) and (1.32), we shall estimate I
the formula:(1.37)$u_{xxt}(s, \cdot)||$. To do this, we use$Q\theta_{xxx}-(T_{\infty}+\theta)\eta_{T}\theta_{xt}-(T_{\infty}+\theta)\eta_{F}u_{xxt}+g_{1}=0$

,

where
$g_{1}=Q_{x}\theta_{xx}+Q_{xx}\theta_{x}-((T_{\infty}+\theta)\eta_{T})_{x}\theta_{t}-((T_{\infty}+\theta)\eta_{F})_{x}u_{xt}$,which follows from the differentiation of (1.27) wi th respect to . Since$x$$(Q\theta_{xxx)}u_{xxt})=<Q\theta_{xx},$ $u_{xxt}>-(Q_{x}\theta_{xx}, u_{xxt})$$- \frac{d}{dt}(Q\theta_{xx}, u_{xxx})+(Q_{t}\theta_{xx}, u_{xxx})+(Q\theta_{xxt}, u_{xxx})$294
where $<u,$ $v>=u(1)v(1)-u(O)v(O)$ , multiplying (1.37) by(1.38)$\int_{0}^{t}e^{2\alpha s}||u_{xxt}(s, \cdot)||^{2}ds\leq\int_{0}^{t}e^{2\alpha s}|<Q\theta_{xx},$$u_{xxt}$

implies that$u_{xxt}>|ds$$+ \int_{0}^{t}e^{2\alpha s}|(Q\theta_{xxt}, u_{xxx})|ds+CR_{\alpha}(t)$.

Choosing $\delta>0$ small enough in (1.31) and using Schwarz’s inequality and (1.32), thesecond term of the right-hand side of (1.38) can be absorbed by the left-hand side of(1.38). Since$< \theta_{xx}(t,\cdot)>2\leq||\theta_{xx}(t,\cdot)^{2}||_{\infty}\leq C\int_{0}^{1}|\overline{\partial}_{x}^{1}(\theta_{xx}(t,\cdot)^{2})|dx$$\leq C\{||\theta_{xx}(t,\cdot)||^{2}+||\theta_{xx}(t,\cdot)||||\theta_{xxx}(t, \cdot)||\}$

where$\overline{\partial}_{x}^{k}v=(v, \partial_{x}v, \ldots\partial_{x}^{k}v)$

and we have used Sobolev’s inequality:$||v||_{\infty} \leq C\int_{0}^{1}|\overline{\partial}_{x}^{1}v(x)|dx$,

we have$\int_{0}^{t}e^{2\alpha s}|<Q\theta_{xx},$$u_{xxt}>|ds$$\leq\mu\int_{0}^{t}e^{2\alpha s}<u_{xxt}(s, \cdot)>^{2}ds+C\mu^{-1}\int_{0}^{t}e^{2\alpha s}<\theta_{xx}(s, \cdot)>^{2}ds$$\leq\mu\int_{0}^{t}e^{2\alpha s}<u_{xxt}(s, \cdot)>2ds+C\mu^{-1}(1+\delta^{-1/2})\int_{0}^{t}e^{2\alpha s}||\theta_{xx}(s, \cdot)||^{2}ds$$+C \mu^{-1}\delta^{1/2}\int_{0}^{t}e^{2\alpha s}||\theta_{xxx}(s, \cdot)||^{2}ds$.

Inserting this inequality into (1.38) and using (1.30) and (1.31), we have (1.36) by.
for a suitably small constantchoosing and in such a way that$\kappa>0$$\mu=\kappa\delta^{1/2}$$\delta$$\mu$

STEP 7. I verify the relation:(1.39)$\int_{0}^{t}e^{2\alpha s}<u_{xxt}(s, \cdot)>^{2}ds$$\leq C\{\int_{0}^{t}e^{2\alpha s}||(D^{3}u, \theta_{xx}, \theta_{xxx}, \theta_{xxt})(s, \cdot)||^{2}ds+R_{\alpha}(t)\}$Differentiating (1.1) with respect to and$t$(1.40)$x$

implies that$u_{xxxt}=S_{F}^{-1}\{u_{xttt}-S_{T}\theta_{xxt}-g_{2}\}$

.

295
where
$g_{2}=2((S_{F})_{x}u_{xxt}+(S_{T})_{x}\theta_{xt})+(S_{F})_{xx}u_{xt}+(S_{T})_{xx}\theta_{t}$.

Put $q(x)=x- \frac{1}{2}$ and then we have$(u_{xxxt}, qu_{xxt})= \frac{1}{4}<u_{xxt}>^{2}-\frac{1}{2}||u_{xxt}||^{2}$;$(S_{F}^{-1}u_{xttt}, qu_{xxt})= \frac{d}{dt}(S_{F}^{-1}u_{xtt}, qu_{xxt})-((S_{F}^{-1})_{t}u_{xtt}, qu_{xxt})$$+ \frac{1}{2}((S_{F}^{-1}q)_{x}u_{xtt}, u_{xtt})-\frac{1}{4}\{(S_{F}^{-1}u_{xtt}^{2})(t, 1)+(S_{F}^{-1}u_{xtt}^{2})(t,0)\}$

Therefore, multiplying (1.40) by(1.41)$qu_{xxt}$.

implies that$\frac{1}{4}\{((1+S_{F}^{-1})u_{xtt}^{2})(t, 1)+((1+S_{F}^{-1})u_{xtt}^{2})(t,’ 0)\}$$\leq C\{||\theta_{xxt}||^{2}+||D^{3}u||^{2}+||g_{2}||^{2}\}+\frac{d}{dt}(S_{F}^{-1}u_{xtt}, qu_{xxt})$.

, integrating the resulting equation over $(0,t)$ and using (1.24),
Multiplying (1.41) by
we have (1.39).
Combining (1.36) and (1.39) and choosing small enough, we arrive at the relation:$e^{2\alpha t}$$\mu$(1.42)$\int_{0}^{t}e^{2\alpha s}||(D^{3}u, \theta_{xx}, \theta_{xxx)}\theta_{xxt})(s, \cdot)||^{2}ds\leq CR_{\alpha}(t)$.

STEP 8. I verify the relation:(1.43)$\int_{0}^{t}e^{2\alpha s}||(D^{2}u, \theta_{t}, \theta_{tt})(s, \cdot)||^{2}ds\leq CR_{\alpha}(t)$.

We already knew the estimate: $||u_{tt}||\leq C||u_{ttx}||$ . The estimat of I
follows from
. Since $S_{t}=0$ on
the formula:as, by (1.18) we havefollows from $S=0$ on$e$$u_{xx}||$$u_{xx}=(S_{F}^{0})^{-1}\{u_{tt}-S_{T}^{0}\theta_{x}-(S_{F}^{0})_{x}u_{x}-(S_{T}^{0})_{x}\theta\}$$\partial\Omega$$\partial\Omega$$||S_{t}||^{2}\leq C||S_{xt}||^{2}\leq C\{||(u_{xxt)}\theta_{xt})(t,\cdot)||^{2}+\sigma||(u_{xt},\theta_{t})(t,\cdot)||^{2}\}$

which combined wi th (1.34) implies that the estimate of 1
and I
follow from the formula:of$||\theta_{t}||$$u_{xt}||$,. Finally, the estimates$\theta_{tt}||$$\theta_{t}=((T_{\infty}+\theta)\eta_{T})^{-1}\{(Q\theta_{x})_{x}-(T_{\infty}+\theta)\eta_{F}u_{xt}\}$and the formula of its differentiation, respectively.
Summing up, we have obtained,$N_{\alpha}(t)^{2}+M_{\alpha}(t)^{2}\leq C\{E_{0}^{2}+(\sigma+\alpha)M_{\alpha}(t)^{2}+\sigma N_{\alpha}(t)^{2}\}$

296
and then choosing$\sigma$and$\alpha$small enough, we have (1.12).
References[1] W. Dan, On a local in time solvability of the Neumann problem of quasilinear hyperbolic parabolic coupled systems, Preprint in 1992.
[2] S. Jiang, Global solutions of the Neumann problem in one-dimensional nonlinear
the7moelasticity, Nonlinear Analysis, TMA. 19 (1992), 107-121.
[3] S. Kawashima and Y. Shibata, On the Neumann problem of one-dimensional nonlinear thenn oelasticity with time-independent extemal forces, to appear in Czech.
Math. J.
[4] J.E. Munos Rivera, Energy decay rates in linear therrreoelasticity, Funkcialaj Ekvacioj, 35 (1992), 19-30.
[5] Y. Shibata, Neumann pro blem for one-dimensional nonlinear themoelasticity, toappear in Banach Center Publication.