Guo, Yung-Jen Lin

Osaka City University Repository

Guo, Y.-J.L.
Osaka J. Math.
47 (2010), 33–40
A NONLOCAL PARABOLIC PROBLEM ARISING
IN LINEAR FRICTION WELDING
YUNG-JEN LIN GUO
(Received September 5, 2008)
Abstract
We study a nonlocal parabolic problem modeling the temperature in a thin region
during linear friction welding for a hard material. We derive the structures of steady
states of this nonlocal problem and its associated approximated problems. Moreover,
some remarks on the parabolic problem are given.
1. Introduction
In this paper, we study the following initial boundary value problem:
ut D uxx  
u p
 R
1
0 u
 p(x , t) dx1C1=p
, 0 < x <1, t > 0,(1.1)
ux (0, t) D 0, ux (1, t) D 1, t > 0,(1.2)
u(x , 0) D u0(x), x  0,(1.3)
where the parameter p > 1 and u0 is a positive smooth function defined on [0, 1).
This model arises in the study of linear friction welding for a hard material (cf. [8]
and its references). In particular, in the real model the parameter p is close to 4.
To study this problem, in [8] they proposed to study the following approximated
problem in bounded intervals:
ut D uxx  

Z R
0
u p(x , t) dx

 1 1=p
u p, 0 < x < R, t > 0,(1.4)
ux (0, t) D 0, u(R, t) D R, t > 0,(1.5)
u(x , 0) D u0(x), 0  x  R,(1.6)
2000 Mathematics Subject Classification. Primary 34K05, 34A34; Secondary 34K60, 34E05.
This work was partially supported by the National Science Council of the Republic of China under
the grant NSC 96-2115-M-003-004.
34 Y.-J.L. GUO
where R is any positive constant. It is convenient to introduce the following transfor-
mation:
(1.7) x D Ry, t D R2s, u(x , t) D Rv(y, s).
Then (1.4)–(1.6) is reduced to the following problem:
vs D vyy   

Z 1
0
v
 p(y, s) dy

 1 1=p
v
 p
, 0 < y < 1, s > 0,(1.8)
vy(0, s) D 0, v(1, s) D 1, s > 0,(1.9)
v(y, 0) D v0(y) WD u0(Ry)R , 0  y  1,(1.10)
where  D (R) WD R1 1=p.
It is well-known that the structure of steady states plays an important role in the
study of parabolic problem. Therefore, the aim of this paper is to study the structure
of steady states of (1.8)–(1.10) with  > 0 as a free parameter. Moreover, we also
study the structure of steady states of (1.1)–(1.3).
In Section 2, we first derive the structure of steady states of (1.8)–(1.10), by using
a method motivated by the works [3, 2]. We show that the problem (1.8)–(1.10) has a
unique steady state for each  > 0. This is quite surprising in comparing with the local
problem [9] and a similar nonlocal problem [5]. For some related works on nonlocal
parabolic problems, we also refer the reader to [1, 4, 5, 6, 7, 8].
Then we prove in Section 3 that a unique steady state for the problem (1.1)–(1.3)
exists for any p > 1. Also, no steady states exist if p 2 (0, 1]. Indeed, by (1.2), when
p 2 (0, 1] the equation (1.1) is reduced to the heat equation. Therefore, this case is not
physically relevant. Finally, some remarks on the global (in time) existence of solution
for (1.4)–(1.6) and some open problems are given in Section 4.
2. Steady states in finite interval
In this section, we shall study the structure of steady states of the approximated prob-
lem (1.4)–(1.6) in a finite interval. By the transformation (1.7), the problem (1.4)–(1.6) is
reduced to the problem (1.8)–(1.10) with  D (R) D R1 1=p for R > 0. In this section,
we always assume that p > 1. With  as a free parameter, the associated steady state of
(1.8)–(1.10) is a solution of
w
00
D 

Z 1
0
w
 p(y) dy

 1 1=p
w
 p
, 0 < y < 1,(2.1)
w
0(0) D 0, w(1) D 1.(2.2)
A NONLOCAL PARABOLIC PROBLEM 35
It is easy to see that any solution of (2.1)–(2.2) must be a solution of
W 00 D W p, 0 < y < 1,(2.3)
W 0(0) D 0, W (1) D 1,(2.4)
for a certain positive constant  . The problem (2.3)–(2.4) has been analyzed by Levine
[9]. It is shown that there exists a positive constant   such that the problem (2.3)–(2.4)
has exactly two solutions when  2 (0,  ); has exactly one solution when  D  ;
and has no solutions when  >  . It is nature to expect that the structure of solutions
of (2.1)–(2.2) is similar to that of (2.3)–(2.4). It turns out that this is not the case as
shown below.
For the structure of solutions of (2.1)–(2.2), we shall modify a method used in
[2, 3]. Let w be a solution of (2.1)–(2.2). Then w0 > 0 in (0, 1] and the minimum
 WD w(0) 2 (0, 1). Multiplying (2.1) by w0 and integrating it over [0, y], using w0 > 0
we deduce that
(2.5) w
0(y)
p

1 p
  w
1 p(y)
D
s
2
p   1
Y (1C1=p)=2.
Then, by integrating (2.5) over y 2 [0, 1], we end up with
G() D
s
2
p   1
Y (1C1=p)=2,(2.6)
where
Y WD
Z 1
0
w
 p(y) dy,(2.7)
G() WD
Z 1

ds
p
F()   F(s) , F(s) WD s
1 p
.(2.8)
In order to eliminate Y in (2.6), we introduce the following transformation:
z WD
R y
0 w
 p(s) ds
Y
, 0  y  1, h(z) WD w (y),  WD 1   p < 0.
Note that y 2 [0, 1] if and only if z 2 [0, 1] and h satisfies the equation
(2.9) h00(z) D  (p   1)Y 1 1=p, z 2 [0, 1].
Moreover, we have
h0(0) D 0, h(1) D 1, h0(z) < 0 for z 2 (0, 1]
36 Y.-J.L. GUO
and  WD h(0) D  2 (1, 1) is the maximum of h. As before, we deduce from (2.9)
the following relation
(2.10) H () WD
Z

1
dh
p
   h
D
p
2(p   1)Y (1 1=p)=2.
Note that H () D 2(   1)1=2 for  > 1. Hence by (2.10) we obtain
(2.11) Y 1 1=p D 2(   1)
(p   1) .
Plugging (2.11) into (2.6), we end up with the following relation between the param-
eters  and  D w(0):
(2.12)  D

2
p   1
1=p
G()(p 1)=p(   1)(pC1)=(2p) WD

2
p   1
1=p
K ()(p 1)=p.
By the transformation
a2 WD 1  

s

1 p
,  D () WD
p
1   p 1,
we compute from (2.8) that
(2.13) G() D 2
p   1

(pC1)=2
Z

0
(1   a2) p=(p 1) da.
It is trivial that G(1 ) D 0 and so K (1 ) D 0. From (2.12) and (2.13), we obtain
(2.14) K () D 2
p   1
(1   p 1)(pC1)=[2(p 1)]L(),
where
L() WD
Z

0
(1   a2) p=(p 1) da,  D
p
1   p 1.
We compute
L 0() D   p   1
2

 2(1   p 1) 1=2 < 0.
A NONLOCAL PARABOLIC PROBLEM 37
Therefore, K 0() < 0 for all  2 (0, 1). For  > 0 close to zero, we have
L() 
Z

1=2
(1   a2) p=(p 1) da

Z

1=2
a(1   a2) p=(p 1) da
D
p   1
2


 1
 

3
4

 1=(p 1)
.
Hence L() !1 as ! 0C.
Since the above procedure in obtaining the relations between w,  and  is re-
versible, we thus have proved the following theorem.
Theorem 1. Given p > 1 fixed. The problem (2.1)–(2.2) has a unique solution
for each  > 0.
3. Steady states in half line
In this section, we study the positive steady states of (1.1)–(1.3), i.e., positive so-
lutions of the following boundary value problem:
U 00 D
U p
 R
1
0 U p dx
1C1=p , 0 < x <1,(3.1)
U 0(0) D 0, U 0(1) D 1,(3.2)
where p is a positive constant. To study this problem, we first study the related bound-
ary value problem:
w
00
D w
 p
, 0 < x <1,(3.3)
w
0(0) D 0, w0(1) D 1,(3.4)
where  is any positive constant. Note that w0 > 0 in (0,1) and w(x)!1 as x !1
for any positive solution w of (3.3)–(3.4).
Multiplying the equation (3.3) by w0, by an integration we obtain that
[w0(x)]2
2
D

1   p
[w1 p(x)   w1 p(0)] for p ¤ 1,(3.5)
[w0(x)]2
2
D [ln w(x)   ln w(0)] for p D 1(3.6)
for all x > 0 for any positive solution w of (3.3)–(3.4).
If p 2 (0, 1), then, by letting x ! 1 in (3.5), we obtain a contradiction. This
implies that (3.3)–(3.4) has no positive solutions for any  > 0, if p 2 (0, 1). Similarly,
38 Y.-J.L. GUO
using (3.6) we can show that (3.3)–(3.4) has no positive solutions for any  > 0, if
p D 1.
For p > 1, set  WD w(0). Then, by letting x !1 in (3.5), we have the identity
(3.7)  D p   1
2

p 1
.
This implies that there exists at most one solution of the problem (3.3)–(3.4) for a
given  > 0. This unique solution is the solution of the initial value problem:
(3.8) w00 D w p, x > 0, w0(0) D 0, w(0) D ,
where  D () is defined by (3.7). We shall denote this solution by w

for a given
. Indeed, it is easily check that w

(x) D w1(x=). In particular, w1 is the solution
of the initial value problem
(3.9) w00 D p   1
2
w
 p
, x > 0, w0(0) D 0, w(0) D 1.
Next, we show that the (local) solution w1 of (3.9) exists globally in [0, 1). In-
deed, if w1 is not global, then there is R <1 such that w1(R ) D1. It follows that
w
00
1 (R ) D 0. Note that w001 > 0 in [0, R). Then
w
0
1(x) D
Z x
0
w
00
1 (y) dy  R max
z2[0, R)
w
00
1 (z) <1
for all x 2 [0, R). This implies that w01(x) is bounded in [0, R), a contradiction. There-
fore, we conclude that w1 exists globally in [0, 1). Notice that w01(1) D 1.
We thus have proved the following proposition.
Proposition 3.1. Let p > 1. For any given  > 0, there exists a unique positive
solution of the boundary value problem (3.3)–(3.4). This unique solution is given by
w

which is the solution of the initial value problem (3.8) with  satisfying the rela-
tion (3.7).
Now, we return to the problem (3.1)–(3.2). Since U 0(1) D 1 for a positive solution
U of (3.1)–(3.2), we see that U (x)  x as x !1. Hence the integral R10 U p(x) dx
is finite. This implies that U D w

for some positive  satisfying
(3.10) p   1
2

p 1

Z
1
0
w
 p

(x) dx
1C1=p
D 1.
We compute that
Z
1
0
w
 p

(x) dx D  p
Z
1
0
w
 p
1

x


dx
D 
1 p
Z
1
0
w
 p
1 (y) dy.
A NONLOCAL PARABOLIC PROBLEM 39
It follows from (3.10) that  satisfies
(3.11) p   1
2

 1C1=p

Z
1
0
w
 p
1 (y) dy
1C1=p
D 1.
The relation (3.11) defines  uniquely. Hence we have the following existence and
uniqueness theorem.
Theorem 2. For any p > 1, there exists a unique positive solution of the bound-
ary value problem (3.1)–(3.2).
For p 2 (0, 1], we have
Z
1
0
U p(x) dx D1
for any solution U of (3.1)–(3.2). Hence a classical solution U of (3.1)–(3.2) exists
only if U 00  0 in [0, 1). This is impossible, since U 0(0) D 0 and U 0(1) D 1. There-
fore, we conclude that there is no solution of (3.1)–(3.2), if p 2 (0, 1].
4. Remarks and discussions
We recall from [9] that the solution of
Vs D Vyy   V p, 0 < y < 1, s > 0,(4.1)
Vy(0, s) D 0, V (1, s) D 1, s > 0,(4.2)
V (y, 0) D v0(y), 0  y  1,(4.3)
with W 

(y)  v0(y)  1 for y 2 [0, 1] and  2 (0,  ), exists globally in time, where
W 

is the minimal solution of (2.3)–(2.4).
Now we suppose that v is a solution of (1.8)–(1.10) with v0 2 (0, 1]. Then we have
g(sI v) WD

Z 1
0
v
 p(y, s) dy

 1 1=p
2 (0, 1)
as long as v exists. Let V be the solution of (4.1)–(4.3) with  D . It follows from
the comparison principle that v  V as long as v exists. Therefore, we conclude that
Theorem 3. Suppose that  2 (0,  ). If W 

(y) < v0(y)  1 for y 2 [0, 1], then
the solution v of (1.8)–(1.10) exists globally in time.
Hence, when R is small enough and W 

(x=R) < u0(x)=R  1 for x 2 [0, R], the
solution of (1.4)–(1.6) exists globally in time.
40 Y.-J.L. GUO
We also recall from [9] that the solution V of (4.1)–(4.3) tends to the maximal
solution WC

of (2.3)–(2.4) as t ! 1, if v0 > W 

. It should be very interesting to
see if such result also holds for the nonlocal problem (1.4)–(1.6). In [8], numerical
simulations indicate that, for p > 1, the solution u of (1.1)–(1.3) exists globally and
tends to the unique steady state as t ! 1. It is also very interesting to see, under
certain condition, whether the solution of (1.1)–(1.3) exists globally and converges to
the unique steady state as t !1. We leave these two questions for the future study.
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Department of Mathematics
National Taiwan Normal University
88, S-4 Ting Chou Road
Taipei 11677
Taiwan
e-mail: yjguo@math.ntnu.edu.tw


A nonlocal parabolic problem arising in linear friction welding

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A nonlocal parabolic problem arising in linear friction welding

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