数理解析研究所講究録
第 836 巻 1993 年 152-169

152

Convergence of Numerical Interface Curves to
Nonlinear Diffusion Equation with Absorption
Kenji Tomoeda\dagger and Tatsuyuki Nakaki\ddagger

1. Introduction.
We are concerned with difference approximations to the following initial value problem
for the nonlinear diffusion equation described by(1.1)$v_{t}=(v^{m})_{xx}-cv^{p}$,$x\in R^{1}$

,

$t>0$

with an initial condition$v(0, x)=v^{0}(x)$ ,(1.2)$x\in R^{1}$,

has compact support. The
where $m(>1),$ $p(>0)$ and $c(\geq 0)$ are constants, andequation of the form (1.1) is known as a simple mathematical model for several physicalphenomena.
The first model with $c=0$ describes the flow of an ideal gas through a homogeneousis the pressure
porous medium, where represents a density of the gas. Physically,of the gas andis the velocity.
The second model with $c>0$ describes the transport of the thermal energy in plasma.
Here means the temperature. The term $-cv^{p}$ is understood as volumetric absorption
caused by radiation.
In both models with $c=0$ and with $c>0$ the most interesting phenomenon is theoccurrence of finite propagation of the initial support. It is already shown that there arethree cases of the behavior of $suppv(t, \cdot)$ .
$v^{0}(x)$$v$$v^{m-1}$$(v^{m-1})_{x}$$v$Case 1. Positivity. For $c=0$ and $m>1$ , or $c>0$ and $p\geq m>1suppv(t, \cdot)$ expands asincreases and $suppv(\infty, \cdot)=R^{1}$ ([1],[3],[4],[8],[9],[15]).
\dagger Osaka Institute of Technology, Asahi-ku-Omiya, Osaka, 535, Japan\ddagger Fukuoka University of Education, Akama, Munakata, 811-41, Japan

$t$

153

$v_{t}=(v^{2})_{XX}$Barenblatt and Pattle’$s$exact solution and interface curve.

$v_{t^{=(V^{2})}xx^{-}}v^{2}$

Bertsch, Kersner and Peletier’s solution.

154
Case 2. Localization. For $c>0$ and $m>p\geq 1suppv(t, \cdot)$ expands asunifonnly bounded with respect to ([7],[9],[11]).

$t$

increases and is$t$$v_{t}=(v^{2})_{xx}-0.5V$

Exact solution (Gurtin and MacCamy).

Case 3. Total Extinction. For $c>0,$ $m>1$ and $0<p<1$
and $v(t, x)$ extincts in finite time ([9],[10],[11]).
$[0, \infty)\cross R^{1}$$v_{t^{=(V^{1.5})}xx}-v^{0.5}$

Kersner t$s$solut ion

$suppv$is compact in155
From a numerical point of view, it is interesting to determine the behavior of $suppv(t, \cdot)$ ,that is, interface curves appearing between $v>0$ and $v=0$ .
Several difference schemes to $(1.1)-(1.2)$ with $c=0$ have been investigated. In [6]Graveleau and Jamet proved the finite propagation of the support by using their difference
scheme. However, their scheme does not give good approximations to the exact interface
curves. DiBenedetto and Hoff’s scheme [5] and Mimura, Nakaki and Tomoeda’s scheme[13] give the convergence of numerical interface curves. It is observed that the numericalinterface curves by the latter scheme are more accurate rather than the former’s.
Numerical computations to $(1.1)-(1.2)$ with $c>0$ are investigated by Rosenau andKamin [16], Mimura, Nakaki and Tomoeda [13] and Nakaki[14]. In [13] the convergence ofnumerical solutions is proved for Cases 1 and 2, $and$ the convergence of numerical interface
curves is so proved for Case 1 and Case 2 with $p=1$ . In Case 3 Rosenau and Kaminnumerically examined the problem of the pulse splitting into several sub-pulses, but thetheoretical results of the numerical scheme are not discussed. In Case 3 with $m+p=2$
it is shown in [14] that not only numerical solutions but also interface curves converge toexact ones under$al$

Condition $A:(v^{0}(x))^{m-1}$ is concave downward on its support.
In this paper we show the convergence of numerical interface curves without Condition A
for Case 3 with $m+p=2$ . However, instead of Condition A we have to impose the following
condition on numerical results.
Condition
(1.3)

$B$

: There exist positive numbers $M,$
$i_{h}(t)<M$

and$T^{*}$

and$\dot{r}_{h}(t)>-M$$h^{*}$

such that$(= \frac{d}{dt})$

anddenote left and rightfor almost all $t\in[0, T^{*}]$ and for all $h\in(0, h^{*})$ . Herenumerical interface curves, respectively, and is a space mesh width.
$\ell_{h}(t)$$r_{h}(t)$$h$We now state the existence and uniqueness of weak solution of $(1.1)-(1.2)$ in Section 2.
In Section 3 we present Mimura, Nakaki and Tomoeda’s scheme for Cases 1 and 2, anddemonstrate some numerical solutions and interface curves. In Section 4 we introduce themodified Mimura, Nakaki and Tomoeda’s scheme for Case 3 with $m+p=2$ and $c>0$ ,and show the convergence of numerical interface curves under the Condition B.

2. Existence and Uniqueness.
To show the convergence of the difference approximation to the exact solution, we
prepare the existence and uniqueness of the weak solution of $(1.1)-(1.2)$ with $m+p\geq 2$
and $c\geq 0$ .

Definition. (Herrero and V\’azquez[8]). A function $v(t, x)$ defined on $S=[0, \infty$ )said to be a weak solution of $(1.1)-(1.2)$ , if$\cross R^{1}$

is

156
(i) $v\in C^{0}(S)\cap L^{\infty}(S)$and$v\geq 0$

on

$S$

;

(ii) for any $x\in R^{1},$ $v(0, x)=v^{0}(x)$ ;(iii) for any function $\phi(x)\in C^{1,2}(S)$ with compact support in
relation holds:
(2.1)

$S$

, the following integral$\parallel_{s}(v^{m}\phi_{xx}+v\phi_{t}-cv^{p}\phi)dxdt+\int_{R^{1}}v(0, x)\phi(0, x)dx=0$.

To show the stability of our difference scheme, we set $u=v^{m-1}$ and rewrite $(1.1)-(1.2)$as$u_{t}=muu_{xx}+a(u_{x})^{2}-(m-1)cu^{q}$ ,

(2.2)

(2.3)

$a= \frac{m}{m-1}$$u(0, x)=u^{0}(x)\equiv(v^{0}(x))^{m-1}$

Definition. A function $u(t, x)$ defined on(i) $u\in C^{0}(S)\cap L^{\infty}(S)$ ,$u_{x}\in L^{\infty}(S)$

$S$

and

,

$q= \frac{m+p-2}{m-1}$,.

is said to be a weak solution of $(2.2)-(2.3)$ , if$u\geq 0$

on

$S$

;

(ii) for any $x\in R^{1},$ $u(0, x)=u^{0}(x)$ ;(iii) for any function $\phi(x)\in C^{1,2}(S)$ with compact support in
relation holds:$S$, the following integral(2.4) $\iint_{S}(u\phi_{t}-muu_{x}\phi_{x}-(m-a)(u_{x})^{2}\phi-(m-1)cu^{q}\phi)dxdt+\int_{R^{1}}u(0, x)\phi(0, x)dx=0$.

be
Theorem 2.1 (Herrero and V\’azquez [8]). Let $m>1,$ $m+p\geq 2,$ $p>0$ and let
a continuous, nonnegative and bounded real function. Then there exists a unique weaksolution of $(1.1)-(1.2)$ and is smooth in the set $\{(t, x):v(t, x)>0\}$ .
$v^{0}$$v$$v$

Remark. Let be the space of$D’$ be its
dual. If a weak solution
$D$

$aU$

$u$

continuous functions with compact support inof $(2.2)-(2.3)$ with $m>1$ satisfies$u_{xx}$,$u_{t}\in D’$,

then $v=u^{1/(m-1)}$ is the unique weak solution of $(1.1)-(1.2)$ (see [6]).

$S$

, and

157
3. Difference Schemes to $(1.1)-(1.2)$ with $m>1$ and $p\geq 1$ .

Our difference scheme approximates the problem $(2.2)-(2.3)$ instead of $(1.1)-(1.2)$ . The

difference scheme is constructed based on splitting the equation (2.2) into three parts:
(3.1)

(3.2)
(3.3)

We denote by$u_{t}$$=$$Pu=muu_{xx}$ ,$u_{t}$$=$$Hu=a(u_{x})^{2}$$u_{t}$$=$$Du=-(m-1)cu^{q}$ ,$V_{h}$,$a=m/(m-1)$ ,$q=(m+p-2)/(m-1)$ .

the set of the nonnegative continuous functions$u_{h}$

with the following

properties:
(i)
(ii)

$u_{h}$$u_{h}$

has compact support

for all$x_{L-1}\equiv l(u_{h})$

$h,$

,

$ih\in(l(u_{h}), r(u_{h}))$$x_{R+1}\equiv r(u_{h})$$(i=L, L+1, \cdots, R-1, R)$ ,.

and $u_{i}=u(x;)$ . Then our difference scheme [13] is described as follows:$=x_{i+1}-x_{i}$Find the sequence
(3.4)

;

is linear on each interval $[x_{i}, x_{i+1}](i\in Z)$ , where$x;=ih$Let$[\ell(u_{h}), r(u_{h})]$$\{u_{h}^{n}\}_{n=0,1,2},\cdots\subset V_{h}$such that$u_{h}^{n+1}=S_{h,k}u_{h}^{n}\equiv(D_{h,k/\nu})^{\nu}\cdot(P_{h,k/\mu})^{\mu}\cdot H_{h,k}u_{h}^{n}$

Here $k\equiv k_{n+1}\equiv t_{n+1}-t_{n}$ is a variable time step,

depending on$k_{n+1}$$\mu\equiv\mu_{n+1}$

and$\cdots$.

$\nu\equiv\nu_{n+1}$

are integers.

Difference operator$H_{h,k}$We define the operator$H_{h,k}$(3.5)

for $n=0,1,2,$$u_{h}’=(H_{h,k}u_{h})(x_{i})$mapping from$=$

$V_{h}$

to$V_{h}$

byexact solution $u(k, x_{i})$ of $u_{t}=Hu$with the initial value $u(0, x)=u_{h}(x)$ .

Let $\{L’, L’+1, \cdots, R’-1, R’\}$ be the set of integers such that$x_{i}=ih$

for all$ih\in(\ell(u_{h}’), r(u_{h}’))$$(i=L’, L’+1, \cdots, R’-1, R’)$ .

158
Then

(3.6)

$u_{h}’=\{\begin{array}{l}u_{i}+a(\delta u_{i})^{2}kifi\in S^{+}=S_{S}^{+}\cup S_{R}^{+}u_{i}+a(\delta u_{-1})^{2}kifi\in S^{-}=S_{S}\cup S_{R}^{-}u_{i}ifi\in S^{0}(Lh-l’)\delta u_{L-1}ifi=L^{/}=L-1(Rh-r^{/})\delta u_{R}ifi=R’=R+10ifi\in Z\backslash \{L,)R’\}\end{array}$

where

and

$\delta u_{i-1}>-\delta u_{i}\}$,and$\delta u_{i-1}\leq-\delta u_{i}\}$,$S_{S}^{+}$$=$$\{i\in\{L, \cdots, R\}:\delta u_{i-1}<6u_{i}$$S_{S}$$=$$\{i\in\{L, \cdots, R\}$:$\delta u_{i-1}<\delta u_{i}$$S_{R}^{+}$$=$$\{i\in\{L, \cdots, R\}$:$\delta u_{i-1}\geq\delta u_{i}>0\}$,$S_{R}^{-}$$=$$\{i\in\{L, \cdots, R\}$:$0>\delta u_{t}-1\geq\delta u_{i}\}$,$=$$\{i\in\{L, \cdots, R\}:\delta u_{-1}\geq 0\geq\delta u_{i}\}$.

$S^{0}$

Stability Condition:$k\equiv k_{n+1}$

is the largest number satisfying the following inequality:(3.7)$a||(u_{h})_{x}||_{\infty}k \leq\min\{h/4, Lh-\ell(u_{h}), r(u_{h})-Rh\}$(3.8)$k\leq Ch^{s}$

Difference operator(3.9)(For simplicity we put $C=1,$ $s=1/2$ ).

$P_{h,k/\mu}$$(P_{h,k/\mu}u_{h})(x_{i})=u;+ \frac{k}{\mu}mu_{i}\delta^{2}u_{i}$$\delta u_{i}=(u_{t+1}-u_{i})/h.$Stability Condition:

$\mu$

,

for all $i\in Z$ ,$\delta^{2}u_{t}=2(\delta u_{i}-\delta u_{i-1})/(h_{t}+h_{i-1})$,

satisfies the following inequalities:(3.10)$m||u_{h}||_{\infty}(k/\mu)[1/h^{2}+2/\{h(h+h_{J}\cdot)\}]\leq 1$(3.11)$4m||(u_{h})_{x}||_{\infty}(k/\mu)/(h+h_{j})\leq 1$

Difference operator(3.12).

for $j=L-1,$$R$,

for $j=L-1,$ .
$R$$D_{h,k/\nu}$$(D_{h,k/\nu}u_{h})(x_{i})=\{[u_{h}-(m-1)c(u_{\Lambda})^{q}(k/\nu)]^{+}\}(x_{i})$,$[f]^{+}= \max\{f, 0\}$.

159
Stability Condition:$\nu$satisfies the following inequality:(3.13)$(k/\nu)(m-1)cq||u_{h}||_{\infty}^{q-1}<1$

We note that the interfaces ofproperty of the operator$H_{h,k}$$D_{h,k1\nu}u_{h}$and$P_{h,k/\mu}u_{h}$.

are same ones of$u_{h}$. From thethe numerical interfaces of solutions given by (3.4) can be

expressed as follows:(3.14)$l_{n+1}=\ell_{n}-a\delta u_{L-1}^{n}k$(3.15)$r_{n+1}=r_{n}-a\delta u_{R}^{n}k$

,

,

where
(3.16)$\ell_{n}\equiv\ell(u_{h}^{n})$,$r_{n}\equiv r(u_{h}^{n})$for $n=0,1,2,$$\cdots$.

To start the scheme (3.4), we take(3.17)$t_{0}=0$,$\ell_{0}=l(u_{h}^{0})$,$r_{0}=r(u_{h}^{0})$,$u_{h}^{0}(x_{i})=u^{0}(x_{i})$and impose the following assumptions on the initial value$u^{0}(x)$,.

Assunption I. $u\in C^{0}(R^{1})\cap BV(R^{1})$ has compact support $[l(u^{0}), r(u^{0})]$ and satisfies$u_{x}^{0}\in L^{\infty}(R^{1})\cap BV(R^{1})$Assumption II.
(3.18)$u^{0}$.

satisfies$u_{xx}^{0}(x)>-C$

,

where

$C$

is some positive constant.

Theorem 3.1 (Stability [13], [18]). Under Assumption I let $m>1,$ $p\geq 1$ and $c>0$ . Then
(3.19)(3.20)$0\leq u_{h}^{n}\leq||u^{0}||_{\infty}$

for all$\lim_{narrow\infty}t_{n}=\infty$.

$n\geq 0$,

160
Moreover, for each $T>0$ , the following estimates(3.21)$1_{||(u_{h}^{n+1}-u_{h}^{n})/k_{n+1}^{u_{h}^{n}}||_{L(R)}^{||_{L_{1}^{1}(R_{1}^{1})}}}^{||(u_{h}^{n})_{x}||_{\infty},||()_{x}}$$\leq^{V((u_{h}^{n}))}C_{1}(T)^{x}$

for all$t_{n+1}\leq T$,

hold for all $n\geq 0$ satisfying $t_{n+1}E[0, T]$ , where $C_{1}(T)$ is a constant depending on

$T$

, but

independent of .
$h$Remark. The conclusion of this theorem is also valid for $c=0$ .
To state the convergence, we define a function $u_{h}(t, x)$ by(3.22)$u_{h}(t, x)=u_{h}^{n}(x)$on$[t_{n},t_{n+1}$

for all $n\geq 0$ .

)Theorem 3.2 (Convergence [13],[18]). Under the same assumptions as stated in Theorem 3.1, $(u_{h})^{1/(m-1)}$ converges uniformly in any bounded domain of$[0, \infty$)$\cross R^{1}$to theunique weak solution of $(1.1)-(1.2)$ .
By piecewise-linearly interpolating $(t_{n}, l_{n})(resp.(t_{n}, r_{n}))(n\geq 0)$ we define the left (resp.
right) numerical interface curve$l_{h}(t)$(resp.

$r_{h}(t)$).

Theorem 3.3 (Numerical Interface Curves [13],[18]). Under Assumptions I and$(m,p)\in\{(m, p) :p\geq m>1\}\cup\{(m, 1) :m>1\}$

Then, for any $T>0$ ,(3.23)$||l_{h}-\ell^{*}||_{L}\infty([0,\tau])arrow 0$

as$harrow 0$,(3.24)$||r_{h}-r^{*}||_{L^{\infty}([0,T])}arrow 0$

as$harrow 0$

,

where

$\ell*$

and$r^{*}$

are the exact interface curves.

Remark. (3.23) and (3.24)$al$so hold for $c=0$ .

.

$\Pi$

let

161

$fi’ v\Leftrightarrow\Gamma\vdash\dot{\tau}n\star\Leftrightarrow rf\neg---\cdot\cdotarrow\cdot,-$

5.0

0.0

$-5.0$

Numerical interface curves toBarenblatt and Pattle’ solution.
$s$: Baklanovskaya’ scheme [2] $h=0.057$
G-J: Graveleau and Jamet scheme [6] $h=0114$
B-H: DiBenedetto and Hoff’s scheme [5] $h=0.125$$N-N-T$ : Mimura, Nakaki and Tomoeda’ scheme $[17, 18]$ $h=0.125$
$B$

$s$

$s$

$x$

162

$v_{t^{=(V^{2})}xx}-V^{2}$

The initial value is the same one
and Peletier‘s solution at $t=0$.

$v_{t}=(v^{2})$xx $-0.5V$

The inlti-al value is the sameat $t=0$ (Gurt in and Mac Camy).

as Bertsch,

Kersner

one as the exact solution

163
4. Difference Schemes to $(1.1)-(1.2)$ with $m+p=2$ .
In this section we assume that $m+p=2,$ $m>1$ and $p>0$ . The solution $(1.1)-$(1.2) extincts in finite time. The front of support may expand $and/or$ shrink. Taking thisproperty into consideration, we approximate (2.2) in the following way [14]:Find the sequence(4.1)$\{u_{h}^{n}\}_{n=0,1,2},\cdots\subset V_{h}$such that$u_{h}^{n+1}=S_{h,k}u_{h}^{n}\equiv(P_{h,k/\mu})^{\mu}\cdot H_{h,k}\cdot D_{h,k}u_{h}^{n}$

The difference operators$H_{h,k}$and$P_{h,k/\mu}$

for $n=0,1,2,$$\cdots$

are given by (3.5) and (3.9), respectively. Since$q=0$ , the difference operator $D_{h,k}$ is written as(4.2)$(D_{h,k}u_{h})(x_{i})=\{[u_{h}-(m-1)ck]^{+}\}(x_{i})$,Stability Condition:
1) $k=k_{n+1}$ is the largest number satisfying the inequality $(3.7)-(3.8)$ with$u_{h}=D_{h,k}u_{h}^{n}$;2) Every connected component of the set $[suppu_{h}]\backslash [suppD_{h,k}u_{h}]$ has at most one point
$x$

3)

$\mu$

such that $x/h$ is an integer;

satisfies the stability conditions $(3.10)-(3.11)$ .

From this stability condition the numerical interface curves can be expressed as fol-

lows:
(4.3)

$l_{n+1}$$= \ell_{n}+\frac{(m-1)ck}{\delta u_{L-1}^{n}}-a\delta u_{L-1}^{n}k$if $x_{L-1}<\ell(D_{h,k}u_{h}^{n})<x_{L}$ ,(4.4)$l_{n+1}$$=$$l_{n}+ \frac{t4_{L}^{n}}{\delta u_{L-1}^{n}}+\frac{(m-1)ck-u_{L}^{n}}{\delta u_{L}^{n}}-a\delta u_{L}^{n}k$if $x_{L}\leq l(D_{h,k}u_{h}^{n})<x_{L+1}$ ,$r_{n+1}$$=$$r_{n}+ \frac{(m-1)ck}{\delta u_{R}^{n}}-a6u_{R}^{n}k$if $x_{R}<r(D_{h,k}u_{h}^{n})<x_{R+1}$ ,

(4.5)

164

(4.6)

$r_{n+1}$

where$\ell_{n}$

$=$

and

$r_{n}+ \frac{u_{R}^{n}}{\delta u_{R}^{n}}+\frac{(m-1)ck-u_{R}^{n}}{\delta u_{R-1}^{n}}-a\delta u_{R-1}^{n}k$if $x_{R-1}<r(D_{h,k}u_{h}^{n})\leq x_{R}$ ,are given by (3.16).

$r_{n}$

Theorem 4.1. Under Assumption I let $m+p=2,$ $m>1$ and $p>0$ . Then (3.19) and(3.21) hold for all $n\geq 0$ , where the constant$C_{1}$

is independent of

$T$

and .
$h$

Theorem 4.2. Under Assumption II and the same assumptions as stated in Theorem 4.1
let Condition$B$

be satisfied. Then there exist Lipschitz continuous functions$l^{*}(t)$and$r^{*}(t)$on $[0, T^{*}]$ such that11$v_{h}-v||_{L^{\infty}(H)}arrow 0$

as$harrow 0$

,

(4.8)

$||\ell_{h}-\ell^{*}||_{L^{\infty}([0,T^{*}])}arrow 0$

as$harrow 0$

,

(4.9)

11

as

$harrow 0$

,

(4.7)

where $H=[0, T^{*}]\cross R^{1},$$(1.2)$. Moreover,$\ell*$

and$r_{h}-r^{*}||_{L^{\infty}([0,T])}arrow 0$$v_{h}\equiv(u_{h})^{1/(m-1)}$$r^{*}$

and $v(t, x)$ is the unique weak solution of $(1.1)-$

become the left and right interface curves, respectively.

165
Numerical interface curves and numerical extinction time with $m=1.5$ ,$c=1$ . The initial value is the same one as$p=0.5$ andKersner’ solution:$s$$v(t, x)=(b_{1}t+b_{2})^{-1/(m-1)}$ $[ \chi^{2}(t)-x^{2}]^{1/(m-1)}$

for

2

1/2

$2/(m+1)$$\lambda(t)=\pm[a_{1}(b_{1}t+b_{2^{)}} -a_{2}(b_{1}t+b_{2}) ]$$c(m-1)^{4}+4m^{2}$$c(m-1)^{2}$$a_{1}=\overline{4m^{2}(m-1)^{2/(m+1)}}$$b_{1}=\underline{2m(m+1)}$

$m-1$

.

’

$a_{2}=\overline{4m^{2}}$$b_{2}=(m-1)$.

.

$|x|\leqq\lambda(t)$

.

,

166
Numerical solutions and interface curves with $m=1.5,$ $p=0.5,$ $c=1$ andfor$v(0, x)=$$\{0for^{2}x^{2}0.75(x-1)^{2_{-}}>14(x^{2}-1)(x^{2}-0.125)$$x^{2}\leqq 1$’

167$\bullet$:Numerical extinction time.

$\star$:Numerical time$\star$:Numerical time

$T_{h}^{*}$

with $M=50$ for each$h$such that$i_{h}$$-\dot{r}_{h}\leqq$$N$$[0, T_{h}^{*}]$.

$T_{h}^{*}$

with $M=25$ for each$h$such that$\dot{2}_{h}$$-\dot{r}_{h}\leqq$

If on $[0, T_{h}^{*}]$

.

on

168

$\bullet;N\iota merical$

extinction time.

$\star:Numerical$

time$T_{h}^{*}$

with$M=50$

for each$h$such that$\dot{p}_{h},$:Numerical time

$T_{h}^{*}$

with$M=25$

for each$h$such that$i_{h}$$\star$.

$-\dot{r}_{h}\leqq M$

on$[0, T_{h}^{*}]$.

$-\dot{r}_{h}\leqq fl$on$[0. T_{h}^{*}]$.

169
References[1] D.G. Aronson, Arch. Rat. Mech. Anal., 37(1970), 1-10.
[2] V.F. Baklanovskaya, Z.Vycisl. Mat.i Mat. Fiz., $1(1961),461-469$ .
[3] G.I. Barenblatt, Prinkl. Math. Meh., 16(1952), 67-78.

[4] M. Bertsch, R. Kersner and L.A. Peletier, Nonlinear Anal., 9(1985), 987-1008.

[5] E. DiBenedetto and D. Hoff, Trans. Amer. Math. Soc., 284(1984), 463-500.
[6] J.L. Graveleau and P. Jamet, SIAM J. Appl. Math., $20(1971),199-223$ .
[7] M.E. Gurtin and R.C. MacCamy, Math. Biosci., 33(1977), 35-49.
[8] M.A. Herrero and V\’azquez, SIAM J. Math. Anal., $18(1987),149-167$ .

[9] A.S. Kalashnikov, Z. Vycisl. Mat.i Mat. Fiz., $14(1974),891-905$ .
[10] R. Kersner, Vestn. Mosk. un-ta, matem. mekhan., $33(1978),44-51$ .
[11] R. Kersner, SIAM J. Appl. Math., 43(1983), 1274-1285.
[12] B.F.Knerr, Trans. Amer. Math. Soc., 249(1979), 409-424.

[13] M. Mimura, T. Nakaki and K. Tomoeda, Japan J. Appl. Math., $1(1984),93-139$ .
[14] T. Nakaki, Hiroshima Math. J., $18(1988),373-397$ .
[15] R.E. Pattle, Quart, J. Mech. Appl. Math., 12(1959), 407-409.

[16] P. Rosenau and S. Kamin, Physica, $8D(1983),273-283$ .
[17] K. Tomoeda and M. Mimura, Hiroshima Math. J.,13(1983), 273-294.

[18] K. Tomoeda, Advances in Computational Methods for Boundary and Interior Layers,

Lecture notes of an international short course held in association with the BAIL III
Conference 1984,88-94.

友枝謙二 (大阪工業大学)
中木達幸 (福岡教育大学)