数理解析研究所講究録
1536 巻 2007 年 151-171

151

Large time behavior of solutions
to a semilinear hyperbolic systemwith relaxation
Yoshihiro UEDA* and Shuichi KAWASHIMA”*Graduate School of Mathematics, Kyushu University
ued@math. kyushu-u. .jp$\mathrm{a}\mathrm{c}$** Facultyof Mathematics, Kyushu University

kawashim@math.kyushu-u.ac.jp

Abstract

We are concerned with the initial value problem for a damped wave equation witha nonlinear convection term which is derived from a semilinear hyperbolic systemwith relaxation. We show the global existence and asymptotic decay of solutions
in $W^{1,p}(1\leq p\leq\infty)$ under smallness condition on the initial data. Moreover, weshow that the solution approaches in $W^{1,p}(1\leq p\leq\infty)$ the nonlinear diffusion
wave expressed in terms of the self-similar solution of the Burgers equation as time
tends to infinity. Our results are based on the detailed pointwise estimates for thefundamental solutions to the linearlized equation.

1

Introduction

We consider a nonlinear relaxation system of the form:$u_{t}+v_{x}=0$,$v_{t}+\mathrm{u}_{x}=f(u)-v$

,

(1.1)

, and $f(u)$ is a smooth function
where and are unknown functions of $t>0$ andof under consideration. If we eliminate from (1.1), we obtain the following damped
wave equation with a nonlinear convection term:$u$$x\in \mathrm{R}$

$v$

$u$

$v$

$u_{u}-u_{xx}+u_{t}+f(u)_{x}=0$

.

(1.2)

We consider the initial value problem for (1.2) with the initial conditions$u(0, x)=u_{0}(x)$,$u_{t}(0, x)=u_{1}(x)$

.

(1.3)

This initial value problem was studied by R. Orive and E. Zuazua [5] when $f(u)=$, so that $f’(0)=0$ . They proved the global existence and asymptotic
with$|\mathrm{u}|^{\gamma-1}u$$\gamma\geq 2$

152

Large time behavior of solu

$t\mathrm{i}ons$

to a $sem$ ilin ear hyperboric system with relaxati on

decay of solutions under smallness condition on the initial data
in
and
in
, they observed that when
. Moreover, under the additional condition$\gamma=2$ , the solution obtained approaches the self-similar solution $z(t, x)$ of the Burgers
equation $z_{t}+(|z|z)_{x}=z_{xx}$ which verifies the integral condition $\int z(t, x)dx=M$ , where$M:= \int(u_{0}+u_{1})(x)dx$ . When $\gamma>2$ , it was also observed in [5] that the asymptotic
profile $z(t, x)$ is given by the heat kernel, i.e., the self-similar solution of the heat equation$z_{t}=z_{xx}$ which verifies $\int z(t, x)dx=M$ with the same.
purposepaper
generalizein thisThe mainis tothe results in [5] to the case where $f(u)$satisfies the so called sub-characteristic condition $|f’(0)|<1$ . In addition, we develop $IP$
theory for including $p=1$ . In fact, under smallness condition on the initial data in, where $1\leq p\leq\infty$ , we prove that the solution exists globallyand inin time and satisfies the decay estimates$u_{0}$$L^{2}\cap L^{1}$$u_{0},$$H^{1}\cap L^{1}$$u_{1}$$u_{1}\in L_{1}^{2}$$\Lambda f$$p$$W^{1,\mathrm{p}}\cap L^{1}$$u_{0}$$IP\cap L^{1}$$u_{1}$$||u(t)||_{L^{q}}\leq C(1+t)^{-\frac{1}{2}(1-\frac{1}{q})}$for any $1\leq q\leq\infty$ ,$||\partial_{x}u(t)||_{L^{\mathrm{p}}}\leq C(1+t)^{-1_{(1-\frac{1}{\mathrm{p}})-\frac{1}{2}}}2$

(1.4)

.

To discuss more detailed large-time behavior of the solution for I $f’(\mathrm{O})|<1$ , we needadditional consideration. To see this, as in $[3,1]$ , we apply the Chapman-Enskog expansion
to (1.1) and derive a viscous conservation law(1.5)$w_{t}+f(w)_{x}=(\mu(w)w_{x})_{x}$as the second order approximation of the expansion, where $\mu(w)=1-(f’(w))^{2}$ . Notethat the sub-characteristic condition 1 $f’(w)|<1$ implies the parabolicity of (1.5). It is
expected that the solution of (1.2) can be approximated by the solution of (1.5) or its
simpler version$w_{t}+( \alpha w+\frac{\beta}{2}w^{2})_{x}=\mu w_{xx}$

,

(1.6)

where $\alpha=f’(0),$ $\beta=f’’(0)$ and $\mu=1-(f’(\mathrm{O}))^{2}$ . When $\beta=f’’(0)>0$ , by the change
of independent and dependent variables $x=y+\alpha t$ and $w=\beta z,$ $(1.6)$ is reduced to the
Burgers equation $z_{t}+(z^{2}/2)_{y}=\mu z_{yy}$ whose asymptotic profile is given by its self-similarsolution (see (2.3) below). Consequently, it is expected that the solution $u(t, x)$ of (1.2) isapproximated by the nonlinear diffusion wave $w(t, x)$ which is a modification of the selfsimilar solution $z(t, y)$ of the Burgers equation and is defined as $w(t, x)=\beta^{-1}z$ ( , x-at).
In fact, under the additional condition, we show that the solution to the
problem (1.2), (1.3) approaches the nonlinear diffusion wave $w(t,x)$ which verifies theintegral condition $\int w(t, x)dx=M$ , where $M:= \int(u_{0}+u_{1})(x)dx$ . More specifically, weshow that$t$$\mathrm{u}_{0},$$u_{1}\in L_{1}^{1}$$||(u-w)(t)||_{L^{q}}\leq C(1+t)^{-_{2}}1-\sigma\iota_{()-_{2}}\iota\iota_{+\epsilon}$

for any $1\leq q\leq\infty$ ,(1.7)$||\partial_{x}(u-w)(t)||_{L^{\mathrm{p}}}\leq C(1+t)^{-\xi}(1-p1_{)-1+\mathrm{g}}$as, where is any fixed positive number.
Before closing this section, we give some notations used in this paper. Let $F[f]$ denote
the Fourier transform and $F^{-1}[f]$ denote the Fourier inverse transform of defined by$tarrow\infty$$\epsilon$$f$$\mathcal{F}[f](\xi)=\hat{f}(\xi):=\int_{-\infty}^{\infty}f(x)e^{-ix\xi}dx$,$F^{-1}[f](x):= \frac{1}{2\pi}\int_{-\infty}^{\infty}f(\xi)\mathrm{e}^{1x\epsilon}d\xi$

.

153

Large time behavior of$sol\mathrm{u}$tions to a semilin ear hyperboric system with relaxati on

For $1\leq p\leq\infty$ , we denote by
the usual Lebesgue space with the norm
denotes the Sobolev space. Let be a nonnegative integer. Thendenote the, letfunctions, equipped with the norm I. For aof
. Let $X$ be a Banach space
space with the norm lfll
weighted
and let be an interval on R. Then $C(I;X)$ denotes the space of continuous functions
on the interval with values in the Banach space $X$ . Also, $L^{\infty}(I;X)$ denotes the spaceoffunctions on with values in $X$ .
$L^{p}=L^{p}(\mathbb{R})$$k$$||\cdot||_{L^{p}}$$W^{k,p}=W^{k,\mathrm{p}}(\mathbb{R})$$L^{\mathrm{p}}$$\in \mathrm{R}$$f||_{W^{k},\nu}$$L^{p}$$L_{\alpha}^{p}=L_{\alpha}^{\mathrm{p}}(\mathrm{R})$$L_{\alpha}^{p}:=||(1+|x|)^{\alpha}f||_{L^{p}}$

$I$

$I$

$I$

$L^{\infty}$

2

Main results

In this section we give statements of our main results in this paper. The first result isconcerning the global existence and optimal decay of solutions to the initial value problem(1.2), (1.3), which can be stated as follows.
Theorem 2.1 Suppose that $|f’(0)|<1$ . Let $1\leq p\leq\infty$ and assume thatand$u_{1}\in L^{p}\cap L^{1}$$u_{0}\in W^{1,p}\cap L^{1}$. Put$E_{0}:=||u_{0}||_{W^{1,\mathrm{p}}}+||u_{0}||_{L^{1}}+||u_{1}||_{L^{p}}+||u_{1}||_{L^{1}}$Then there is a positive constantsuch that if(1.2), (1.3) has a unique global solution $u(t, x)$ with$\delta_{0}$$E_{0}\leq\delta_{0}$$u\in C([0, \infty);W^{1,p}\cap L^{1})$.

$||u(t)||_{L^{q}}\leq CE_{0}(1+t)^{-\frac{1}{2}(1-\frac{1}{q})}$,.

, then the initial value problem

Moreover, the solution satisfies$||\partial_{x}u(t)||_{L^{p}}\leq CE_{0}(1+t)^{-\frac{1}{2}(1-\frac{1}{p})-\frac{1}{2}}$for any with
$q$

$1\leq q\leq\infty$

and

$C$

(2.1)

,

is a constant.

Remark 2.2 When $p=\infty$ , we should replace the solution space by $C([0, \infty);L^{1})\cap$$L^{\infty}((0, \infty);W^{1,\infty})$.

In order to state our second main result concerning the large-time behavior of the solution obtained in Theorem 2.1, we define the nonlinear diffusion wave for (1.2). Consider
the self-similar solution to the Burgers equation$z_{t}+(z^{2}/2)_{x}=\mu z_{xx}$

(2.2)

,

. We denote
where $\mu=1-(f’(0))^{2}$ , which is a solution of the form
condition
integral$z(t,x)=Z(t,x;\mu,M)$bythe self-similar solution which satisfies the$\int z(t,x)dx=M$ , where $M$ is a parameter. This self-similar solution is given explicitly as$z(t,x)=t^{-\frac{1}{2}}\phi(_{7\iota}^{x})$$Z(t,x; \mu, M)=\sqrt{\frac{\mu}{t}}\frac{(e^{M}2\mu 1)e^{-y^{2}}}{\sqrt{\pi}+(e^{4}2\mu 1)\int_{y}^{\infty}e^{-\xi^{2}}d\xi}=$,$y= \frac{x}{\sqrt{4\mu t}}$

.

(2.3)

154

Large time behavior of$sol\mathrm{u}$tions to a emilinear hyperboric system with relaxa tion$\mathrm{s}$We then define $W(t, x)$ by$W(t,x)=\beta^{-1}Z$( , x-at;$t$$\mu,\beta M$

),

(2.4)

where $\alpha=f’(0),$ $\beta=f’’(0)$ and $\mu=1-(f’(\mathrm{O}))^{2}$ . Here we assumed that $\beta=f’’(\mathrm{O})>0$ .
We see that this $W(t, x)$ has the conserved quantity $\int W(t, x)dx=M$ and satisfies (1.6),i.e.,$w_{t}+( \alpha w+\frac{\beta}{2}w^{2})_{x}=\mu w_{xx}$

(2.5)

,

which is an approximation to the viscous conservation law (1.5) derived form (1.1) byapplying the Chapman-Enskog expansion. We call $W(t, x)$ defined by (2.4) the nonlinear
diffusion wave for (1.2) if the parameter $M$ is chosen as $M= \int(u_{0}+u_{1})(x)dx$ .
The nonlinear diffusion wave defined above gives the large-time description of thesolution obtained in Theorem 2.1.
Theorem 2.3 Suppose that $|f’(0)|<1$ and $f”(0)>0$ . Let $1\leq p\leq\infty$ and assume thatand
. Let $u(t, x)$ be the global solution of the problem (1.2),
1. constructed in Theorem 2.1, and let $W(t, x)$ be the nonlinear diffusion wave definedby (2.4) Utth $M= \int(u_{0}+u_{1})(x)dx$ . Put $w(t, x)=W(t+1, x)$ and$u_{0}\in W^{1,p}\cap L_{1}^{1}$$($$u_{1}\in L^{p}\cap L_{1}^{1}$

$S)$

.

$E_{1}:=||u_{0}||_{W^{1,p}}+||u_{0}||_{L_{1}^{1}}+||u_{1}||_{L^{\mathrm{p}}}+||\mathrm{u}_{1}||_{L_{1}^{1}}$Then, for any withthere is a positive constant such that if
is given in Theorem 2.1), then we have the following asymptotic relations:$\epsilon$$0< \epsilon<\frac{1}{2}f$$E_{0}\leq\delta_{1}$$\delta_{1}$(where$E_{0}$$||(u-w)(t)||_{L^{q}}\leq CE_{1}(1+t)^{-}2\mathrm{q}2\iota_{(1-}\iota_{)-}\iota_{+\epsilon}$$||\partial_{x}(u-w)(t)||_{L^{p}}\leq CE_{1}(1+t)^{-\frac{1}{2}(1-\frac{1}{p})-1+\epsilon}$

for any$q$

Utth $1\leq q\leq\infty$ and

$C$

,

(2.6)

,

is a constant.

Remark 2.4 A straightfo nvard computation using (2.4) and 2.
$($

$S)$

yields$||\partial_{x}^{l}w(t)||_{L^{q}}\leq C|M|(1+t)-\mathrm{i}(1-\mathrm{q})\iota-\tau\iota$

for any(2.7), where $M= \int(u_{0}+u_{1})(x)dx$ . More precisely, whenand $l=0,1,$$M\neq 0,$
behaves exactly likeinas
. Therefore, the estimate(2.6) gives meaningful asymptotic relations for, provided that $M\neq 0$ .
$1\leq q\leq\infty$$\theta_{x}w(t, x)$$\cdots$$t^{-;\iota\iota}(1-_{q})-f$$L^{q}$$tarrow\infty$$tarrow\infty$

3

Fundamental solutions

The aim of this section is to study the fundamental solutions to the linearized equationof(1.2):$u_{u}-u_{xx}+u_{t}+\alpha u_{x}=0$ ,(3.1)
where $\alpha=f’(0)$ . To this end, we consider (3.1) with the initial data$u(\mathrm{O}, x)=u_{0}(x)$,$u_{t}(0, x)=u_{1}(x)$

.

(3.2)

155

Large time behavior of solu tions to a semilinear hyperboric system with relaxation

We take the Fourier transform, obtaining$\hat{u}_{tt}+\hat{u}_{t}+(\xi^{2}+\alpha i\xi)\hat{u}=0$a,$(0,\xi)=\hat{u}_{0}(\xi),\hat{u}_{t}(0,\xi)=\hat{u}_{1}(\xi)$

(3.3)

.

The characteristic equation of (3.3) is $\lambda^{2}+\lambda+(\xi^{2}+ai\xi)=0$ and the eigenvalues are$\lambda_{1}(\xi)=\frac{1}{2}(-1+\sqrt{1-4(\xi^{2}+\alpha i\xi)})$,$\lambda_{2}(\xi)=\frac{1}{2}(-1-\sqrt{1-4(\xi^{2}+\alpha i\xi)})$

.

(3.4)

The problem (3.3) is then solved as\^u $(t,\xi)=\hat{G}(t, \xi)(\hat{u}_{0}(\xi)+\hat{u}_{1}(\xi))+\hat{H}(t,\xi)\hat{u}_{0}(\xi)$ ,

(3.5)

where
$\hat{G}(t,\xi)=\frac{1}{\lambda_{1}(\xi)-\lambda_{2}(\xi)}(e^{\lambda_{1}(\xi)t}-e^{\lambda_{2}(\xi)t})$

,
(3.6)

$\hat{H}(t, \xi)=\frac{1}{\lambda_{1}(\xi)-\lambda_{2}(\xi)}((1+\lambda_{1}(\xi))e^{\lambda_{2}(\zeta)t}-(1+\lambda_{2}(\xi))e^{\lambda_{1}(\xi)t})$.

We take the Fourier inverse transform of (3.5). This yields the solution formula of thelinearized problem (3.1), (3.2):$u(t)=G(t)*(u_{0}+u_{1})+H(t)*u_{0}$ ,(3.7)

where $G(t,x)$ and $H(t,x)$ denote the Fourier inverse transforms of$\hat{G}(t,\xi)$and$\hat{H}(t,\xi)$in(3.6), respectively:$G(t,x):=F^{-1}[\hat{G}(t, \cdot)](x)$,$H(t, x):=F^{-1}[\hat{H}(t, \cdot)](x)$

(3.8)

,

the convolution with respect to . We call $G(t, x)$ and $H(t, x)$ the fundamental solutions of linearized damped wave equation (3.1).
We are interested in the asymptotic expressions of the fundamental solutions together
with their detailed pointwise estimates. To state the results, we introduce the modified
heat kernel:$\mathrm{a}\mathrm{n}\mathrm{d}*\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{s}$$x$$G_{0}(t,x)= \frac{1}{\sqrt{4\pi\mu t}}e^{-(x-at)^{2}/4\mu t}$

(3.9)

,

where $a=f’(\mathrm{O})$ and $\mu=1-(f’(0))^{2}$ , which is the fundamental solution to the linear
heat equation $w_{t}+\alpha w_{x}=\mu w_{xx}$ . Then the result for $G(t, x)$ can be stated as follows.

Theorem 3.1 Let $\alpha=f’(0)$ and $\mu=1-(f’(0))^{2}$ , and assume that $|\alpha|<1$ . For eachnonnegative integer the fundamental solution $G(t, x)$ can be expressed as$l_{f}$$G(t,x)=G_{0}(t,x)+G_{\infty}^{(l)}(t,x)+R^{(l)}(t,x)=G_{\infty}^{(l)}(t, x)+R_{\infty}^{(l)}(t,x)$

is the modified heat kemel in
as follows: We have
and

Here

$G_{0}(t, x)$$(S.\mathit{9})_{f}$and$G_{\infty}^{(l)}(t, x)$.

is the singular part given$G_{\infty}^{(0)}(t,x)\equiv 0$$\theta_{x}G_{\infty}^{(l)}(t,x)=\sum_{k=0}^{l-1}\{e^{-\kappa t}P_{k}(t)\partial_{x}^{l-k-1}\delta(x+t)+e^{-\nu t}Q_{\mathrm{k}}(t)\partial_{x}^{l-k-1}\delta(x-t)\}$(3.10)

156

Large time behavior of solu tions to a semilinear hyper\’ooric system with relaxation

are some polynomials ofandanddenotes the Dirac delta function. The remainder terms
verify the following pointwise estimates:, whereforof degree , and$\kappa=(1+\alpha)/2,$ $\nu=(1-\alpha)/2,$$l\geq 1$

$k$

$t$

$R_{\infty}^{(l)}(t, x)$$Q_{k}(t)$$P_{k}(t)$$R^{(1)}(t, x)$$\delta$$|\partial_{x}^{l}R^{(l)}(t, x)|\leq Ct^{-\frac{1+1}{2}(1}+t)^{-\frac{1}{2}}e^{-\mathrm{c}(x-\alpha t)^{2}/t}+Ce^{-\mathrm{c}(t+|x|)}$,(3.11)$|\partial_{x}^{l}R_{\infty}^{(l)}(t,x)|\leq C(1+t)^{-\frac{l+1}{2}}e^{-c(x-\alpha t)^{2}/t}+Ce^{-\mathrm{c}(t+|x|)}$for$l\geq 0$, where

$C$

and

$c$

are positive constants.

This theorem shows that the fundamental solution $G(t, x)$ can be well approximatedby the modified heat kernel $G_{0}(t, x)$ as.
We have a similar expression also for $H(t, x)$ .
$tarrow\infty$

Theorem 3.2 Assume the same condition as in Theorem 3.1. For each $l\geq 0$ , we canempress $H(t, x)$ as.
$H(t, x)=H_{\infty}^{(l)}(b,x)+S_{\infty}^{(l)}(t, x)$

Here the singular part

$H_{\infty}^{(l)}(t, x)$

is given as(3.12)$\partial_{x}^{l}H_{\infty}^{(l)}(t, x)=\sum_{k=0}^{\iota}\{e^{-\kappa t}\tilde{P}_{k}(t)\partial_{x}^{l-k}\delta(x+i)+e^{-\nu t}\tilde{Q}_{k}(t)\partial_{x}^{l-k}\delta(x-t)\}$, where and are the same as in Theorem 3.1,
andare some
polynomials of of degree , and denotes the Dirac delta function. The remainder term
satisfies the follounng pointwise estimate:for$l\geq 0$$\kappa$$k$$\tilde{Q}_{k}(t)$$\tilde{P}_{k}(t)$

$\mu$

$t$

$\delta$(3.13)$|\partial_{x}^{l}S_{\infty}^{(l)}(t,x)|\leq C(1+t)^{-*}e^{-\mathrm{c}(x-\alpha t)^{2}/t}+Ce^{-\mathrm{c}(t+|x|)}$for$l\geq 0$, where

$C$

and

$c$

are positive constants.

As a corollary of the above pointwise estimates of the fundamental solutions, we havethe following $If-L^{q}$ estimates for solutions to the linearized equation (3.1).
Corollary 3.3 Assume the same condition as in Theorem 3.1 and let $1\leq q\leq p\leq\infty$ .
Then we have the following
estimates:$L^{\mathrm{p}}-L^{q}$,$||G(t)*\phi||_{L^{\mathrm{p}}}\leq C(1+t)^{-\mathrm{z}^{(}}\mathrm{e}^{-\frac{1}{p})}||\phi||_{L\emptyset}11$$||\theta_{x}G(t)*\phi||_{L^{p}}\leq C(1+t)^{-\frac{1}{2}(\frac{1}{q}-\frac{1}{\mathrm{p}})-\frac{l}{2}}||\phi||_{L^{q}}+Ce^{-ct}||\phi||_{W^{1-1,p}}$,$l\geq 1$,$|| \partial_{x}^{l}H(t)*\phi||_{L^{p}}\leq C(1+t)^{-f}(q11-\frac{1}{\mathrm{p}})-l\not\simeq||\phi||_{L^{q}}+Ce^{-\mathrm{c}t}||\phi||_{W^{l,\mathrm{p}}}$,$l\geq 0$.

(3.14)

and(3.15)

Moreover, the solution operator $G(t)*is$ appronimated by $G_{0}(t)*in$ the following sense:$||(G-G_{0})(t)*\phi||_{L^{\mathrm{p}}}\leq Ct^{-\frac{1}{2}(\frac{1}{q}-\frac{1}{p})}(1+t)^{-\frac{1}{2}}||\phi||_{L^{q}}$,$||\theta_{x}(G-G_{0})(t)*\phi||_{L^{p}}\leq Ct^{-\#(_{q}^{1}-\frac{1}{\mathrm{p}})-\#}(1+t)^{-\frac{1}{2}}||\phi||_{L^{q}}+Ce^{-c\mathrm{t}}||\phi||_{W^{l-1,p}}$

Here

$C$

and

$c$

,

are some positive constants.

We can prove Corollary 3.3 by using Theorem 3.1, 3.2, and omit her .
$e$$l\geq 1$.

(3.16)

157

Large time behavior of soluti$\mathrm{o}\mathrm{n}s$to a semilinear hyperboric system with relaxation

Fundamental solution in Fourier space4

in (3.6)
andIn this section, under the condition $|f’(0)|<1$ , we consider
and derive their pointwise estimates, which are crucial in the proof of Theorems 3.1 and
C. We divide our computations
3.2. Here 4 is regarded as a complex variable, i.e.,, the middle frequency
into three parts corresponding to the low frequency region$r_{0}\leq|\xi|\leq
K_{0}$,
respectively.
highfrequency
We omit theregionregionand theproof in this section.
In the low frequency region we have:$\hat{G}(t, \xi)$$\hat{H}(t, \xi)$$\xi\in$$|\xi|\leq r_{0}$$|\xi|\geq K_{0}$Lemma 4.1 There is a positive constant
have the follouring expressions:$r_{0}$

such that for any$\hat{G}(t, \xi)=\hat{G}_{0}(t, \xi)+\hat{R}_{0}(t,\xi)$$\hat{G}_{0}(t, \xi)=e^{-(\alpha i\xi+\mu\xi^{2})t}$,$\xi\in \mathbb{C}$,$\hat{H}(t, \xi)=e^{-(\alphaxi+\mu\xi^{2})t}\hat{H}_{1}(t, \xi)+e^{-t}\hat{H}_{2}(t,\xi)$$\alpha=f’(\mathrm{O}),$$\mu=1-(f’(0))^{2}$ ,,$|\hat{H}_{1}(t, \xi)|\leq C|\xi|e^{C|\xi|^{3}t}$$\leq r_{0}$

, we

(4.1)

.

(4.2)

and

$|\hat{R}_{0,1}(t, \xi)|\leq C|\xi|(1+|\xi|^{2}t)e^{C|\xi|^{3}t}$

for IEI$\leq r_{0}$$\hat{R}_{0}(t,\xi)=e^{-(\alpha i\xi+\mu\xi^{2})t}\hat{R}_{0,1}(t,\xi)+e^{-t}\hat{R}_{0,2}(t,\xi)$

and

Here

with ICI,$|\hat{R}_{0,2}(t, \xi)|\leq Ce^{C|\xi|t}$

,

(4.3)

$|\hat{H}_{2}(t, \xi)|\leq Ce^{C|\xi|t}$, where $C$ is a positive constant.

Remark 4.2 For$\hat{G}(t, \xi)$, we have another $e\varphi ression$ :(4.4)$\hat{G}(t,\xi)=e^{-(\alpha i\xi+\mu\xi^{2})t}\hat{G}_{1}(t,\xi)+e^{-\iota}\hat{G}_{2}(t, \xi)$with $\hat{G}_{1}(t,\xi)=1+\hat{R}_{0,1}(t,\xi)$ and$\hat{G}_{2}(t,\xi)=\hat{R}_{0,2}(t,\xi)$$|\hat{G}_{1}(t,\xi)|\leq Ce^{C|\xi|^{3}t}$,satisfying$|\hat{G}_{2}(t,\xi)|\leq Ce^{C|\xi|t}$

.

(4.5)

Next we consider in the high frequency region.

Lemma 4.3 For each nonnegative integer , there is a positive constant, we have the following expressions:
any
with
$l$

$\xi\in \mathbb{C}$$|\xi|\geq K_{0}$$\hat{G}(t, \xi)=\hat{G}_{\infty}^{(l)}(t,\xi)+\hat{R}_{\infty}^{(l)}(t,\xi)$Heresuch that for$K_{0}$$\hat{G}_{\infty}^{(0)}(t, \xi)\equiv 0$,$\hat{H}(t,\xi)=\hat{H}_{\infty}^{(l)}(t,\xi)+\hat{S}_{\infty}^{(l)}(t,\xi)$

.

(4.6)

,
,

$\hat{G}_{\infty}^{(l)}(t,\xi)=\sum_{k=0}^{l-1}\{e^{-(\kappa-*)t}P_{\mathrm{k}}(t)+e^{-(\nu+:\xi\rangle t}Q_{k}(t)\}(i\xi)^{-k-1}$$l\geq 1$,(4.7)$\hat{R}_{\infty}^{(l)}(t,\xi)=\{e^{-(\kappa-1\xi)t}P_{l}(t)+e^{-(\nu+*\xi)t}Q_{\iota}(t)\}’(i\xi)^{-l-1}$$+e^{-(\kappa-\mathrm{t}\xi)t}\hat{R}_{\infty,1}^{(l)}(t,\xi)+e^{-(\nu+i\xi)t}\hat{R}_{\infty,2}^{(l)}(t,\xi)$,$l\geq 0$,

158

Large time behavior of solu tions to a semilinear hyperboric system with relaxation
and,$l\geq 0$$\hat{H}_{\infty}^{(l)}(t, \xi)=\sum_{k=0}^{\iota}\{e^{-(\kappa-i\xi)t}\tilde{P}_{k}(t)+e^{-(\nu+i\xi)t}\overline{Q}_{k}(t)\}(i\xi)^{-k}$

,
(4.8)

$\hat{S}_{\infty}^{(l)}(t,\xi)=\{e^{-(\kappa-i\xi)t}\tilde{P}_{l}(t)+e^{-(\nu+i\xi)t}\tilde{Q}_{l}(t)\}(i\xi)^{-l-1}$$+e^{-(\kappa-t)t}\hat{S}_{\infty,1}^{(l)}(t,\xi)+e^{-(\nu+:\xi)t}\hat{S}_{\infty,2}^{(l)}(t,\xi)$

where $\kappa=(1+\alpha)/2,$ $\nu=(1-\alpha)/2$ with $\alpha=f’(0)$ , andpolynomials of of degree . Moreover, we have$P_{k}(t),$,$l\geq 0$$Q_{k}(t),\tilde{P}_{k}(t)$,and$\tilde{Q}_{k}(t)$

are

$k$

$t$

$|\hat{R}_{\infty,1}^{(l)}(t,\xi)|+|\hat{R}_{\infty,2}^{(l)}(t,\xi)|\leq C|\xi|^{-\iota-2}(1+t)^{l+1}e^{C|\xi|^{-1}t}$

,

(4.9)

$|\hat{S}_{\infty,1}^{(l)}(t,\xi)|+|\hat{S}_{\infty,2}^{(l)}(t,\xi)|\leq C|\xi|^{-l-2}(1+t)^{\iota+2}e^{C|\xi|^{-1}t}$for$|\xi|\geq K_{0}$, where$C$

is a positive constant.

In the middle frequency region, as in [2], we derive the corresponding estimates by
employing the energy method in the Fourier space.

. Then, for any $r>0$ , there etzsts a
Lemma 4.4 We Utte $\xi=\eta+i\zeta$ , where$if|\eta|\geq
r$ anddepending on such that
positive constant, then we havethe following estimates:$\eta,$$\sigma(r)$$\zeta\in \mathbb{R}$$|\hat{G}(t,\xi)|\leq C(1+|\eta|)^{-1}e^{-c\rho(\eta)t}$

where

5

$\rho(\eta)=\frac{\eta^{2}}{1+\eta^{2}}$, and$|\zeta|\leq\sigma(r)$

$r$

$C$

and

$c$

,

$|\hat{H}(t,\xi)|\leq Ce^{-\mathrm{c}\rho(\eta)t}$(4.10),

are positive constants independent of .
$r$

Proof of pointwise estimates

In this section, following $[4,2]$ , we give the proof of Theorems 3.1 and 3.2 concerning thepointwise estimates of the fundamental solutions.

Proof of Theorem 3.1. For each nonnegative integer , we express$l$$\hat{G}(t,\xi)=\hat{G}_{0}(t,\xi)+\hat{G}_{\infty}^{(l)}(t,\xi)+\hat{R}^{(l)}(t,\xi)$$\hat{G}$in (3.6) as

,

(5.1)

where
and
are given explicitly in (4.1) and (4.7), respectively, andby (5.1). We write the Fourier inverse transform of (5.1) as$\hat{G}_{0}$$\hat{G}_{\infty}^{(l)}$$G(t,x)=G_{0}(t,x)+G_{\infty}^{(l)}(t,x)+R^{(\mathrm{t})}(t,x)$.

$\hat{R}^{(l)}$is defined(5.2)

Here the first two terms on the right hand side of (5.2) can be given explicitly. In thisproof, we consider the derivative
andof the remainder terms.
$\theta_{x}R^{(l)}(t, x)$$\partial_{x}^{l}R_{\infty}^{(l)}(t, x)$

Lemma 5.1 For each $l\geq 0$ , we have the following estimate:$|\theta_{x}R^{(1)}(t,x)|\leq C(1+t)^{-*}e^{-\mathrm{c}(x-\alpha t)^{2}/\iota}+Ce^{-c(t+|x|)}$for$t\geq 1$, where$C$

and$\mathrm{c}$are positive constants.

(5.3)

159

Large time behavior of solu tions to a semilinear hyperboric system with relaxation

Proof. We have$\theta_{x}R^{(1)}(t,x)=\mathcal{F}^{-1}[(i\xi)^{\iota}\hat{R}^{(l)}(t, \cdot)](x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}(i\xi)^{\iota}\hat{R}^{(l)}(t,\xi)e^{i\xi x}d\xi$$(\xi=\eta+i\zeta)$$= \frac{1}{2\pi}\int_{-\infty}^{\infty}(i\xi)^{l}\hat{R}^{(l)}(t,\xi)e^{1\xi x}d\eta$,where, thanks to the Cauchy integral theorem, we have changed the path of integrationfrom the real axis to the straight line $\xi=\eta+i\zeta$ (with a small fixed specified later) whichis parallel to the real axis. We divide the above integral into three parts corresponding$r\leq|\eta|\leq K$ and $|\eta|\geq K$ , respectively, where $r>0$ and $K>0$ areto the regions
constants which will be specified later. Now we recall the relations$\zeta$$|\eta|\leq r,$$\hat{R}^{(l\rangle}=\hat{R}_{0}-\hat{G}_{\infty}^{(l)}$,$\hat{R}^{(l)}=\hat{G}-\hat{G}_{0}-\hat{G}_{\infty}^{(l)}$,$\hat{R}^{(l)}=\hat{R}_{\infty}^{(l)}-\hat{G}_{0}$,which follow from (4.1), (4.6) and (5.1). We then substitute these three relations into$r\leq|\eta|\leq K$ , and $|\eta|\geq K$ , respectively.
the above integral over the regions
Consequently, we obtain$|\eta|\leq r,$$2\pi d_{x}R^{(l)}(t,x)$$= \int_{|\eta|\leq r}(i\xi)^{\iota}\hat{R}_{0}e^{i\xi x}d\eta+\int_{\mathrm{r}\leq|\eta|\leq K}(i\xi)^{\iota}\hat{G}e^{1\xi x}d\eta$(5.4)$+ \int_{|\eta|\geq\kappa}(i\xi)^{l}\hat{R}_{\infty}^{(l)}e^{\dot{\iota}\xi x}d\eta-\int_{|\eta|\leq K}(i\xi)^{l}\hat{G}_{\infty}^{(l)}e^{\mathrm{g}_{x}}d\eta-\int_{|\eta|\geq r}(i\xi)^{l}\hat{G}_{0}e^{1\xi x}d\eta$$=:I_{1}+I_{2}+I_{3}-I_{4}-I_{5}$where $\xi=\eta+i\zeta$ . We choose$\zeta$.

according to the point $(t, x)$ as follows:$\zeta=\delta(x-\alpha t)/t$$\zeta=\delta$$\zeta=-\delta$

if

if$|x-\alpha t|/t\leq 1$$|x-\alpha t|/t\geq 1$

if$|x-\alpha t|/t\geq 1$,

and $x- at>0$ ,and $x- at<0$ ,(5.5)

is a small constant which will be specified later. Note that in any case we have
. For the moment, we assume that and are so small that,are the constants
andandwhile $K$ is so large that $K\geq K_{0}$ , wherein Lemmas 4.1, 4.3 and 4.4, respectively.
Case 1. Consider the case where $|x-\alpha t|/t\leq 1$ . In this case we take $\zeta=\delta(x-at)/t$ by(5.5) so that $\xi=\eta+i\delta(x-\alpha t)/t$ .
First, we rewrite the term by using (4.1) as

where$\delta>0$$|\xi|^{2}\leq|\eta|^{2}+\delta^{2}$$r$$\delta\leq\sigma(r)$$r^{2}+\delta^{2}\leq r_{0}^{2}$$\delta$$r_{0},$$K_{0}$$\sigma(r)$$I_{1}$$I_{1}= \int_{|\eta|\leq\prime}e^{-\mu\xi^{2}}{}^{t}(i\xi)^{l}\hat{R}_{0,1}e^{*\xi(x-\alpha t)}.d\eta+e^{-t}\int_{|\eta|\leq t}(i\xi)^{l}\hat{R}_{0,2}e^{i\xi x}d\eta=:I_{1,1}+I_{1,2}$.

and then use the simple reWe substitute the first pointwise estimate in (4.3) into
and ${\rm Re}(i\xi(x-\alpha t))=-\delta(x-\alpha t)^{2}/t$ . This
$I_{1,1}$$1\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}-{\rm Re}(\mu\xi^{2}t)=-\mu\eta^{2}t+\mu\delta^{2}(x-\alpha t)^{2}/t$

gives

,

$|I_{1,1}| \leq C\int_{|\eta|\leq r}|e^{-\mu\xi^{2}t}||\xi|^{l+1}(1+|\xi|^{2}t)e^{C|\xi|^{3}t}|e^{*(x-\alpha t)}|d\eta\leq C(1+t)^{-\frac{l+2}{2}}e^{-\gamma(x-\alpha t)^{2}/t}$160

Large time behavior of solu tions to a semilinear hyperboric system With relaxa on$t\mathrm{i}$.
provided that and are suitably small, where is a positive constant such that
Similarly, using the second pointwise estimate in (4.3) and the relation ${\rm Re}(i\xi x)\leq-\delta(x-$$at)^{2}/t+|a|\delta t$ , we have$r$$\delta$$\gamma<\delta$$\gamma$$|I_{1,2}| \leq Ce^{-t}\int_{|\eta|\leq r}|\xi|^{\iota}e^{C|\xi|t}|e^{i\xi x}|d\eta\leq Ce^{-\mathrm{c}t}e^{-\delta(x-\alpha t)^{2}/t}$provided that

we have us$e\mathrm{d}$,and are suitably small, where is a positive constant with $c<1$ . Here
. Thus we havethe inequality$r$$\delta$$c$$|\xi|^{2}\leq|\eta|^{2}+\delta^{2}$$|I_{1}|\leq C(1+t)^{-^{l}A_{2}l}e^{-\gamma(x-\alpha t)^{2}/\mathrm{t}}$

.

(5.6)

Next we estimate . When $r\leq|\eta|\leq K$ , we have from (4.10) that,provided that is suitably small and $K$ is suiatbly large, where is a positive constant
independent of and $K$ . Therefore, noting that ${\rm Re}(i\xi x)\leq-\delta(x-\alpha t)^{2}/t+|a|\delta t$ , weobtain,(5.7)$|\hat{G}|\leq Ce^{-c\mathrm{o}r^{2}t}$$I_{2}$$r$$c_{0}$$r$$|I_{2}| \leq\int_{r\leq|\eta|\leq K}|\xi|^{\mathrm{t}}|\hat{G}||e^{i\xi x}|d\eta\leq Ce^{-a}e^{-\delta(x-\alpha t)^{2}/\iota}$

provided that$c<c_{0}r^{2}$

For.

$I_{3}$$\delta$

is suitably small depending on , where$r$, we use the expression of$\hat{R}_{\infty}^{(l)}$in (4.7) and write$c$$I_{3}$

is a positive constant withas$I_{3}=e^{-\kappa t} \int_{|\eta|\geq K}\{P_{l}(t)(i\xi)^{-1}+(i\xi)^{l}\hat{R}_{\infty,1}^{(l)}\}e^{i\xi(x+t)}d\eta$$+e^{-\nu t} \int_{|\eta|\geq K}\{Q_{l}(t)(i\xi)^{-1}+(i\xi)^{l}\hat{R}_{\infty,2}^{(l)}\}e^{1\xi(x-t)}d\eta=:I_{3}^{+}+I_{3}^{-}$

Moreover, we rewrite.

$I_{3}^{+}\mathrm{a}\mathrm{e}$$I_{3}^{+}=e^{-\kappa t}P_{l}(t) \int_{|\eta 1\geq K}(i\eta)^{-1}e^{i\xi(x+t)}d\eta+e^{-\kappa t}P_{l}(t)\int_{|\eta|\geq K}((i\xi)^{-1}-(i\eta)^{-1})e^{1\xi(x+t)}d\eta$$+e^{-\kappa t} \int_{|\eta|\geq K}(i\xi)^{l}\hat{R}_{\infty,1}^{(l)}e^{i\xi(x+t)}d\eta=:I_{3,1}^{+}+I_{3,2}^{+}+I_{3,3}^{+}$

We estimate each term as follows. For$I_{3,1}^{+}$.

, we see that

$I_{3,1}^{+}=e^{-\kappa t}P_{l}(t)e^{-\delta(x-\alpha t)(x+\mathrm{t})/t} \int_{|\eta|\geq K}(i\eta)^{-1}e^{i\eta(x+t)}d\eta$because $i\xi(x+t)=-\delta(x-\alpha t)(x+t)/t+i\eta(x+t)$ . Here we observe thatwith $c_{1}=1+\alpha$ , and that$e^{-\delta(x-\alpha t)(x+t)/t}\leq$$e^{-\delta(x-\alpha t)^{2}/t}e^{c_{1}\delta t}$,$\int_{|\eta|\geq K}(i\eta)^{-1}e^{1\eta(x+t)}d\eta=2\int_{K}^{\infty}\frac{\sin\eta(x+t)}{\eta}d\eta=2\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(x+t)\int_{|x+t\{K}^{\infty}\frac{\sin y}{y}dy$which is uniformly bounded. Consequently, we obtain$|I_{3,1}^{+}|\leq C(1+t)^{l}e^{-\kappa t}e^{-\delta(x-\alpha t)^{2}/t}e^{c_{1}\delta t}\leq Ce^{-ct}e^{-\delta(x-\alpha t\rangle^{2}/t}$

,

161

Large time behavior of solu tion to a semilinear hyperboric system with relaxation$s$

provided that
we haveis suitably small, where is a positive constant with $c<\kappa$ . Also, for$\delta$$(i \xi)^{-1}-(i\eta)^{-1}=\frac{1}{i\eta-\delta(x-\alpha t)/t}-\frac{1}{i\eta}=\frac{\delta(x-\alpha t)/t}{i\eta(i\eta-\delta(x-\alpha t)/t)}=O(|\eta|^{-2})$

and$I_{3,2}^{+}$$c$$|e^{i\xi(x+t)}|\leq e^{-\delta(x-\alpha t)^{2}/t}e^{c_{1}\delta t}$,,with $c_{1}=1+a$ . Hence we obtain$|I_{3,2}^{+}| \leq C(1+t)^{l}e^{-nt}\int_{|\eta|\geq K}|(i\xi)^{-1}-(i\eta)^{-1}||e^{1\xi(x+t)}|d\eta\leq Ce^{-\mathrm{c}t}e^{-\delta(x-\alpha t)^{2}/t}$

for suitably small , where $0<c<\kappa$ . Similarly, making use of the pointwise estimate ofin (4.9), we have$\delta$$\hat{R}_{\infty,1}^{(l)}$,$|I_{3,3}^{+}| \leq C(1+t)^{l+1}e^{-\kappa t}\int_{|\eta|\geq K}|\xi|^{-2}e^{C|\xi|^{-1}t}|e^{1\xi(x+t)}|d\eta\leq Ce^{-\mathrm{c}t}e^{-\delta(x-\alpha t)^{2}/t}$provided that is suitably small and $K$ is suitably large, where $0<c<\kappa$ . Summarizing all. Another term can be estimatedthese computations, we havejust in the same way. Thus we arrive at the estimate$\delta$$|I_{3}^{+}|\leq Ce^{-ct}e^{-\mathit{5}(x-\alpha t)^{2}/t}$$|I_{3}|\leq Ce^{-\mathrm{c}t}e^{-\delta(x-\alpha t)^{2}/t}$

The fourth term

$I_{4}$$I_{3}^{-}$

.

(5.8)

can be treated more easily. We have from (4.7) that

$I_{4}= \sum_{k=0}^{l-1}\{e^{-\kappa \mathrm{t}}P_{k}(t)\int_{|\eta|\leq K}(i\xi)^{\mathrm{t}-k-1}e^{1\epsilon(x+t)}d\eta+e^{-\nu t}Q_{k}(t)\int_{|\eta|\leq K}(i\xi)^{l-k-1}e^{l\xi(x-t)}d\eta\}$

Here we note thatletting$|e^{i\xi(x\pm t)}|\leq e^{-\delta(x-\alpha t)^{2}/\iota_{e^{c_{1}\delta t}}}$$\kappa_{1}=\min\{\kappa, \nu\}$, we have.

with $c_{1}= \max\{1+\alpha, 1-\alpha\}$ . Therefore,$|I_{4}| \leq C(1+t)^{l-1}e^{-\kappa_{1}t}\int_{|\eta|\leq K}(1+|\xi|)^{l}(|e^{j\xi(x+t)}|+|e^{\mathrm{g}(x-t)}|)d\eta\leq Ce^{-ct}e^{-\delta(x-\alpha t)^{2}/t}$.

for suitably small 6, where $0<c<\kappa_{1}$ .
Finally, we estimate the term which is rewritten by using the expression of(4.1) as$I_{5}$$I_{5}= \int_{|\eta|\geq\prime}(i\xi)^{l}e^{-\mu\xi^{2}}{}^{t}e^{i\xi(x-\alpha t)}d\eta$$(5.9)$$\hat{G}_{0}$in.

We have$|I_{5}| \leq\int_{|\eta|\geq \mathrm{r}}|\xi|^{l}|e^{-\mu\xi^{2}t}||e^{1\xi(x-\alpha t)}|d\eta\leq Ct^{-\frac{l+1}{2}}e^{-ct}e^{-\gamma(x-\alpha t)^{2}/t}$(5.10),

provided that is suitably small, where $0<\gamma<\delta$ .
All these computations from (5.6) to (5.10) prove the desired estimate (5.3) for $|x-$$\delta$$\alpha t|/t\leq 1$

.

where
Case 2. Next we consider the case where $|x-\alpha t|/t\geq 1$ and $x-\alpha t>0$ . (The$|x-\alpha t|/t\geq 1$ and $x- at<0$ can be treated just in the same way and we omit this final
case.) In this case we take.
by (5.5) so that$c\mathrm{a}s\mathrm{e}$$\zeta=\delta$$\xi=\eta+i\delta$

162

Large time behavior of$sol\mathrm{u}t\mathrm{i}on\mathrm{s}$to a semilin ear hyperboric system with relaxa tion

. Also,.

$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{w}\mathrm{e}\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}I_{1,1},\mathrm{w}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}{\rm Re}(\xi^{2})=\eta^{2}-\mathit{6}^{2}\mathrm{a}\mathrm{n}\mathrm{d}{\rm Re}(i\xi(x-\alpha t))=-\delta|x-\alpha t|$

we$\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}e^{-\delta|x-\alpha t|}\leq e^{-\delta t/2}e^{-\delta|x-\alpha t|/2}\mathrm{b}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{o}\mathrm{f}|x-\alpha t|\geq t$,$|I_{1,1}| \leq C\int_{|\eta|\leq\tau}|e^{-\mu\xi^{2}t}||\xi|^{l+1}(1+|\xi|^{2}t)e^{C|\xi|^{8}t}|e^{1\xi(x-\alpha t)}|d\eta\leq Ce^{-\gamma_{1}t}e^{-\delta|x-\alpha t|/2}\leq Ce^{-\gamma(t+|x|)}$

provided that and are suitably small, where
noting that ${\rm Re}(i\xi x)=-\delta x\leq-\delta|x-\alpha t|+|\alpha|\delta t$, we have$\delta$$0<\gamma<\gamma_{1}<\mathit{6}/2.\mathrm{F}\mathrm{o}\mathrm{r}$$r$

the term$I_{1,2}$,,$|I_{1,2}| \leq Ce^{-t}\int_{|\eta|\leq r}|\xi|^{l}e^{C|\xi|t}|e^{1\xi x}|d\eta\leq Ce^{-\mathrm{c}t}e^{-\delta|x-\alpha t|}\leq Ce^{-\gamma(t+|x|)}$

provided that
we have$\delta$andare suitably small, where $0<\mathrm{c}<1$ and$r$$|I_{1}|\leq Ce^{-\gamma(t+|x|)}$.

$I_{4}$$I_{3}$. Thus(5.11)

Similarly, for the termand , we can replace the factor
and obtainand (5.9) by$I_{2},$$0< \gamma<\min\{c,\mathit{6}\}$$\delta|x-\alpha t|^{2}/t$in (5.7), (5.8)$\delta|x-\alpha t|$$|I_{2}|,$$|I_{3}|,$$|I_{4}|\leq Ce^{-\mathrm{c}t}e^{-\delta|x-\alpha t|}\leq Ce^{-\gamma(t+|x|)}$,(5.12)

provided that and are suitably small and $K$ is suitably large, where
positive constant and $0< \gamma<\min\{c, \delta\}$ . Also, for the term , we have$\delta$$r$$c$

is a certain$I_{5}$$|I_{5}| \leq\int_{|\eta|\geq\prime}|\xi|^{\iota}|e^{-\mu\xi^{2}t}||e^{i\xi(x-\alpha t)}|d\eta\leq Ct^{-\frac{l\neq 1}{2}}e^{-\gamma_{1}t}e^{-\delta|x-\alpha t|/2}\leq Ct^{-\frac{l\neq 1}{2}}e^{-\gamma(t+|x|)}$,(5.13)

provided that is suitably small, where $0<\gamma<\gamma_{1}<\delta/2$ . All these observations show
the desired estimate (5.3) for $|x-\alpha t|/t\geq 1$ and hence the proof of Lemma 5.1 is complete.
$\delta$$\square$given in Lemma 5.1 contains the additional
The pointwise estimate ofin (5.13)). For the proof of Theorem 3.1 we must
singularity at $t=0$ (see the term
remove this singularity. To this end, we recall (4.6) and write$\theta_{x}R^{(l)}(t,x)$$I_{5}$(5.14)$\hat{G}(t,\xi)=\hat{G}_{\infty}^{(l)}(t,\xi)+\hat{R}_{\infty}^{(l)}(t,\xi)$for each $l\geq 0$ , wheretransform of (5.14) as$\hat{G}_{\infty}^{(l)}(t,\xi)$

is given explicitly in (4.7). We write the Fourier inverse$G(t,x)=G_{\infty}^{(l)}(t,x)+R_{\infty}^{(l)}(t,x)$

,

where
was given explicitly. We show that the remainder term
the pointwise estimate given in (3.11):$\partial_{x}^{l}G_{\infty}^{(l)}(t,x)$(5.15)$R_{\infty}^{(l)}(t, x)$satisfies

Lemma 5.2 For each $l\geq 0$ , we have the following pointunse estimate:$|d_{x}R_{\infty}^{(l)}(t, x)|\leq C(1+t)^{-\frac{l+1}{2}}e^{-\mathrm{c}(x-\alpha t)^{2}/\iota}+Ce^{-\mathrm{c}(t+|x|)}$for any $t>0$ , where

$C$

and

$c$

are positive constants.

(5.16)

163

Large time behavior of$sol\mathrm{u}$tions to a emilinear hyperboric system with relaxation$s$

Proof. We have as the counterpart of (5.4) that$2 \pi\partial_{x}^{l}R_{\infty}^{(l)}(t, x)=\int_{-\infty}^{\infty}(i\xi)^{\iota}\hat{R}_{\infty}^{(l)}(t,\xi)d\eta$$= \int_{|\eta|\leq r}(i\xi)^{l}\hat{G}e^{*\xi x}.d\eta+\int_{r\leq|\eta|\leq K}(i\xi)^{l}\hat{G}e^{1\zeta x}d\eta$(5.17)$+ \int_{|\eta|\geq K}(i\xi)^{l}\hat{R}_{\infty}^{(l)}e^{\dot{*}\xi x}d\eta-\int_{|\eta 1\leq K}(i\xi)l\hat{G}_{\infty}(l)edt\epsilon x\eta$$=:J_{1}+\sqrt 2+J_{\mathrm{a}}+J_{4}$,in the regions
where $\xi=\eta+i\zeta$ . Here we have used the relation$r\leq|\eta|\leq
K$(5.4). In the present
incompareitwith,. To estimate the term weandcase, it suffices to use the expression (4.4) of instead of the expression (4.1) of . Thissuggests that all the estimates for in the proof of Lemma 5.1 are valid also for if weby . In particular, as thereplace the exponent $l+1$ appearing in the estimates forcounterpart of (5.6), we have$\hat{R}_{\infty}^{(l)}=\hat{G}-\hat{G}_{\infty}^{(l)}$$|\eta|\leq r$$I_{1}$$J_{1}$$\hat{R}_{0}$$\hat{G}$$J_{1}$$I_{1}$$l$$I_{1}$$|I_{1}|\leq C(1+t)^{\frac{l+1}{2}}e^{-\gamma(x-\alpha t)^{2}/t}$

for $|x-\alpha t|/t\leq 1$ . The other terms in (5.17) are just the same as those in (5.4), namely,. (Here we do not have any term like having the
we have $J_{2}=I_{2},$ $J_{3}=I_{3}$ andadditional singularity at $t=0.$ ) These observations give the desired estimate (5.16). This
complet the proof of Lemma 5.2.
Now, in order to complete the proof of Theorem 3.1, we show the estimate (3.11) for. Namely, for each $l\geq 0$ , we show that
$J_{4}=I_{4}$$e$$I_{5}$$\square$$\partial_{x}^{l}R^{(\mathrm{t})}(t, x)$$|\theta_{x}R^{(l)}(t, x)|\leq Ct^{-\frac{l+1}{2}(1}+t)^{-\frac{1}{2}}e^{-c(x-\alpha t)^{2}/\iota}+Ce^{-\mathrm{c}(t+|x|)}$(5.18)and estimate the rightfor any $t>0$ . To see this, we recall the relation
hand side of this equality. For the first term, we apply the estimate (5.16). For the second. Thusterm, by a straightforward computation, we have
we obtain$R^{(1)}=R_{\infty}^{(l)}-G_{0}$$|\partial_{x}^{l}G_{0}(t, x)|\leq o_{t^{-\#}}e^{-\mathrm{c}(x-\alpha t)^{2}/t}$$|d_{x}R^{(l)}(t,x)|\leq Ct^{-\frac{l+1}{2}}e^{-(x-\alpha t)^{2}/t}+Ce^{-\mathrm{c}(t+|x|)}$.

(5.19)

A combination of the estimates (5.3) for $t\geq 1$ and (5.19) for $0<t\leq 1$ yields the desired
estimate (5.18). This completes the proof of Theorem 3.1.
The proof of Theorem 3.2 is similar to that of Lemma 5.2 and omitted here.
$\square$6

Global existence and decay

In this section we study the initial value problem (1.2), (1.3) and prove the global existence
result stated in Theorem 2.1. First, we rewrite the equation (1.2) as$u_{u}-u_{xx}+u_{t}+\alpha u_{x}=-g(u)_{x}$

,

(6.1)

164

Large time behavior of solutions to a $sem$ ilinear hyperboric system With relaxation

where $\alpha=f’(0)$ and $g(u):=f(u)-f(\mathrm{O})-f’(\mathrm{O})u=O(u^{2})$ . Then, applying the Duhamel
principle, we transform the problem (1.2) (or (6.1)), (1.3) into the integral equation$u(b)=G(t)*(u_{0}+u_{1})+H(t)*u_{0}- \int_{0}^{t}G(t-s)*g(u)_{x}(s)ds$ ,(6.2)

where $G(t, x)$ and $H(t, x)$ are the fundamental solutions to the linearized equation (3.1)
and are defined in (3.8).
We want to solve the above integral equation by applying the contraction mappingbyprinciple. For this purpose, we define the mapping$\Phi[u]$$\Phi[u](t):=G(t)*(u_{0}+u_{1})+H(t)*u_{0}-\int_{0}^{t}G(t-s)*g(u)_{x}(s)ds$

and put$\Phi_{0}(t):=G(t)*(u_{0}+u_{1})+H(t)*u_{0}$

.

(6.3)

(6.4)

Let us consider in the Banach space $X$ defined as follows: For $1\leq p<\infty$ ,$X:=\{u\in C([0, \infty);W^{1,\mathrm{p}}\cap L^{1});||u||_{X}<\infty\}$,$||u|| \mathrm{x}:=\sup_{t\geq 0}||u(t)||_{L^{1}}+\sup_{t\geq 0}(1+t)^{\frac{1}{2}(1-\frac{1}{p})+_{f}^{1}}||\partial_{x}u(t)||_{L^{\mathrm{p}}}$

(6.5)

.

and for $p=\infty$ ,$X:=\{u\in C([0, \infty);L^{1})\cap L^{\infty}((0, \infty);W^{1,\infty});||u||_{X}\leq\infty\}$$||u||x:= \sup_{\iota\geq 0}||u(t)||_{L^{1}}+\sup_{\iota\geq 0}(1+t)||\partial_{x}u(t)||_{L\infty}$

,
(6.6)

.

It is also useful to introduce$||u||_{Y}:= \sup_{t\geq 0}||u(t)||_{L^{1}}+\sup_{\iota\geq 0}(1+t)I||u(t)1$

I .
$\iota\infty$(6.7)

Notice that
(6.8)$||u(t)||_{L^{q}}\leq||u||_{Y}(1+t)^{-\frac{1}{2}(1-\frac{1}{q})}$for each with $1\leq q\leq\infty$ , which follows from the inequality. Also, we see that $||u||_{Y}\leq C_{*}||u||_{X}$ , where $C_{*}\geq 1$ is theand the definition of
constant appearing in the Gagliardo-Nirenberg inequality
with$||u||_{L^{q}}\leq||u||_{L\infty}^{1-1/q}||u||_{L^{1}}^{1/q}$$q$$||u||_{Y}$$||u_{\mathrm{I}}^{1}|_{\iota\infty}\leq C_{*}||\partial_{x}u||_{L^{p}}^{\theta}||u||_{L^{1}}^{1-\theta}$$\theta=1/(2-1/p)$.

Let us introduce a closed convex subset$S_{R}$

of $X$ by$S_{R}:=\{u\in X;||u||_{X}\leq R\}$

,

(6.9)

where $R>0$ is a parameter which will be determined later. We wish to show that for asuitably chosen $R,$ becomes a contraction mapping of . To this end, we prepare thefollowing:$\Phi$$S_{R}$

165

Large time behavior of solutions to a semilinear hyperboric system with relaxa tion

Lemma 6.1 (i) Let
we have$1\leq p\leq\infty$

and assume that$u_{0}\in W^{1,p}\cap L^{1}$and$u_{1}\in L^{p}\cap L^{1}$(6.10)$||\Phi_{0}||_{X}\leq C_{0}E_{0}$for some positive constant, whereis given in Theorem 2.1.
(ii) LetFor any given positive number $M$ , we suppose that$M$
for $t\geq 0$ . Then we have$C_{0}$. Then$E_{0}$$u,$ $v\in X.$$||\Phi[u]-\Phi[v]||_{X}\leq C_{1}(||u||_{X}+||v||_{X})||u-v||_{X}$$||u(t)||_{L}\infty,$,$||v(t)||_{L\infty}\leq$(6.11)

where $C_{1}=C_{1}(M)$ is a positive constant depending on $M$ .

Proof. We obtain the proof of (i) by using Corollary 3.3, and omit here. Let us show (ii).
It follows from (6.3) that.

(6.12)$||\partial_{x}(g(u)-g(v))||_{L^{\mathrm{p}}}\leq C\{(||u||\iota\infty+||v||\iota\infty)||\partial_{x}(u-v)||_{L^{p}}$(6.13)$\Phi[u](t)-\Phi[v](t)=-\int_{0}^{t}\partial_{x}G(t-s)*(g(u)-g(v))(s)ds$

Here we claim that$||g(u)-g(v)||_{L^{q}}\leq C(||u||_{L\infty}+||v||_{L\infty})||u-v||_{L^{q}}$,$+(||\partial_{x}u||_{L^{p}}+||\partial_{x}v||_{L^{p}})||u-v||\iota\infty\}$,$||v||_{L}\infty\leq M$ , where$q\leq\infty$ , and $C=C(M)$ denotes aprovided that$M$
constant depending on . This follows from the fact that $g(u)=O(u^{2})$ and hence$g(u)-g(v)=a(u, v)(u-v)$ with a function $a(u, v)=O(|u|+|v|)$ . Consequently, we havein terms ofandthat$||u||_{L}\infty,$$||\cdot||_{X}$$1\leq \mathrm{p},$$||\cdot||_{\mathrm{Y}}$$||(g(u)-g(v))(t)||_{L^{q}}\leq C(||u||_{Y}+||v||_{\mathrm{Y}})||u-v||_{\mathrm{Y}}(1+t)^{-\mathrm{B}\mathrm{r}2}1(1-1)-1$,$||\partial_{x}(g(u)-g(v))(t)||_{L^{p}}\leq C(||u||_{X}+||v||_{X})||u-v||_{X}(1+t)^{-\tau^{(1-1_{)-1}}}1p$,(6.14)

where $C=C(M)$ . Now, we take thenorm of (6.12) and apply (3.14) with $l=1$ ,$p=q=1$ . Then, using the first estimate in (6.14), we have$L^{1}$$||( \Phi[u]-\Phi[v])(t)||_{L^{1}}\leq\int_{0}^{t}||\partial_{x}G(t-s)*(g(u)-g(v))(s)||_{L^{1}}ds$(6.15),$\leq C(M)|[u, v]|_{Y}\int_{0}^{t}(1+t-s)^{-\frac{1}{2}}(1+s)^{-1}2ds\leq C(M)|[u,v]|_{\mathrm{Y}}$where we wrote $|[u, v]|_{Y}:=(||u||_{\mathrm{Y}}+||v||_{Y})||u-v||_{Y}$ . Next, we want to estimate thederivative of (6.12). To this end, we decompose the integral on the right hand side of (6.12)
into two pars and write $\Phi[u]-\Phi[v]=\Psi_{1}+\Psi_{2}$ , whereandare corresponding to the
$[0,
t/2]$
$[t/2,t]$, we apply (3.14) with, respectively. For the term
integrations overand$\Psi_{1}$$\Psi_{2}$$\partial_{x}\Psi_{1}$

166

Large

$ti\mathrm{m}e$behavior of solutions to a semilinear hyperboric system with relaxation$l=2,$ $q=1$ and then make use of (6.14). Then, writing $|[u, v]|_{X}:=(||u||_{X}+||v||_{X})||u-v||_{X}$ ,we obtain$|| \partial_{x}\Psi_{1}(t)||_{L^{\mathrm{p}}}\leq\int_{0}^{t/2}||\partial_{x}^{2}G(t-s)*(g(u)-g(v))(s)||_{L^{p}}ds$,$\leq C(M)|[u, v]|_{X}(1+t)-\mathrm{i}(1-p21_{)-}1$

Similarly, for the termyields$\partial_{x}I_{2}$, we apply (3.14) with $l=1,$ $q=p$ and then use (6.14). This$|| \partial_{x}\Psi_{2}(t)||_{L^{\mathrm{p}}}\leq\int_{t/2}^{t}||\partial_{x}G(t-s)*\partial_{x}(g(\mathrm{u})-g(v))(s)||_{L^{\mathrm{p}}}ds$$\leq C(M)|[u, v]|_{X}(1+t)^{-I\mathrm{p}}1(1-\perp)-\pi 1$.

Thus we have shown that$||\partial_{x}(\Phi[u]-\Phi[v])(t)||_{L^{p}}\leq C(M)|[u, v]|_{X}(1+t)^{-\frac{1}{2}(1-\frac{1}{p})-\pi}1$.

(6.16)The desired estimate (6.11) follows from (6.15) and (6.16), and hence the proof of Lemma
6.1 is complete.
Proof of Theorem 2.1. We determine the parameter by $R:=2C_{0}E_{0}$ , whereisthe positive constant in (6.10). For this choice of , we suppose that $v\in S_{R}$ . Then,$||u||_{X}\leq C_{*}R$ (the same for), where
we have $||u||_{X}\leq R$ and hence
is the constant appeared in the previous Gagliardo-Nirenberg inequality. Therefore, wehave from (6.11) that$\square$$R$$C_{0}$$R$$u,$$||u||_{\mathrm{Y}}\leq C.$$C_{*}\geq 1$$v$,$||\Phi[\mathrm{u}]-\Phi[v]||_{X}\leq C_{1}(||u||_{X}+||v||_{X})||u-v||_{X}\leq 2C_{1}R||u-v||_{X}=4C_{0}C_{2}(E_{0})E_{0}||u-v||_{X}$where the constant $C_{1}=C_{1}(M)$ in (6.11) is evaluated at $M=C_{*}R=2C_{\mathrm{r}}C_{0}E_{0}$ and is. Consequently, we havedenoted by$C_{2}(E_{0})$$|| \Phi[u]-\Phi[v]||_{X}\leq\frac{1}{2}||u-v||_{X}$provided that
(6.17), we have$E_{0}$

is so small that

$4C_{0}C_{2}(E_{0})E_{0} \leq\frac{1}{2}$$\Phi[0]=\Phi_{0}$(6.17). On the other hand, letting $v=0$ in$||\Phi[u]-\Phi[0]||_{X}\leq R/2$

Therefore, noting that,.

and using (6.10), we obtain(6.18)$||\Phi[u]||_{X}\leq||\Phi_{0}||_{X}+||\Phi[u]-\Phi[0]||_{X}\leq C_{0}E_{0}+R/2=R$

Thus we have shown by (6.17) and (6.18) that is a contraction mapping of , provided. Hence we can conclude that the mapping admits a unique fixedthat, namely, we have $u=\Phi[u]$ . This fixed point verifies the estimate (2.1)
point inand is the desired global solution to the problem (1.2), (1.3). Thus the proof of Theorem
2.1 is complete.
$\Phi$$S_{R}$$4C_{0}C_{2}(E_{0})E_{0} \leq\frac{1}{2}$$u$$S_{R}$$\Phi$$u$$\square$167

Large time behavior of solutions to a semilinear hyperboric system with

7

axation$\mathrm{r}el$

Asymptotic behavior

The aim of this section is to prove Theorem 2.3 concerning the asymptotic profile of thesolution to the problem (1.2), (1.3).
We denote by $W(t, x)$ be the nonlinear diffusion wave defined by (2.4) with $M=$$\int(u_{0}+u_{1})(x)dx$ and put $w(t, x)=W(t+1, x)$ . Then this $w(t, x)$ solves (2.5) and hence
the integral equation

.

(7.1)

$w(t)=G_{0}(t)*w_{0}- \frac{\beta}{2}\int_{0}^{t}G_{0}(t-s)*(w^{2})_{x}(s)ds$

Here $G_{0}(t,x)$ is the fundamental solution to the linearized equation of (2.5) and is givenby (3.9), and $w_{0}(x):=W(1, x)$ is a rapidly decreasing function satisfying $\int w_{0}(x)dx=$$M= \int(u_{0}+u_{1})(x)dx$ and$||w_{0}||_{W^{1,p}}+||w_{0}||_{L_{1}^{1}}\leq C|M|\leq C||u_{0}+u_{1}||_{L^{1}}\leq CE_{0}$

.

(7.2)

Let $u(t, x)$ be the global solution to the problem (1.2), (1.3) which was constructed in
Theorem 2.1 as a solution to the integral equation (6.2). In order to study the difference$u(t,x)-w(t, x)$ , we subtract (7.1) from (6.2), obtaining$(u-w)(t)=(G-G_{0})(t)*(u_{0}+u_{1})+G_{0}(t)*(u_{0}+u_{1}-w_{0})$$+H(t)*u_{0}- \int_{0}^{t}G(t-s)*(g(u)-\beta u^{2}/2)_{x}(s)ds$(7.3)$- \frac{\beta}{2}\int_{0}^{t}(G-G_{0})(t-s)*(u^{2})_{x}(s)ds-\frac{\beta}{2}\int_{0}^{t}G_{0}(t-s)*(u^{2}-w^{2})_{x}(s)ds$$=:I_{1}+I_{2}+I_{3}+I_{4}+I_{5}+I_{6}$.

We want to estimate the right hand side of (7.2). To do that, we need the following
estimate for the solution operator $G_{0}(t)*$ .
Lemma 7.1$([\mathit{2}J)$$L^{p_{-}}L^{q}$

Let $1\leq q\leq p\leq\infty$, and let $l\geq 0$ be an integer. Then we have$||\partial_{x}^{\iota}G_{0}(t)*\phi||_{L^{p}}\leq C\iota^{-\frac{1}{2}(_{qp})-t_{||\phi||_{L^{q}}}}\iota_{-\perp}$

(7.4)

.

Also, if $\int\phi(x)dx=0$ , then we have$||d_{x}G_{0}(t)*\phi||_{L^{p}}\leq Ct^{-\frac{1}{2}(1-\frac{1}{\mathrm{p}})-\frac{\iota}{2}}(1+t)^{-\frac{1}{2}}||\phi||_{L_{1}^{1}}$

Here

$C$

and

$c$

.

(7.5)

are positive constants.

The proof is given in Iguchi, Kawashima [2], and is omitted here.
Now we estimate (7.3) by introducing the following quantities:$M(t):= \sup_{0\leq s\leq t}(1+s)^{\mathrm{z}^{-e}}||(u-w)(s)||_{L^{1}}1$

where$\epsilon$,

is any fixed constant such that,

$N(t):= \sup_{0\leq s\leq t}(1+s)^{\frac{1}{2}(1-\frac{1}{\mathrm{p}})+1-\epsilon}||\partial_{x}(u-w)(s)||_{L^{\mathrm{p}}}$$0< \epsilon<\frac{1}{2}$

.

(7.6)

Proof of Theorem 2.3. The proof consists of three claims below. First, we show theestimate:$\mathrm{f}\mathrm{o}\mathbb{I}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}L^{1}$

168

Large time behavior of solutions to a semilinear $hyp$ er oric system with relaxation$\mathrm{b}$Claim 7.2 There is a positive constant
we have$||$$\delta_{1}(\epsilon)$depending on such that if$\epsilon$$(u-v)(t)||_{L^{1}}\leq CE_{1}(1+t)^{-\mathrm{B}^{+e}}1$$E_{0}\leq \mathit{6}_{1}(\epsilon)$, then.

(7.7)

It suffices to estimate each term on the right hand side of (7.3). For the term
have from (3.16) that$||I_{1}||_{L^{1}}\leq C(1+t)^{-\frac{1}{2}}||u_{0}+\mathrm{u}_{1}||_{L^{1}}\leq CE_{0}(1+t)^{-\frac{1}{2}}$$I_{1}$, we.

Also, since $\int(u_{0}+u_{1}-w_{0})(x)dx=0$ , we have from (7.5) that$||I_{2}||_{L^{1}}\leq C(1+t)^{-\frac{1}{2}}||u_{0}+u_{1}-w_{0}||_{L_{1}^{1}}\leq CE_{1}(1+i)^{-\}}$

where we used (7.2). For$I_{3}$, we apply (3.15) to obtain$||I_{3}||_{L^{1}}\leq C(1+t)^{-\frac{1}{2}}||u_{0}||_{L^{1}}\leq CE_{0}(1+t)^{-\frac{1}{2}}$Next, we estimate$I_{4}$,.

by applying (3.14) with $l=1,$ $p=q=1$ as$||I_{4}||_{L^{1}} \leq\int_{0}^{t}||\partial_{x}G(t-s)*(g(u)-\beta u^{2}/2)(s)||_{L^{1}}ds\leq CE_{0}^{3}(1+t)^{-\#}\log(2+t)$,where we have used the fact that $g(u)-\beta u^{2}/2=O(|u|^{3})$ and the estimate (2.1). The
term can be estimated similarly. In fact, we have from (3.16) with $l=1,$ $p=q=1$$I_{6}$

that$||I_{5}||_{L^{1}} \leq C\int_{0}^{t}||\partial_{x}(G-G_{0})(t-s)*(u^{2})(s)||_{L^{1}}ds\leq CE_{0}^{2}(1+t)^{-\frac{1}{2}}\log(2+t)$,where we used (2.1). (A more delicate computation can give the present estimate without
the factor $\log(2+t)$ but we omit it.) Finally, we estimate by applying (7.4) with $l=1$ ,$p=q=1$ . We obtain$I_{6}$$||I_{6}||_{L^{1}} \leq C\int_{0}^{t}||\partial_{x}G_{0}(t-s)*(u^{2}-w^{2})(s)||_{L^{1}}ds\leq C(\epsilon)E_{0}M(t)(1+t)^{-_{2}}\iota_{+\epsilon}$

for some constantdepending on . Here we have used the inequality$||u+w||_{L}\infty||u-w||_{L^{1}}$ together with the estimates (2.1) and (2.7) and the definition of$M(t)$ in (7.6). Summarizing all these estimates, we arrive at$C(\epsilon)$$||u^{2}-w^{2}||_{L^{1}}\leq$$\epsilon$$||(u-w)(t)||_{L^{1}}\leq CE_{1}(1+b)^{-\perp}2+CE_{0}^{2}(1+t)^{-\}}\log(2+t)+C(\epsilon)E_{0}M(t)(1+t)^{-_{2}}\iota_{+e}$. (7.8)Since $\log(2+t)\leq C(\epsilon)(1+t)^{e}$ , this yields the inequality $M(t)\leq CE_{1}+C(\epsilon)E_{0}^{2}+$, from which follows the desired estimate $M(t)\leq CE_{1}$ if
is so small that
. Thus we have shown the estimate (7.7).
Second, we derive the following
estimate:$C(\epsilon)E_{0}M(t)$$E_{0}$$L^{1}$$C( \epsilon)E_{0}\leq\frac{1}{2}$$L^{\infty}$

Claim 7.3 We have$||(u-v)(t)||_{L}\infty\leq CE_{1}(1+t)^{-1+\epsilon}$provided that

$E_{0}\leq \mathit{6}_{2}(\epsilon)$with a suitably small$\delta_{2}(\epsilon)$

.

,

(7.9)

169

Large time behavior of solutions to a semilinear hyperboric system with relaxation
For the term

$I_{1}$, we apply (3.16) with $p=\infty,$ $q=1$ and obtain$||I_{1}||\iota\infty\leq Ct^{-\frac{1}{2}}(1+i)^{-\frac{1}{2}}||u_{0}+u_{1}||_{L^{1}}\leq CE_{0}t^{-\frac{1}{2}}(1+t)^{-\frac{1}{2}}$Also, for$I_{2}$.

, we apply (7.5) to obtain$||I_{2}||_{L\infty}\leq Ct^{-_{2}}(1\iota+t)^{-\frac{1}{2}}||u_{0}+u_{1}-w_{0}||_{L_{1}^{1}}\leq CE_{1}\theta^{-\frac{1}{2}}(1+t)^{-\frac{1}{2}}$.

Similarly, applying (3.15) with $p=\infty,$ $q=1$ , we have$||I_{3}||_{\iota\infty}\leq C(1+t)^{-1}||u_{0}||_{L^{1}}+Ce^{-ct}||u_{0}||_{L}\infty\leq CE_{0}(1+t)^{-1}$

Next, we estimate$I_{4}$.

by applying (3.14) with $l=1,$ $p=\infty,$ $q=1$ as,$||I_{4}||_{L^{\infty}} \leq\int_{0}^{t}||\partial_{x}G(t-s)*(g(u)-\beta u^{2}/2)(s)||\iota\infty ds\leq CE_{0}^{3}(1+t)^{-1}\log(2+t)$where we used (2.1). Similarly, we estimate$q=1$ . We obtain$I_{5}$

by applying (3.16) with $l=0,$ $p=\infty$ ,,$||I_{5}||_{L} \infty\leq C\int_{0}^{t}||(G-G_{0})(t-s)*\partial_{x}(u^{2})(s)||\iota\infty ds\leq CE_{0}^{2}(1+t)^{-1}\log(2+t)$

and thewithwhere we have used the inequality
estimate (2.1). Finally, we estimate . We apply (7.4) with $l=1,$ $p=\infty,$ $q=1$ andthen with $l=1,$ $p=q=\infty$ . A combination of the resulting two estimates gives$||\partial_{x}(u^{2})||_{L^{1}}\leq C||u||_{L^{r}}||\partial_{x}u||_{L^{\mathrm{p}}}$$\frac{1}{p}+\frac{1}{f}=1$$I_{6}$$||I_{6}||_{L\infty} \leq C\int_{0}^{t}||\partial_{x}G_{0}(t-s)*(u^{2}-w^{2})(s)||_{L\infty}ds\leq C(\epsilon)E_{0}E_{1}(1+t)^{-1+\epsilon}$depending on . Here we have used the inequalities Il$||u+w||_{L}\infty||u-w||_{L^{1}}$ andand the estimates (2.1), (2.7)
and (7.7). Since $\log(2+t)\leq C(\epsilon)(1+t)^{\epsilon}$ , these observations show thatfor some constant$C(\epsilon)$$u^{2}-w^{2}||_{L^{1}}\leq$$\epsilon$$||u^{2}-w^{2}||_{L^{\infty}}\leq||u||_{L}^{2}\infty+||w||_{L}^{2}\infty$$||(u-w)(t)||_{\iota\infty}\leq CE_{1}t^{-\frac{1}{2}}(1+b)^{-\frac{1}{2}}+C(\epsilon)E_{0}E_{1}(1+t)^{-1+\epsilon}$

Therefore, assuming that

$C(\epsilon)E_{0}\leq 1$.

(7.10), we obtain$||(u-w)(t)||_{L}\infty\leq CE_{1}t^{-\mathrm{z}}(11+t)^{-_{2}}\iota_{+e}$This combined with (2.1) and (2.7) gives the desired estimate (7.9).
It remains to prove the following estimate for the derivative:

Claim 7.4 We have$||\partial_{x}(u-v)(t)||_{L^{\mathrm{p}}}\leq CE_{1}(1+t)^{-\frac{1}{2}(1-\frac{1}{p})-1+\epsilon}$provided that

$E_{0}\leq \mathit{6}_{3}(\epsilon)$with a suitably small$\delta_{3}(\epsilon)$.

,(7.11)

170

Large time behavior of solutions to a semilinear hyperboric system with relaxation

In the following we put. Notice thatfor $1\leq p\leq\infty$ . For the, we apply (3.16) with $l=1,$ $q=1$ and then with $l=1,$ $q=p$ , and combine
termthem to obtain$0 \leq\gamma\leq\frac{1}{2}$$\gamma=\frac{1}{2}(1-\frac{1}{p})$$\partial_{x}I_{1}$$||\partial_{x}I_{1}||_{L^{p}}\leq Ct^{-_{2}}(1\iota+t)^{-\gamma-\frac{1}{2}}(||u_{0}+u_{1}||_{L^{1}}+||u_{0}+u_{1}||_{L^{p}})\leq CE_{0}t^{-_{2}}(1\iota+t)^{-\gamma-\frac{1}{2}}$.

Also, for, we apply (7.5) with $l=1$ and then (7.4) with $l=1,$ $q=p$ . A combinationof the resulting two estimates gives$\partial_{x}I_{2}$$||\partial_{x}I_{2}||_{L^{p}}\leq Ct^{-\frac{1}{2}}(1+t)^{-\gamma-\frac{1}{2}}(||u_{0}+u_{1}-w_{0}||_{L_{1}^{1}}+||u_{0}+u_{1}-w_{0}||_{L^{p}})\leq CE_{1}\iota^{-\#}(1+t)^{-\gamma-\frac{1}{2}}$.

Similarly, applying (3.15) with $l=1,$ $q=1$ , we have$||\partial_{x}I_{3}||_{L^{\mathrm{p}}}\leq C(1+t)^{-\gamma-1}||u_{0}||_{L^{1}}+Ce^{-\mathrm{c}\ell}||u_{0}||_{W^{1,p}}\leq CE_{0}(1+t)^{-\gamma-1}$.

Next, we we want to estimate the derivatives$j=4,5,6$ . To this end, we$I_{j}=I_{j,1}+I_{j,2}$
decompose each integral into two parts and write, whereandarecorresponding to the integrations over $[0, t/2]$ and $[t/2, t]$ , respectively. Now, for the term
, we apply (3.14) with $l=2,$ $q=1$ , obtaining$\partial_{x}I_{j},$$I_{j,1}$$I_{j}$$I_{j,2}$$\partial_{x}I_{4,1}$$|| \partial_{x}I_{4,1}||_{L^{p}}\leq\int_{0}^{t/2}||\partial_{x}^{2}G(t-s)*(g(u)-\beta u^{2}/2)(s)||_{L^{p}}ds\leq CE_{0}^{3}(1+t)^{-\gamma-1}\log(2+b)$,where we have used the estimates $||(g(u)-\beta u^{2}/2)(s)||_{L^{1}}\leq CE_{0}^{3}(1+s)^{-1}$ andwhich follow from (2.1). Also, applying(3.14) with $l=1,$ $q=p$ , we have$||\theta_{x}(g(u)-$$\beta u^{2}/2)(s)||_{L^{\mathrm{p}}}\leq CE_{0}^{3}(1+s)^{-\gamma-1-\frac{\iota}{2}}(l=0,1)$.

$|| \partial_{x}I_{4,2}||_{L^{\mathrm{p}}}\leq\int_{\iota/2}^{t}||\partial_{x}G(t-s)*\partial_{x}(g(u)-\beta u^{2}/2)(s)||_{L^{p}}ds\leq CE_{0}^{3}(1+t)^{-\gamma-1}$

On the other hand, for the term
, we apply (3.16) with $l=2,$ $q=1$ and then with
$l=1,$ $q=1$ , and combine them to obtain$\partial_{x}I_{5,1}$$|| \partial_{x}I_{6,1}||_{L^{\mathrm{p}}}\leq C\int_{0}^{t/2}||\partial_{x}^{2}(G-G_{0})(t-s)*(u^{2})(s)||_{L^{p}}ds\leq CE_{0}^{2}(1+t)^{-\gamma-1}$,

where we have used the estimates 1
and$CE_{0}^{2}(1+t)^{-\gamma-\frac{l+1}{2}}(l=0,1)$. Also, applying (3.16) with $l=1,$ $q=p$ , we have$(u^{2})(s)||_{L^{1}}\leq CE_{0}^{2}(1+t)^{-\frac{1}{2}}$$||\theta_{x}(u^{2})(s)||_{L^{\mathrm{p}}}\leq$$|| \partial_{x}I_{5,2}||_{L^{p}}\leq C\int_{\iota/2}^{t}||\partial_{x}(G-G_{0})(t-s)*\partial_{x}(u^{2})(s)||_{L^{\mathrm{p}}}ds\leq CE_{0}^{2}(1+t)^{-\gamma-1}\log(2+t)$.

Finally, we consider. For the term
, we apply (7.4) with $l=2$ ,and$q=1$ and then with $l=1,$ $q=p$ , and combine them to obtain$\partial_{x}I_{6,1}$$\partial_{x}I_{6,2}$$\partial_{x}I_{6,1}$$|| \partial_{x}I_{6,1}||_{L^{\mathrm{p}}}\leq C\int_{0}^{t/2}||\partial_{x}^{2}G_{0}(t-s)*(u^{2}-w^{2})(s)||_{L^{p}}ds$$\leq C\int_{0}^{t/2}(t-s)^{-\#}(1+t-s)^{-\gamma-\mathrm{i}}(||(u^{2}-w^{2})(s)||_{L^{1}}+||\partial_{x}(u^{2}-w^{2})(s)||_{L^{p}})ds$

.

171

Large

$t\mathrm{i}m\mathrm{e}$behavior of solutions to a semilinear hyperboric system With relaxation

Here we observe that $||u^{2}-w^{2}||_{L^{\mathit{1}}}\leq||u+w||_{L}\infty||u-w||_{L^{1}}$ and. Therefore, making use of (2.1), (2.7) and (7.7), we obtain$||\partial_{x}(u^{2} - w^{2})||_{L^{p}}\leq$$||\partial_{x}(u^{2})||_{L^{\mathrm{p}}}+||\partial_{x}(w^{2})||_{L^{\mathrm{p}}}$$|| \partial_{x}I_{6,1}||_{L^{\mathrm{p}}}\leq CE_{0}E_{1}\int_{0}^{t/2}(t-s)^{-\frac{1}{2}}(1+t-s)^{-\gamma-\frac{1}{2}}(1+s)^{-1+\epsilon}ds\leq C(\epsilon)E_{0}E_{1}(1+t)^{-\gamma-1+e}$

for a constant$C(\epsilon)$depending . Also, applying (7.4) with $l=1,$ $q=p$ , we have$\epsilon$.

$|| \partial_{x}I_{6,2}||_{L^{p}}\leq C\int_{t/2}^{t}||\partial_{x}G_{0}(t-s)*\partial_{x}(u^{2}-w^{2})(s)||_{L^{p}}ds\leq C\int_{\ell/2}^{t}(t-s)^{-_{2}}||\partial_{x}(u^{2}-w^{2})(s)||_{L^{p}}ds\iota$

Here we observe that$||\partial_{x}(u^{2}-w^{2})||_{L^{\mathrm{p}}}\leq||u+w||\iota\infty||\partial_{x}(u-w)||_{L^{\mathrm{p}}}+||\partial_{x}(u+w)||_{L^{p}}||u-w||\iota\infty$.

We know from (2.1) and the definition of $N(t)$ in (7.6) that the first term here is boundedby. Also, using (2.1) and (7.9), we can majorize theterm by. Consequently, we obtain$CE_{0}N(t)s^{-\frac{1}{2}}(1+s)^{-\gamma-1+\epsilon}$$8\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}$$CE_{0}E_{1}(1+s)^{-\gamma-_{l}^{3}+\epsilon}$$|| \partial_{x}I_{6,2}||_{L^{\mathrm{p}}}\leq C(E_{0}N(t)+E_{0}E_{1})\int_{t/2}^{t}(t-s)^{-\frac{1}{2}}s^{-\frac{1}{2}}(1+s)^{-\gamma-1-e}ds$$\leq C(E_{0}N(t)+E_{0}E_{1})(1+t)^{-\gamma-1-\epsilon}$.

We can summarize all the above computations as$||\partial_{x}(u-w)(t)||_{L^{\mathrm{p}}}\leq CE_{1}t^{-1}2(1+t)^{-\gamma-\frac{1}{2}}+C(\epsilon)E_{0}E_{1}(1+t)^{-\gamma-1+e}+CE_{0}N(t)(1+t)^{-\gamma-1+e}$.

(7.12)

This yields $N(t)\leq CE_{1}+C(\epsilon)E_{0}E_{1}+CE_{0}N(t)$ , from which we can deduce the desired
estimate $N(t)\leq CE_{1}$ for suitably small , say,. Thus we obtain$E_{0}$$E_{0}\leq \mathit{6}_{3}(\epsilon)$.

$||\partial_{x}(u-w)(t)||_{L^{p}}\leq CE_{1}t^{-\frac{1}{2}}(1+t)^{-\gamma-\#+\epsilon}$

which together with (2.1) and (2.7) yields the desired estimate (7.11). This completes theproof of Theorem 2.3.
$\square$

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Math. Phys., 172 (1995), 39-55.
[2] T. Iguchi and S. Kawashima, On space-time decay properties of solutions tohyperbolic-elliptic coupled systems, Hiroshima Math. J., 32 (2002), 119-308.

[3] T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108(1987), 153-175.
[4] T.-P. Liu and Y. Zeng, Large time behavior of solutions for general quasilinear
hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125, no.
599, (1997).
[5] R. Orive and E. Zuazua, Long-time behavior of solutions to a non-linear hyperbolic
relaxation system, (preprint 2003).