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Asymptotic Behavior and Convergence of Solutions of a Semilinear Transport Equation with Delay
Journal of Mathematical Analysis and Applications 254, 464᎐483 Ž2001. doi:10.1006rjmaa.2000.7189, available online at http:rrwww.idealibrary.com on
Asymptotic Behavior and Convergence of Solutions of a Semilinear Transport Equation with Delay He Mengxing 1 and Luo Ronggui Department of Mathematics and Physics, Wuhan Uni¨ ersity of Technology, Wuhan Hubei 430070, People’s Republic of China Submitted by G. F. Webb Received May 20, 1999
This paper analyses the behavior and convergence of the solutions for a generalized singular transport equation which arises as a model of the blood production system by using the characteristic theory of first order partial differential equations and iterative methods. 䊚 2001 Academic Press Key Words: characteristic theory; iterative method; asymptotic behavior; regularity; convergence.
1. INTRODUCTION The blood production model
⭸u ⭸t
q
⭸ ⭸x
Ž xu . s u Ž t y , ␣ x . Ž 1 y u Ž t y , ␣ x . .
Ž 1.1.
was proposed by Rey and Mackey in w4x, where uŽ t, x . is the population density of cells with respect to maturity x at time t and , ␣ , are parameters satisfying ) 0, 0 - ␣ F 1, ) 0. The maturity variable x has values in w0, 1x. In w3x, Dyson et al. generalized Ž1.1. to the more general form
⭸u ⭸t
q
⭸ ⭸x
Ž g Ž x . u. s f Ž uŽ t y , ␣ x . . .
Ž 1.2.
For simplicity, this paper still considers Eq. Ž1.2., but there are two points different from w3x. Ži. We transform the differential equation Ž1.2. into an integral equation by making use of the characteristic theory of first order 1
This work was supported by the National Science Foundation of China. 464
0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
465
SEMILINEAR TRANSPORT EQUATION
partial differential equations. This method not only has obvious geometric significance, but also may apply to multidimensional problems and systems of differential equations. Žii. We will stress the existence and asymptotic behavior of nontrivial steady state solutions, regularity, and convergence of solutions. The content of this paper is organized as follows: In Section 2 the characteristic line method is presented; in Section 3 the asymptotic behavior of the solution is discussed; in Section 4 the regularity of solutions is studied; in Section 5 the existence and stability of the nontrivial steady state solution is discussed. The convergence of solutions as the time lag approaches zero is investigated in Section 6. The generalization of the above results is presented in the final part of this article.
2. CHARACTERISTIC LINE METHOD Consider the initial problem
⭸u ⭸t
q
⭸ ⭸x
t ) 0, x g w 0, 1 x ,
Ž g Ž x . u. s f Ž uŽ t y , ␣ x . . ,
Ž 2.1.
t g w y , 0 x , x g w 0, 1 x ,
u Ž t , x . s Ž x, t . ,
where f Ž y . and Ž t, x . are continuous, and g Ž x . satisfies g g C 1 Ž w 0, 1 x . ,
g Ž 0 . s 0,
g Ž x . ) 0 for 0 - x F 1
1
and
H0
dx gŽ x.
s ⬁.
Ž H.
According to the characteristic theory of first order partial differential equations, the ordinary differential system dx dt du dt
s gŽ x. , s yg Ž x . u q f Ž u Ž t y , ␣ x . .
Ž 2.2.
is called the characteristic equation of Ž2.1.. Geometrically, the integral surface of Ž2.1. can be identified with an integral curve of Ž2.2.. We defined h Ž x . s exp
ž
x
H1
d gŽ .
/
.
Ž 2.3.
466
MENGXING AND RONGGUI
Obviously, hŽ1. s 1, hŽ0. s 0; moreover, hŽ x . is an monotone increasing function. From the first equation of Ž2.2. and Ž2.3., we have dh h
s dt.
Ž 2.4.
Given initial conditions h Ž x . < ts0 s h Ž x 0 . , u Ž t , x . < ts0 s Ž 0, x 0 . ,
0 F x 0 F 1,
Ž 2.5.
it is easy to see from Ž2.4. and Ž2.5. that x Ž t . s hy1 Ž h Ž x 0 . e t . ,
Ž 2.6.
x 0 s hy1 Ž h Ž x . eyt . .
Considering the initial condition Ž2.6. and the second equation of Ž2.2., the relation t
H0 g ⬘ Ž x Ž . . d
u Ž t , x . s exp y
ž
t
/
Ž 0, x 0 .
t
H0 exp H g ⬘ Ž x Ž . . d
q
ž
f u y , ␣ x Ž . . . d Ž 2.7.
/ŽŽ
is achieved. Then, substituting Ž2.6. into Ž2.7., we obtain the integral equation u Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
= f Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . . d , uŽ t , x . s Ž t , x . ,
t g w y , 0 x ,
/
t)0, xg w 0, 1 x ,
x g w 0, 1 x ,
Ž 2.8.
which was and obtained in w3x by using the linear operator semigroup. The continuous solution of the integral equation Ž2.8. is called a mild solution of the differential equation Ž2.1.. The existence and uniqueness of a mild solution of Ž2.8. is easily proved by using the step method. According to the characteristic theory of first order partial differential equations, when f, are once continuous differentiable and g is twice
467
SEMILINEAR TRANSPORT EQUATION
continuous differentiable, Ž2.8. is equivalent to Ž2.1. and the solution of Ž2.8. is exactly the integral surface of Ž2.1..
3. ASYMPTOTIC BEHAVIOR OF SOLUTION We discuss two cases for time lag ; that is, ) 0 and s 0. 3.1. The time lag ) 0 We introduce the following assumptions: ŽH 1 . f Ž0. s 0 and there is a constant ␦ ) 0 such that < f Ž y1 . y f Ž y 2 . < F L < y1 y y 2 < , as < y 1 < F ␦ , < y 2 < F ␦ , where the positive number L is replaced by LŽ ␦ ., which is an increasing function of ␦ ; ŽH 2 . < Ž t, x .< F for t g wy , 0x, x g w0, 1x, where 0 - F ␦ ; ŽH 3 . I s inf g ⬘Ž x .4 ) 0 for x g w0, 1x, and there is a small enough ) 0 such that L - Iey . Now we discuss the asymptotic nature of solutions. We adopt an iterative method, and choose the initial iterative
¡ Ž0, h
u
Ž0.
y1
Ž t , x . s~
Ž h Ž x . eyt . . exp
¢ Ž t , x . ,
y
t
y1
H0 g ⬘ Ž h
ž
t g w y , 0 x ,
Ž h Ž x . eyŽ ty . . . d
x g w 0, 1 x .
t G 0,
/
,
x g w 0, 1 x ,
Ž 3.1. Define the kth approximation uŽ k . Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
= f Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . . d , uŽ k . Ž t , x . s Ž t , x . ,
t g w y , 0 x ,
x g w 0, 1 x ,
From Ž3.1., it is easy to see that < uŽ0. Ž t , x . < F
/
tG0, xg w 0, 1 x ,
k s 1, 2, . . . .
Ž 3.2.
468
MENGXING AND RONGGUI
for t G y , x g w0, 1x. When k s 1, from Ž3.2. we have t ylŽ ty .
L uŽ0. Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . d
t ylŽ ty .
Ž LrI . I d
uŽ1. Ž t , x . F eyI t q
H0 e
F eyI t q
H0 e
F eyI t q Ž LrI . Ž 1 y eyI t . F eyI t Ž 1 y Ž LrI . . q Ž LrI . F for t G y , x g w0, 1x. Using an induction argument, it is not difficult to verify that the inequality < uŽ k . Ž t , x . < F
Ž 3.3.
holds for t G y , x g w0, 1x, k s 1, 2, . . . . From Ž3.2., obviously uŽ1. y uŽ0. s
t
t
y1
H0 exp yH g ⬘ Ž h
ž
Ž h Ž x . eyŽ ty . . . d
/
=f Ž uŽ0. Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d s
t
t
y1
H0 exp yH g ⬘ Ž h
ž
Ž h Ž x . eyŽ ty . . . d
/
Ž t y q . ey Ž ty q .
=f Ž uŽ0. Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d , and it follows that < uŽ1. y uŽ0. < F
t
H0 exp Ž y Ž I y . Ž t y . q . =Ley Ž ty q . uŽ0. Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . d .
When G 0, y G 0, we have < uŽ0. Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . < F eyl Ž y . F ey Ž y . . Thus < uŽ1. y uŽ0. < F
L
t
y Ž ty q .
H0 exp Ž y Ž I y . Ž t y . q . Ž 1 y . e = Ž I y . ey Ž y . d
F ey t
ž
L 1y
/
F a ey t ,
469
SEMILINEAR TRANSPORT EQUATION
where a s LrŽ I y . - 1. By using induction, we obtain for any positive integral k the inequality < uŽ k . y uŽ ky1. < F a k ey t
Ž 3.4.
for t G 0, x g w0, 1x. This shows that the sequence uŽ n.4 is uniformly convergent on w0, T x = w0, 1x, where T is any positive number. Let lim k ª⬁ uŽ k . Ž t, x . s uŽ t, x ., and it is easily seen that uŽ t, x . is a unique continuous solution of Ž2.8.. Since uŽ k . Ž t , x . F Ž 1 q a q a2 q ⭈⭈⭈ a k . ey t for t G 0, x g w0, 1x, it follows that uŽ t , x . F
ž
1 1ya
/
ey t .
Ž 3.5.
This shows uŽ t, x . tending to zero exponentially as t tends to infinity uniformly for 0 F x F 1. Summarizing the above discussions, we have THEOREM 3.1. If assumptions ŽH., ŽH 1 . ᎐ ŽH 3 . are satisfied, then Ž2.8. has a unique solution uŽ t, x ., which tends to zero exponentially as t tends to infinity uniformly for x g w0, 1x. 3.2. The time lag s 0 Consider the following problem:
⭸u
⭸
t ) 0, Ž g Ž x . u. s f Ž uŽ t , ␣ x . . , ⭸t ⭸x u < ts0 s Ž x, t . , t g w y , 0 x , x g w 0, 1 x . q
x g w 0, 1 x ,
Ž 3.6.
Then the corresponding integral equation can be written as u Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
/
= f Ž u Ž , ␣ hy1 Ž h Ž x . eyŽyty . . . . d ,
tG0,
xg w 0, 1 x .
Ž 3.7.
470
MENGXING AND RONGGUI
If assumptions ŽH., ŽH 1 ., ŽH 2 . hold and there is small enough ) 0 such that I )Lq , Ž 3.8. then the estimate Ž3.5. is also satisfied for the solution of Ž3.7.. Because the proof is analogous to the proof of Theorem 1, we do not repeat it.
4. REGULARITY OF SOLUTION In this section, we consider
⭸u
⭸
t ) 0, Ž g Ž x . u. s f Ž uŽ t y , ␣ x . . , ⭸t ⭸x u Ž t , x . s Ž x, t . , t g w y , 0 x , x g w 0, 1 x , q
x g w 0, 1 x ,
Ž 4.1.
where is a small parameter. Obviously, Ž1.1. is special form of Ž4.1.. We further suppose that f, are once continuously differentiable, f Ž0. / 0, g g C 2 Žw0, 1x., and there is a positive ␥ such that g Ž x . ( x for 0 F x F ␥ . From Section 2, there exists a unique uŽ t, x . satisfying the integral equation u Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
t
q
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
= f Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . . d , t g w y , 0 x ,
uŽ t , x . s Ž t , x . ,
/ Ž 4.2.
t)0, xg w 0, 1 x ,
x g w 0, 1 x .
Namely uŽ t, x . is a mild solution of Ž4.1.. Differentiating Ž4.1. with respect to x on w0, x = w0, 1x, we get
⭸u ⭸x
s
⭸ ⭸x
Ž 0, hy1 Ž h Ž x . eyt . . hy1 Ž h Ž x . eyt . ⬘h⬘ Ž x . eyt
= exp y
t
y1
H0 g ⬘ Ž h
ž
Ž h Ž x . eyŽ ty . . . d
/
q Ž 0, hy1 Ž h Ž x . eyt . . =
t
½H
0
g ⬙ Ž hy1 Ž h Ž x . eyŽ ty . . . hy1 Ž h Ž x . eyŽ ty . . ⬘h⬘ Ž x . eyt d
= exp y
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
/
5
Ž 4.3.
471
SEMILINEAR TRANSPORT EQUATION
q
t
H0
½
y
t
y1
H g ⬙ Ž h
= exp y
t
y1
H g ⬘ Ž h
ž
hy1 Ž h Ž x . eyŽ ty . . ⬘h⬘ Ž x . eyt d
Ž h Ž x . eyŽ ty . . .
Ž h Ž x . eyŽ ty . . . d
5
/
= f Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d q
t
t
y1
H0 exp yH g ⬘ Ž h = =
ž
⭸f
⭸x
/
Ž Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . .
⭸u ⭸
Ž h Ž x . eyŽ ty . . . d
Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . .
=␣ hy1 Ž h Ž x . eyŽ ty . . ⬘h⬘ Ž x . eyŽ ty . d . This shows that if w hy1 Ž hŽ x . eyt .x⬘hy1 eyt is bounded for x g w0, 1x, then all terms of right hand on Ž4.1. are L1 integrable on w0, x. According to the definition of hŽ x ., it is easy to see that if w hy1 Ž hŽ x . eyt .x⬘hy1 eyt is bounded for 0 F x F ␥ , then it is also bounded for x g w0, 1x. Noting that h⬘Ž x . s hŽ x .rg Ž x ., when 0 F x F ␥ , we have hy1 Ž h Ž x . eyt . ⬘h1 eyt s s
g Ž h Ž x . eyt . h Ž h Ž x . eyt . h Ž x . eyt h Ž h Ž x . eyt .
s eyt exp
x
= =
h Ž x . eyt gŽ x. g Ž h Ž x . eyt . gŽ x.
d
y exp
H1hŽ x . e
ž ž / ž Ž H ž / ž H1
d
x
s eyt exp ( eyt
gŽ .
h Ž x . eyt
x yt
hŽ x . e
=
gŽ .
hŽ x . x
=
yt
d gŽ .
g h Ž x . eyt . gŽ x.
/
=
g Ž h Ž x . eyt . gŽ x.
/
/
s eyt .
Thus, integral in Ž4.3. exists for t g w0, x, x g w0, 1x, and from w2x, it is easy to see that ⭸ uŽ t, x .r⭸ x exists for t g w0, x, x g w0, 1x. Using the method
472
MENGXING AND RONGGUI
of steps, we obtain that ⭸ ur⭸ x exists for t G , x g w0, 1x, and from Ž4.1., ⭸ ur⭸ x exists for t G , x g w0, 1x. Suppose that
THEOREM 4.1.
Ž1. f g C R ., f Ž0. / 0, and g C 1 Žwy , 0x = w0, 1x.; Ž2. ŽH. holds; Ž3. g g C 2 Žw0, 1x., I s inf g ⬘Ž x .4 for x g w0, 1x, and there exists constant ␥ ) 0 such that g Ž x . ( x for 0 F x F ␥ . Then ⭸ ur⭸ t and ⭸ ur⭸ x exists and, moreo¨ er, satisfy Ž4.1.. 1Ž
Now consider
⭸u
⭸
Ž g Ž x . u. s f Ž uŽ t , ␣ x . . , ⭸t ⭸x u Ž t , x . < ts0 s Ž x . , x g w 0, 1 x , q
t ) 0,
x g w 0, 1 x ,
Ž 4.4.
which corresponds to the case s 0 in Ž4.1.. Obviously, a similar conclusion to Theorem 4.1 holds and the proof is also the same.
5. EXISTENCE AND STABILITY OF NONTRIVIAL STEADY STATE SOLUTION Consider d
Ž gŽ x. . s f Ž Ž ␣ x. . , dx limq Ž x . s Ž 0 . ,
x g w 0, 1 x ,
Ž 5.1.
xª0
where the initial value Ž0. satisfies
Ž 0 . s f Ž Ž 0 . . rg ⬘ Ž 0 . .
Ž 5.2.
A nontrivial solution of Ž5.1. is called a nontrivial steady state solution of Ž4.1. and Ž4.4., because the steady state problems of both are the same. It is easy to see that solution of Ž5.1. can be written as
Ž x. s Choose C s y
1
Ž . ž
g x
g Ž 1. C q
x
H1 f Ž Ž ␣ . . d
/
.
0 gŽ x. 1
H f Ž Ž ␣ x .. dx, and we get
Ž x. s
x
H f Ž Ž ␣ . . d , gŽ x. 1
x g w 0, 1 x .
Ž 5.3.
473
SEMILINEAR TRANSPORT EQUATION
We now prove the nontrivial solution of Ž5.1. exists and it is unique. Define the set S s w Ž x . < w g C w 0, 1 x , w Ž 0 . s f Ž w Ž 0 . . rg ⬘ Ž 0 . ,
½
sup < w < F ␦ 0FxF1
5
and define the mapping
s Tw s
x
H f Ž w Ž ␣ . . d gŽ x. 0
Ž 5.4.
on S. Obviously, T is continuous for w g S. Noting that x Ž x. s f Ž w Ž ␣ . . d , gŽ x. 0 we then have x < Ž x . < F f Ž w Ž ␣ . . d gŽ x. 0
H
H
F F
x
gŽ x.
gŽ x.
L
H0
f Ž w Ž ␣ . . y f Ž 0 . d q
L sup < w < x q
gŽ x.
x
H0 f Ž 0. d
< f Ž 0. < x
< f Ž 0. <
F Ž L q f Ž 0 . . ␦rI. I I Thus, when 0 - F IrŽ L q < f Ž0.<., < < F ␦ , it follows that the mapping T : S ª S. From Ž5.4. we have x 1 y 2 s f Ž w 1 Ž ␣ . . y f Ž w 2 Ž ␣ . . d . gŽ x. 0 F
␦q
gŽ x.
H
It follows that <1 y 2 < F F
x
H gŽ x. 0 L gŽ x.
f Ž w 1 Ž ␣ . . y f Ž w 2 Ž ␣ . . d
sup < w 1 y w 2 < x 0FxF1
F L sup < w 1 y w 2 < 0FxF1
F
L I
sup < w 1 y w 2 < . 0FxF1
1
žH
0
g ⬘ Ž x . d
/
474
MENGXING AND RONGGUI
This shows that T is a contraction mapping, so that there exists a unique Ž x . g S satisfying
Ž x . s T ;
Ž 5.5.
namely, from Ž5.4. there exists a unique solution
Ž x. s
x
H f Ž Ž ␣ . . d , gŽ x. 0
x g w 0, 1 x .
Obviously, Ž x . g C w0, 1x and Ž x . / 0. Noting that g ⬘ q g ⬘ s f Ž Ž ␣ x .. and Ž0. s Ž rg⬘Ž0.. f Ž Ž0.., we can define ⬘Ž0. s 0, and then ⬘Ž x . exists on 0 F x F 1 and satisfies Ž5.1.. Summarizing the above discussions, we have THEOREM 5.1. If the assumptions of Theorem 4.1 are satisfied and equality Ž5.2. holds, then from Ž4.1. and Ž4.4., there exists a unique nontri¨ ial steady state solution. In the following we discuss the stability of the steady state solution. Before doing this, we show that if sup < < F F ␦ for Ž t, x . g wy , 0x = w0, 1x, then the inequality < uŽ t , x . < F
Ž 5.6.
holds for t G y , x g w0, 1x, where uŽ t, x . is a solution of Ž4.1.. From Ž4.2., we have u Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
/
= f Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . . y f Ž 0 . q f Ž 0 . d , and it follows that u Ž t , x . F eyI t q q
t yIŽ ty .
H0 e
t yI Ž ty . <
H0 e
L u Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . d
f Ž 0. < d .
Thus, when u Ž y , ␣ hy1 Ž h Ž x . eyŽ ty . . . s Ž y , ␣ hy1 Ž h Ž x . eyŽ ty . . . ,
Ž t , x . g w y , 0 x = w 0, 1 x ,
475
SEMILINEAR TRANSPORT EQUATION
then < u Ž t , x . < F eyI t q q
t yIŽ ty .
L Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . d
H0 e
< f Ž 0. < I
Ž 1 y eyI t .
½
F eyI t 1 y
I
ž
< f Ž 0. <
Lq
q
/ ž I
Lq
< f Ž 0. <
/5
F . Using the method of steps, the estimate Ž5.6. is obtained. Let w s u y , and from Ž4.1. and Ž5.1., we have
⭸w ⭸t
q
⭸ ⭸x
Ž g Ž x . w . s f Ž uŽ t y , ␣ x . . y f Ž Ž ␣ x . . , t ) 0,
x g w 0, 1 x ,
Ž 5.7.
w Ž t , x . s Ž x, t . y Ž x . s Ž t , x . , t g w y , 0 x ,
x g w 0, 1 x ;
namely
⭸w ⭸t
q
⭸ ⭸x
Ž gw . s F Ž u Ž t y , ␣ x . . w Ž t y , ␣ x . , t g w y , 0 x ,
wŽ t, x. s Ž t, x. ,
Ž 5.8.
x g w 0, 1 x ,
t ) 0,
x g w 0, 1 x .
where F s H0t f ⬘Ž Ž x . q Ž uŽ t y , ␣ x . y Ž x ... d , and w Ž t, x . satisfies w Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d y1
= F Ž uŽ y , ␣ h
/
yŽy ty .
Ž hŽ x . e
Ž 5.9.
...
= w Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . d , w Ž t , x . s Ž t , x . ,
/
t)0,
Ž t , x . g w y , 0 x , xg w 0, 1 x .
xg w 0, 1 x ,
476
MENGXING AND RONGGUI
In the following we prove stability. We adopt an iterative method, and choose the initial approximation w Ž0. Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . = exp y
t
Ž h Ž x . eyŽ ty . . . d
y1
H0 g ⬘ Ž h
ž
tg w y , 0 x ,
wŽ t, x. s Ž t, x. ,
/
tG0, xg w 0, 1 x Ž 5.10.
,
xg w 0, 1 x .
Define the kth approximation w Ž k . Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
/ Ž 5.11.
= F Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . = w Ž ky1. Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . d , t)0,
xg w 0, 1 x ,
Ž t , x . g w y , 0 x , xg w 0, 1 x .
wŽk. Ž t , x . s Ž t , x . ,
Noting that sup < < F 2 for t g wy , 0x, x g w0, 1x, from Ž5.10., it is easy to see that w Ž0. Ž t , x . s
½
t G 0, x g w 0, 1 x , t g w y , 0 x , x g w 0, 1 x .
2 eyI t , 2 ,
Ž 5.12.
From Ž5.11. and by similar arguments to Section 3, we get < w Ž k . Ž t , x . < F Ž 1 q a q a2 q ⭈⭈⭈ qa k . 2 ey t
Ž 5.13.
< w Ž k . y w Ž ky1. < F a k 2 ey t
Ž 5.14.
and for t G 0, x g w0, 1x, k s 0, 1, 2, . . . . This shows that the sequence w Ž k .4 is convergent uniformly on an arbitrary compact region w0, T x = w0, 1x. Let lim k ª⬁w Ž k . s w Ž t, x ., and it is not difficult to verify that w Ž t, x . is a unique solution of Ž5.9.. From Ž5.13., obviously wŽ t, x. F
ž
2 1ya
/
ey t
Ž 5.15.
477
SEMILINEAR TRANSPORT EQUATION
for t G 0, x g w0, 1x, and it follows that w Ž t, x . tends to zero exponentially as t tends to q⬁ uniformly with respect to x g w0, 1x. THEOREM 5.2. Let the hypothesis of Theorem 4.1 be satisfied. Then inequality Ž5.15. holds for t G 0, x g w0, 1x; namely, uŽ t, x . tends to Ž x . exponentially as t tends to q⬁. For the problem Ž4.4., the corresponding conclusion holds also.
6. CONVERGENCE OF SOLUTION In this section, with Ž0, x . s Ž x ., we investigate whether the mild solution of Ž4.1. converges to a mild solution of Ž4.4. as the time lag tends to 0. Let T˜ be a fixed positive number and let n be a positive integer so large ˜ F . Put n s Trn. ˜ that Trn We denote by n the restriction of on wy , 0x = w0, 1x and let u nŽ t, x . be a mild solution of the problem
⭸u
⭸
Ž g Ž x . u . s f Ž u Ž t y n , ␣ x . . , n G 0, x g w 0, 1 x , Ž 6.1. ⭸t ⭸x u Ž t , x . s n Ž x, t . , t g w yn , 0 x , x g w 0, 1 x ; q
namely, u nŽ t, x . satisfies u n Ž t , x . s n Ž 0, hy1 Ž h Ž x . eyt . . = exp y q
t
y1
H0 g ⬘ Ž h
ž
t
t
Ž h Ž x . eyŽ ty . . . d
y1
H0 exp yH g ⬘ Ž h
ž
/
Ž h Ž x . eyŽ ty . . . d
/
Ž 6.2.
= f Ž u n Ž y n , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d , tG0, xg w 0, 1 x , u Ž t , x . s n Ž t , x . ,
tg w yn , 0 x , xg w 0, 1 x ,
First, we estimate u n . For t g w0, yn x, x g w0, 1x, from Ž6.2., we have u n Ž t , x . s n Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
/
= f Ž n Ž y n , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d ,
Ž 6.3.
478
MENGXING AND RONGGUI
or t
u n Ž t , x . s n Ž 0, hy1 Ž h Ž x . eyt . . exp y q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
y1
H0 g ⬘ Ž h
ž
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
/
= f Ž n Ž y n , ␣ hy1 Ž h Ž x . eyŽyty . . . . yf Ž 0 . q f Ž 0 . d .
Ž 6.4.
It follows that < u n < F eyI t q q
t yIŽ ty .
L n Ž y n ␣ hy1 Ž h Ž x . eyŽ ty . . . d
H0 e
t yI Ž ty . <
H0 e
F eyI t q
½
L I
F eyI t 1 y
f Ž 0. < d
Ž 1 y eyI t . q I
ž
< f Ž 0. <
< f Ž 0. <
Lq
I
/
q
Ž 1 y eyI t .
I
ž
Lq
< f Ž 0. <
⬘
/5
F , where 0 - - IrŽ L q < f Ž0.
for t G yn ,
x g w 0, 1 x ,
n s 0, 1, 2, . . . . Ž 6.5.
Thus, the sequence u n4 is uniformly bounded on t G 0, x g w0, 1x. We now estimate ⭸ ur⭸ n. Differentiating Ž6.3. with respect to x, we have
⭸ un ⭸x
s
⭸ ⭸x
n Ž 0, hy1 Ž h Ž x . eyt . . hy1 Ž h Ž x . eyt .
=h⬘ Ž x . eyt exp y
ž
t
y1
H0 g ⬘ Ž h
qn Ž 0, hy1 Ž h Ž x . eyt . .
X x
Ž h Ž x . eyŽ ty . . . d
/
479
SEMILINEAR TRANSPORT EQUATION t
= y
y1
H0 g ⬙ Ž h
½
Ž h Ž x . eyŽ ty . . .
hy1 Ž h Ž x . eyŽ ty . .
X x
=h⬘ Ž x . eyt d t
=exp y
y1
H0 g ⬘ Ž h
ž
t
q
H0
½
t
y
Ž h Ž x . eyŽ ty . . . d y1
H g ⬙ Ž h
/
Ž h Ž x . eyŽ ty . . .
hy1 Ž h Ž x . eyŽ ty . . =h⬘ Ž x . eyt d
t
=exp y
y1
H g ⬘ Ž h
ž
5
Ž h Ž x . eyŽ ty . . . d
X x
5
/
= f Ž n Ž y n , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d t
q
t
y1
H0 exp yH0 g ⬘ Ž h
= =
⭸f ⭸x
⭸ ⭸x
ž
Ž h Ž x . eyŽ ty . . . d
/
Ž n Ž y n , ␣ hy1 Ž h Ž x . eyŽ ty . . . .
n Ž y n , ␣ hy1 Ž h Ž x . eyŽyty . . .
=␣ hy1 Ž h Ž x . eyŽ ty . .
X x
h⬘ Ž x . eyŽ ty . d
s I1 q I2 q I3 q I4 .
Ž 6.6.
X Notice that , xX , g ⬙, and w hy1 Ž hŽ x . eyt .x x h⬘Ž x . eyt are bounded, Ž nŽ t y n , ␣ hy1 Ž hŽ x . eyt . eyŽ ty . .., ⭸⭸ xf Ž nŽ t y n , ␣ hy1 Ž hŽ x . eyt . eyŽ ty . .. are uniformly bounded, and then there exist constants ⬘ ) 0, L⬘ ) 0, L⬙ ) 0 such that
< I1 < q < I2 < F ⬘eyI t , < I3 < F < I4 < F
t yIŽ ty .
H0 e
L⬙
t yI Ž ty .
H0 e
⭸n ⭸x
L⬘ d ;
Ž y n ␣ hy1 Ž h Ž x . eyŽ ty . . .
d .
480
MENGXING AND RONGGUI
It follows that
⭸n ⭸x
F ⬘eyI t q q
t yIŽ ty .
L⬘ d
H0 e
t yIŽ ty .
L⬙
H0 e
F ⬘eyI t q
L⬘ I
½
F eyI t 1 y
I
⭸n ⭸x
Ž y n ␣ hy1 Ž h Ž x . eyŽ ty . . .
Ž 1 y eyI t . q L⬙ q
ž
L⬘
⬘
/
L⬙ I
q
I
d
Ž 1 y eyI t .
ž
L⬙ q
L⬘
⬘
/5
F ⬘ for t g w0, n x, x g w0, 1x. Using the step method and induction, we obtain
⭸ un ⭸x
F ⬘
x g w 0, 1 x .
for t G 0,
Ž 6.7.
From Ž6.1., Ž6.5., and Ž6.7., it is not difficult to see that
⭸ un ⭸x
F ⬙
x g w 0, 1 x ,
for t G 0,
Ž 6.8.
where ⬙ is a positive constant which is independent of n. Equation Ž6.1. and estimates Ž6.5., Ž6.7., Ž6.8. show that the sequence u n4 is uniformly bounded and equicontinuous on an arbitrary compact region w0, T x = w0, 1x. Then, we can choose a subsequence u n j 4 which is uniformly convergent on w0, T x = w0, 1x. Let lim jª⬁ u n j s uŽ t, x ., and it is easily verified that uŽ t, x . satisfies u Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH0 g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
/
= f Ž u Ž , ␣ hy1 Ž h Ž x . eyŽyty . . . . d ,
tG0,
xg w 0, 1 x .
Ž 6.9. Namely, uŽ t, x . is a mild solution of Ž4.4.. Again, since the solution of Ž6.9. is unique, it follows that the sequence u n4 uniformly converges to uŽ t, x ..
481
SEMILINEAR TRANSPORT EQUATION
Summarizing the above discussions, we obtain THEOREM 6.1. Let the hypothesis of Theorem 4.1 be satisfied and suppose 0 - - IrŽ L q < f Ž0.<., Ž0, x . s Ž x .. Then the mild solution of Ž4.1. con¨ erges to the mild solution of Ž4.4. as the time lag n ª 0 Ž n ª q⬁.. 7. GENERALIZATION OF PROBLEM We only discuss two cases. 7.1. Multidimensional Case We consider
⭸u ⭸t
q
m
⭸
Ý
⭸ xi
is1
t ) 0, x i g w 0, 1 x ,
Ž gi Ž x i . u. s f Ž uŽ t y , ␣ x . . ,
u Ž t , x . s Ž x, t . ,
t g w y x , x i g w 0, 1 x , i s 1, 2, . . . , m,
Ž 7.1.
where x s Ž x 1 , . . . , x m ., ␣ s Ž ␣ 1 , . . . , ␣ m .T , 0 - ␣ i - 1. For f, g i , and , we suppose that ŽH. f, are continuous, and g i is continuously differentiable and satisfies g i Ž 0 . s 0,
g i Ž x i . ) 0, 0 - x i - 1,
1
H0
dx i gi Ž xi .
s ⬁.
As we have seen above, the characteristic equations of Ž7.1. are dx i dt du dt
s gi Ž xi . , m
sy Ý
Ž 7.2. g Xi
Ž x i . u q f Ž uŽ t y , ␣ x . . .
is1
Define h i Ž x i . s expŽ H1x d irg i Ž i .., and then h i Ž1. s 1, h i Ž0. s 0, and h i is a monotone increasing function and satisfies dh i hi
s dt.
Ž 7.3.
482
MENGXING AND RONGGUI
Consider the initial conditions h i Ž xUi . < ts0 s h i Ž x i0 . ,
0 F x i0 F 1,
u Ž t , x . < ts0 s Ž 0, x 0 . ,
Ž 7.4.
0 F x i0 F 1,
It is known from Ž7.3. and Ž7.4. that h i Ž x i Ž t .. s h i Ž x i0 . e t. So t x i Ž t . s hy1 i Ž h Ž x i0 . e . ,
Ž 7.5.
yt x i0 s hy1 ., i Ž hŽ xi . e
where i s 1, 2, . . . , m. Consider the initial condition Ž7.4. and the second equation of Ž7.2. to obtain the relation m
t X
u Ž t , x . s Ž 0, x 0 . exp Ý y is1 t
q
H0
m
exp y Ý
ž
is1
H0 g Ž x Ž s . . ds Ž 0, x
ž
i
i
t X
H0 g Ž x Ž s . . ds i
i
0
.
/
f u Ž s y , ␣ x Ž s . . . ds. Ž 7.6.
/Ž
Then, substituting Ž7.4., Ž7.5. into Ž7.6., we obtain u Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . m
= exp y Ý
ž
q
is1
t H0 g Ž h Ž h Ž x . e
m
t
H0 exp
ž
yÝ is1
X i
y1 i
i
yŽ ty .
i
. . d
t H g Ž h Ž h Ž x . e X i
y1 i
i
i
yŽ ty .
/
. . d
/
Ž 7.7.
= f Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d , t ) 0, uŽ t , x . s Ž t , x . ,
t g w y , 0 x ,
x i g w 0, 1 x ,
x g w 0, 1 x ,
Ž h1Ž x 1 . eyt ., . . . , hy1 Ž Ž . yt ... where hy1 Ž hŽ x . eyt . s Ž hy1 1 m hm xm e 7.2. System of Equation Case We consider
⭸ ui ⭸t
q
⭸ ⭸ xi
Ž g i Ž x i . u i . s fi Ž uŽ t y , ␣ x . . ,
u i Ž t , x . s i Ž x, t . ,
t g w y x ,
t ) 0, x g w 0, 1 x ,
Ž 7.8.
x g w 0, 1 x ,
where i s 1, 2, . . . , n, u s Ž u1 , . . . , u n ., x s Ž x 1 , . . . , x n ., ␣ s Ž ␣ 1 , . . . , ␣ n .T .
483
SEMILINEAR TRANSPORT EQUATION
It is well known that the characteristic equation is dx i dt du i dt
s gi Ž xi . , s
yg Xi u
Ž 7.9.
q
f iX
Ž uŽ t y , ␣ x . . ,
i s 1, 2, . . . , n,
where f, , and g satisfy again condition ŽH.. By means of the above method, we can transform Ž7.8. into the integral equation u i Ž t , x . s i Ž 0, hy1 Ž h Ž x . eyt . . t H0 g Ž h Ž h Ž x . e
= exp y
ž
X i
y1 i
i
yŽ ty .
i
t t H0 exp yH g Ž h Ž h Ž x . e
q
ž
X i
y1 i
i
i
. . d
yŽ ty .
/
. . d
/
Ž 7.10.
=f i Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d , t ) 0, u i Ž t , x . s i Ž t , x . ,
t g w y , 0 x ,
x i g w 0, 1 x , x i g w 0, 1 x ,
For Ž7.1. and Ž7.8., we can prove, under suitable conditions, that corresponding results hold. Because the methods are similar, we omit them. ACKNOWLEDGMENT The author appreciates the anonymous referee for his valuable comments and helpful suggestions on this paper.
REFERENCES 1. P. Brunowsky and J. Komornik, Ergodicity exactness of the shift on C w0, ⬁. and the semiflow of a first-order partial differential equation, J. Math. Anal. Appl. 104 Ž1984., 235᎐245. 2. J. Dyson, R. Villella-Bressan, and G. F. Webb, A singular transport equation modeling a proliferating maturity structured cell population, Canad. Appl. Math. Quart. 1 Ž1996., 65᎐95. 3. J. Dyson, R. Villella-Bressan, and G. F. Webb, A semilinear transport equation with delay, J. Integral Equations Appl. 8Ž1. Ž1996., 31᎐59. 4. A. O. Rey and M. C. Mackey, Multistability and boundary layer development in a transport equation with delayed arguments, Canad. Appl. Math. Quart. 1 Ž1993., 61᎐81. 5. G. F. Webb, Autonomous nonlinear functional differential equations and nonlinear semigroups, J. Math. Anal. Appl. 45 Ž1974., 1᎐12. 6. G. F. Webb, Periodic and chaotic behavior in structural models of cell population dynamics, to appear.
Asymptotic Behavior and Convergence of Solutions of a Semilinear Transport Equation with Delay He Mengxing 1 and Luo Ronggui Department of Mathematics and Physics, Wuhan Uni¨ ersity of Technology, Wuhan Hubei 430070, People’s Republic of China Submitted by G. F. Webb Received May 20, 1999
This paper analyses the behavior and convergence of the solutions for a generalized singular transport equation which arises as a model of the blood production system by using the characteristic theory of first order partial differential equations and iterative methods. 䊚 2001 Academic Press Key Words: characteristic theory; iterative method; asymptotic behavior; regularity; convergence.
1. INTRODUCTION The blood production model
⭸u ⭸t
q
⭸ ⭸x
Ž xu . s u Ž t y , ␣ x . Ž 1 y u Ž t y , ␣ x . .
Ž 1.1.
was proposed by Rey and Mackey in w4x, where uŽ t, x . is the population density of cells with respect to maturity x at time t and , ␣ , are parameters satisfying ) 0, 0 - ␣ F 1, ) 0. The maturity variable x has values in w0, 1x. In w3x, Dyson et al. generalized Ž1.1. to the more general form
⭸u ⭸t
q
⭸ ⭸x
Ž g Ž x . u. s f Ž uŽ t y , ␣ x . . .
Ž 1.2.
For simplicity, this paper still considers Eq. Ž1.2., but there are two points different from w3x. Ži. We transform the differential equation Ž1.2. into an integral equation by making use of the characteristic theory of first order 1
This work was supported by the National Science Foundation of China. 464
0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
465
SEMILINEAR TRANSPORT EQUATION
partial differential equations. This method not only has obvious geometric significance, but also may apply to multidimensional problems and systems of differential equations. Žii. We will stress the existence and asymptotic behavior of nontrivial steady state solutions, regularity, and convergence of solutions. The content of this paper is organized as follows: In Section 2 the characteristic line method is presented; in Section 3 the asymptotic behavior of the solution is discussed; in Section 4 the regularity of solutions is studied; in Section 5 the existence and stability of the nontrivial steady state solution is discussed. The convergence of solutions as the time lag approaches zero is investigated in Section 6. The generalization of the above results is presented in the final part of this article.
2. CHARACTERISTIC LINE METHOD Consider the initial problem
⭸u ⭸t
q
⭸ ⭸x
t ) 0, x g w 0, 1 x ,
Ž g Ž x . u. s f Ž uŽ t y , ␣ x . . ,
Ž 2.1.
t g w y , 0 x , x g w 0, 1 x ,
u Ž t , x . s Ž x, t . ,
where f Ž y . and Ž t, x . are continuous, and g Ž x . satisfies g g C 1 Ž w 0, 1 x . ,
g Ž 0 . s 0,
g Ž x . ) 0 for 0 - x F 1
1
and
H0
dx gŽ x.
s ⬁.
Ž H.
According to the characteristic theory of first order partial differential equations, the ordinary differential system dx dt du dt
s gŽ x. , s yg Ž x . u q f Ž u Ž t y , ␣ x . .
Ž 2.2.
is called the characteristic equation of Ž2.1.. Geometrically, the integral surface of Ž2.1. can be identified with an integral curve of Ž2.2.. We defined h Ž x . s exp
ž
x
H1
d gŽ .
/
.
Ž 2.3.
466
MENGXING AND RONGGUI
Obviously, hŽ1. s 1, hŽ0. s 0; moreover, hŽ x . is an monotone increasing function. From the first equation of Ž2.2. and Ž2.3., we have dh h
s dt.
Ž 2.4.
Given initial conditions h Ž x . < ts0 s h Ž x 0 . , u Ž t , x . < ts0 s Ž 0, x 0 . ,
0 F x 0 F 1,
Ž 2.5.
it is easy to see from Ž2.4. and Ž2.5. that x Ž t . s hy1 Ž h Ž x 0 . e t . ,
Ž 2.6.
x 0 s hy1 Ž h Ž x . eyt . .
Considering the initial condition Ž2.6. and the second equation of Ž2.2., the relation t
H0 g ⬘ Ž x Ž . . d
u Ž t , x . s exp y
ž
t
/
Ž 0, x 0 .
t
H0 exp H g ⬘ Ž x Ž . . d
q
ž
f u y , ␣ x Ž . . . d Ž 2.7.
/ŽŽ
is achieved. Then, substituting Ž2.6. into Ž2.7., we obtain the integral equation u Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
= f Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . . d , uŽ t , x . s Ž t , x . ,
t g w y , 0 x ,
/
t)0, xg w 0, 1 x ,
x g w 0, 1 x ,
Ž 2.8.
which was and obtained in w3x by using the linear operator semigroup. The continuous solution of the integral equation Ž2.8. is called a mild solution of the differential equation Ž2.1.. The existence and uniqueness of a mild solution of Ž2.8. is easily proved by using the step method. According to the characteristic theory of first order partial differential equations, when f, are once continuous differentiable and g is twice
467
SEMILINEAR TRANSPORT EQUATION
continuous differentiable, Ž2.8. is equivalent to Ž2.1. and the solution of Ž2.8. is exactly the integral surface of Ž2.1..
3. ASYMPTOTIC BEHAVIOR OF SOLUTION We discuss two cases for time lag ; that is, ) 0 and s 0. 3.1. The time lag ) 0 We introduce the following assumptions: ŽH 1 . f Ž0. s 0 and there is a constant ␦ ) 0 such that < f Ž y1 . y f Ž y 2 . < F L < y1 y y 2 < , as < y 1 < F ␦ , < y 2 < F ␦ , where the positive number L is replaced by LŽ ␦ ., which is an increasing function of ␦ ; ŽH 2 . < Ž t, x .< F for t g wy , 0x, x g w0, 1x, where 0 - F ␦ ; ŽH 3 . I s inf g ⬘Ž x .4 ) 0 for x g w0, 1x, and there is a small enough ) 0 such that L - Iey . Now we discuss the asymptotic nature of solutions. We adopt an iterative method, and choose the initial iterative
¡ Ž0, h
u
Ž0.
y1
Ž t , x . s~
Ž h Ž x . eyt . . exp
¢ Ž t , x . ,
y
t
y1
H0 g ⬘ Ž h
ž
t g w y , 0 x ,
Ž h Ž x . eyŽ ty . . . d
x g w 0, 1 x .
t G 0,
/
,
x g w 0, 1 x ,
Ž 3.1. Define the kth approximation uŽ k . Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
= f Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . . d , uŽ k . Ž t , x . s Ž t , x . ,
t g w y , 0 x ,
x g w 0, 1 x ,
From Ž3.1., it is easy to see that < uŽ0. Ž t , x . < F
/
tG0, xg w 0, 1 x ,
k s 1, 2, . . . .
Ž 3.2.
468
MENGXING AND RONGGUI
for t G y , x g w0, 1x. When k s 1, from Ž3.2. we have t ylŽ ty .
L uŽ0. Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . d
t ylŽ ty .
Ž LrI . I d
uŽ1. Ž t , x . F eyI t q
H0 e
F eyI t q
H0 e
F eyI t q Ž LrI . Ž 1 y eyI t . F eyI t Ž 1 y Ž LrI . . q Ž LrI . F for t G y , x g w0, 1x. Using an induction argument, it is not difficult to verify that the inequality < uŽ k . Ž t , x . < F
Ž 3.3.
holds for t G y , x g w0, 1x, k s 1, 2, . . . . From Ž3.2., obviously uŽ1. y uŽ0. s
t
t
y1
H0 exp yH g ⬘ Ž h
ž
Ž h Ž x . eyŽ ty . . . d
/
=f Ž uŽ0. Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d s
t
t
y1
H0 exp yH g ⬘ Ž h
ž
Ž h Ž x . eyŽ ty . . . d
/
Ž t y q . ey Ž ty q .
=f Ž uŽ0. Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d , and it follows that < uŽ1. y uŽ0. < F
t
H0 exp Ž y Ž I y . Ž t y . q . =Ley Ž ty q . uŽ0. Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . d .
When G 0, y G 0, we have < uŽ0. Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . < F eyl Ž y . F ey Ž y . . Thus < uŽ1. y uŽ0. < F
L
t
y Ž ty q .
H0 exp Ž y Ž I y . Ž t y . q . Ž 1 y . e = Ž I y . ey Ž y . d
F ey t
ž
L 1y
/
F a ey t ,
469
SEMILINEAR TRANSPORT EQUATION
where a s LrŽ I y . - 1. By using induction, we obtain for any positive integral k the inequality < uŽ k . y uŽ ky1. < F a k ey t
Ž 3.4.
for t G 0, x g w0, 1x. This shows that the sequence uŽ n.4 is uniformly convergent on w0, T x = w0, 1x, where T is any positive number. Let lim k ª⬁ uŽ k . Ž t, x . s uŽ t, x ., and it is easily seen that uŽ t, x . is a unique continuous solution of Ž2.8.. Since uŽ k . Ž t , x . F Ž 1 q a q a2 q ⭈⭈⭈ a k . ey t for t G 0, x g w0, 1x, it follows that uŽ t , x . F
ž
1 1ya
/
ey t .
Ž 3.5.
This shows uŽ t, x . tending to zero exponentially as t tends to infinity uniformly for 0 F x F 1. Summarizing the above discussions, we have THEOREM 3.1. If assumptions ŽH., ŽH 1 . ᎐ ŽH 3 . are satisfied, then Ž2.8. has a unique solution uŽ t, x ., which tends to zero exponentially as t tends to infinity uniformly for x g w0, 1x. 3.2. The time lag s 0 Consider the following problem:
⭸u
⭸
t ) 0, Ž g Ž x . u. s f Ž uŽ t , ␣ x . . , ⭸t ⭸x u < ts0 s Ž x, t . , t g w y , 0 x , x g w 0, 1 x . q
x g w 0, 1 x ,
Ž 3.6.
Then the corresponding integral equation can be written as u Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
/
= f Ž u Ž , ␣ hy1 Ž h Ž x . eyŽyty . . . . d ,
tG0,
xg w 0, 1 x .
Ž 3.7.
470
MENGXING AND RONGGUI
If assumptions ŽH., ŽH 1 ., ŽH 2 . hold and there is small enough ) 0 such that I )Lq , Ž 3.8. then the estimate Ž3.5. is also satisfied for the solution of Ž3.7.. Because the proof is analogous to the proof of Theorem 1, we do not repeat it.
4. REGULARITY OF SOLUTION In this section, we consider
⭸u
⭸
t ) 0, Ž g Ž x . u. s f Ž uŽ t y , ␣ x . . , ⭸t ⭸x u Ž t , x . s Ž x, t . , t g w y , 0 x , x g w 0, 1 x , q
x g w 0, 1 x ,
Ž 4.1.
where is a small parameter. Obviously, Ž1.1. is special form of Ž4.1.. We further suppose that f, are once continuously differentiable, f Ž0. / 0, g g C 2 Žw0, 1x., and there is a positive ␥ such that g Ž x . ( x for 0 F x F ␥ . From Section 2, there exists a unique uŽ t, x . satisfying the integral equation u Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
t
q
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
= f Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . . d , t g w y , 0 x ,
uŽ t , x . s Ž t , x . ,
/ Ž 4.2.
t)0, xg w 0, 1 x ,
x g w 0, 1 x .
Namely uŽ t, x . is a mild solution of Ž4.1.. Differentiating Ž4.1. with respect to x on w0, x = w0, 1x, we get
⭸u ⭸x
s
⭸ ⭸x
Ž 0, hy1 Ž h Ž x . eyt . . hy1 Ž h Ž x . eyt . ⬘h⬘ Ž x . eyt
= exp y
t
y1
H0 g ⬘ Ž h
ž
Ž h Ž x . eyŽ ty . . . d
/
q Ž 0, hy1 Ž h Ž x . eyt . . =
t
½H
0
g ⬙ Ž hy1 Ž h Ž x . eyŽ ty . . . hy1 Ž h Ž x . eyŽ ty . . ⬘h⬘ Ž x . eyt d
= exp y
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
/
5
Ž 4.3.
471
SEMILINEAR TRANSPORT EQUATION
q
t
H0
½
y
t
y1
H g ⬙ Ž h
= exp y
t
y1
H g ⬘ Ž h
ž
hy1 Ž h Ž x . eyŽ ty . . ⬘h⬘ Ž x . eyt d
Ž h Ž x . eyŽ ty . . .
Ž h Ž x . eyŽ ty . . . d
5
/
= f Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d q
t
t
y1
H0 exp yH g ⬘ Ž h = =
ž
⭸f
⭸x
/
Ž Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . .
⭸u ⭸
Ž h Ž x . eyŽ ty . . . d
Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . .
=␣ hy1 Ž h Ž x . eyŽ ty . . ⬘h⬘ Ž x . eyŽ ty . d . This shows that if w hy1 Ž hŽ x . eyt .x⬘hy1 eyt is bounded for x g w0, 1x, then all terms of right hand on Ž4.1. are L1 integrable on w0, x. According to the definition of hŽ x ., it is easy to see that if w hy1 Ž hŽ x . eyt .x⬘hy1 eyt is bounded for 0 F x F ␥ , then it is also bounded for x g w0, 1x. Noting that h⬘Ž x . s hŽ x .rg Ž x ., when 0 F x F ␥ , we have hy1 Ž h Ž x . eyt . ⬘h1 eyt s s
g Ž h Ž x . eyt . h Ž h Ž x . eyt . h Ž x . eyt h Ž h Ž x . eyt .
s eyt exp
x
= =
h Ž x . eyt gŽ x. g Ž h Ž x . eyt . gŽ x.
d
y exp
H1hŽ x . e
ž ž / ž Ž H ž / ž H1
d
x
s eyt exp ( eyt
gŽ .
h Ž x . eyt
x yt
hŽ x . e
=
gŽ .
hŽ x . x
=
yt
d gŽ .
g h Ž x . eyt . gŽ x.
/
=
g Ž h Ž x . eyt . gŽ x.
/
/
s eyt .
Thus, integral in Ž4.3. exists for t g w0, x, x g w0, 1x, and from w2x, it is easy to see that ⭸ uŽ t, x .r⭸ x exists for t g w0, x, x g w0, 1x. Using the method
472
MENGXING AND RONGGUI
of steps, we obtain that ⭸ ur⭸ x exists for t G , x g w0, 1x, and from Ž4.1., ⭸ ur⭸ x exists for t G , x g w0, 1x. Suppose that
THEOREM 4.1.
Ž1. f g C R ., f Ž0. / 0, and g C 1 Žwy , 0x = w0, 1x.; Ž2. ŽH. holds; Ž3. g g C 2 Žw0, 1x., I s inf g ⬘Ž x .4 for x g w0, 1x, and there exists constant ␥ ) 0 such that g Ž x . ( x for 0 F x F ␥ . Then ⭸ ur⭸ t and ⭸ ur⭸ x exists and, moreo¨ er, satisfy Ž4.1.. 1Ž
Now consider
⭸u
⭸
Ž g Ž x . u. s f Ž uŽ t , ␣ x . . , ⭸t ⭸x u Ž t , x . < ts0 s Ž x . , x g w 0, 1 x , q
t ) 0,
x g w 0, 1 x ,
Ž 4.4.
which corresponds to the case s 0 in Ž4.1.. Obviously, a similar conclusion to Theorem 4.1 holds and the proof is also the same.
5. EXISTENCE AND STABILITY OF NONTRIVIAL STEADY STATE SOLUTION Consider d
Ž gŽ x. . s f Ž Ž ␣ x. . , dx limq Ž x . s Ž 0 . ,
x g w 0, 1 x ,
Ž 5.1.
xª0
where the initial value Ž0. satisfies
Ž 0 . s f Ž Ž 0 . . rg ⬘ Ž 0 . .
Ž 5.2.
A nontrivial solution of Ž5.1. is called a nontrivial steady state solution of Ž4.1. and Ž4.4., because the steady state problems of both are the same. It is easy to see that solution of Ž5.1. can be written as
Ž x. s Choose C s y
1
Ž . ž
g x
g Ž 1. C q
x
H1 f Ž Ž ␣ . . d
/
.
0 gŽ x. 1
H f Ž Ž ␣ x .. dx, and we get
Ž x. s
x
H f Ž Ž ␣ . . d , gŽ x. 1
x g w 0, 1 x .
Ž 5.3.
473
SEMILINEAR TRANSPORT EQUATION
We now prove the nontrivial solution of Ž5.1. exists and it is unique. Define the set S s w Ž x . < w g C w 0, 1 x , w Ž 0 . s f Ž w Ž 0 . . rg ⬘ Ž 0 . ,
½
sup < w < F ␦ 0FxF1
5
and define the mapping
s Tw s
x
H f Ž w Ž ␣ . . d gŽ x. 0
Ž 5.4.
on S. Obviously, T is continuous for w g S. Noting that x Ž x. s f Ž w Ž ␣ . . d , gŽ x. 0 we then have x < Ž x . < F f Ž w Ž ␣ . . d gŽ x. 0
H
H
F F
x
gŽ x.
gŽ x.
L
H0
f Ž w Ž ␣ . . y f Ž 0 . d q
L sup < w < x q
gŽ x.
x
H0 f Ž 0. d
< f Ž 0. < x
< f Ž 0. <
F Ž L q f Ž 0 . . ␦rI. I I Thus, when 0 - F IrŽ L q < f Ž0.<., < < F ␦ , it follows that the mapping T : S ª S. From Ž5.4. we have x 1 y 2 s f Ž w 1 Ž ␣ . . y f Ž w 2 Ž ␣ . . d . gŽ x. 0 F
␦q
gŽ x.
H
It follows that <1 y 2 < F F
x
H gŽ x. 0 L gŽ x.
f Ž w 1 Ž ␣ . . y f Ž w 2 Ž ␣ . . d
sup < w 1 y w 2 < x 0FxF1
F L sup < w 1 y w 2 < 0FxF1
F
L I
sup < w 1 y w 2 < . 0FxF1
1
žH
0
g ⬘ Ž x . d
/
474
MENGXING AND RONGGUI
This shows that T is a contraction mapping, so that there exists a unique Ž x . g S satisfying
Ž x . s T ;
Ž 5.5.
namely, from Ž5.4. there exists a unique solution
Ž x. s
x
H f Ž Ž ␣ . . d , gŽ x. 0
x g w 0, 1 x .
Obviously, Ž x . g C w0, 1x and Ž x . / 0. Noting that g ⬘ q g ⬘ s f Ž Ž ␣ x .. and Ž0. s Ž rg⬘Ž0.. f Ž Ž0.., we can define ⬘Ž0. s 0, and then ⬘Ž x . exists on 0 F x F 1 and satisfies Ž5.1.. Summarizing the above discussions, we have THEOREM 5.1. If the assumptions of Theorem 4.1 are satisfied and equality Ž5.2. holds, then from Ž4.1. and Ž4.4., there exists a unique nontri¨ ial steady state solution. In the following we discuss the stability of the steady state solution. Before doing this, we show that if sup < < F F ␦ for Ž t, x . g wy , 0x = w0, 1x, then the inequality < uŽ t , x . < F
Ž 5.6.
holds for t G y , x g w0, 1x, where uŽ t, x . is a solution of Ž4.1.. From Ž4.2., we have u Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
/
= f Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . . y f Ž 0 . q f Ž 0 . d , and it follows that u Ž t , x . F eyI t q q
t yIŽ ty .
H0 e
t yI Ž ty . <
H0 e
L u Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . d
f Ž 0. < d .
Thus, when u Ž y , ␣ hy1 Ž h Ž x . eyŽ ty . . . s Ž y , ␣ hy1 Ž h Ž x . eyŽ ty . . . ,
Ž t , x . g w y , 0 x = w 0, 1 x ,
475
SEMILINEAR TRANSPORT EQUATION
then < u Ž t , x . < F eyI t q q
t yIŽ ty .
L Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . d
H0 e
< f Ž 0. < I
Ž 1 y eyI t .
½
F eyI t 1 y
I
ž
< f Ž 0. <
Lq
q
/ ž I
Lq
< f Ž 0. <
/5
F . Using the method of steps, the estimate Ž5.6. is obtained. Let w s u y , and from Ž4.1. and Ž5.1., we have
⭸w ⭸t
q
⭸ ⭸x
Ž g Ž x . w . s f Ž uŽ t y , ␣ x . . y f Ž Ž ␣ x . . , t ) 0,
x g w 0, 1 x ,
Ž 5.7.
w Ž t , x . s Ž x, t . y Ž x . s Ž t , x . , t g w y , 0 x ,
x g w 0, 1 x ;
namely
⭸w ⭸t
q
⭸ ⭸x
Ž gw . s F Ž u Ž t y , ␣ x . . w Ž t y , ␣ x . , t g w y , 0 x ,
wŽ t, x. s Ž t, x. ,
Ž 5.8.
x g w 0, 1 x ,
t ) 0,
x g w 0, 1 x .
where F s H0t f ⬘Ž Ž x . q Ž uŽ t y , ␣ x . y Ž x ... d , and w Ž t, x . satisfies w Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d y1
= F Ž uŽ y , ␣ h
/
yŽy ty .
Ž hŽ x . e
Ž 5.9.
...
= w Ž y , ␣ hy1 Ž h Ž x . eyŽyty . . . d , w Ž t , x . s Ž t , x . ,
/
t)0,
Ž t , x . g w y , 0 x , xg w 0, 1 x .
xg w 0, 1 x ,
476
MENGXING AND RONGGUI
In the following we prove stability. We adopt an iterative method, and choose the initial approximation w Ž0. Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . = exp y
t
Ž h Ž x . eyŽ ty . . . d
y1
H0 g ⬘ Ž h
ž
tg w y , 0 x ,
wŽ t, x. s Ž t, x. ,
/
tG0, xg w 0, 1 x Ž 5.10.
,
xg w 0, 1 x .
Define the kth approximation w Ž k . Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
/ Ž 5.11.
= F Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . = w Ž ky1. Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . d , t)0,
xg w 0, 1 x ,
Ž t , x . g w y , 0 x , xg w 0, 1 x .
wŽk. Ž t , x . s Ž t , x . ,
Noting that sup < < F 2 for t g wy , 0x, x g w0, 1x, from Ž5.10., it is easy to see that w Ž0. Ž t , x . s
½
t G 0, x g w 0, 1 x , t g w y , 0 x , x g w 0, 1 x .
2 eyI t , 2 ,
Ž 5.12.
From Ž5.11. and by similar arguments to Section 3, we get < w Ž k . Ž t , x . < F Ž 1 q a q a2 q ⭈⭈⭈ qa k . 2 ey t
Ž 5.13.
< w Ž k . y w Ž ky1. < F a k 2 ey t
Ž 5.14.
and for t G 0, x g w0, 1x, k s 0, 1, 2, . . . . This shows that the sequence w Ž k .4 is convergent uniformly on an arbitrary compact region w0, T x = w0, 1x. Let lim k ª⬁w Ž k . s w Ž t, x ., and it is not difficult to verify that w Ž t, x . is a unique solution of Ž5.9.. From Ž5.13., obviously wŽ t, x. F
ž
2 1ya
/
ey t
Ž 5.15.
477
SEMILINEAR TRANSPORT EQUATION
for t G 0, x g w0, 1x, and it follows that w Ž t, x . tends to zero exponentially as t tends to q⬁ uniformly with respect to x g w0, 1x. THEOREM 5.2. Let the hypothesis of Theorem 4.1 be satisfied. Then inequality Ž5.15. holds for t G 0, x g w0, 1x; namely, uŽ t, x . tends to Ž x . exponentially as t tends to q⬁. For the problem Ž4.4., the corresponding conclusion holds also.
6. CONVERGENCE OF SOLUTION In this section, with Ž0, x . s Ž x ., we investigate whether the mild solution of Ž4.1. converges to a mild solution of Ž4.4. as the time lag tends to 0. Let T˜ be a fixed positive number and let n be a positive integer so large ˜ F . Put n s Trn. ˜ that Trn We denote by n the restriction of on wy , 0x = w0, 1x and let u nŽ t, x . be a mild solution of the problem
⭸u
⭸
Ž g Ž x . u . s f Ž u Ž t y n , ␣ x . . , n G 0, x g w 0, 1 x , Ž 6.1. ⭸t ⭸x u Ž t , x . s n Ž x, t . , t g w yn , 0 x , x g w 0, 1 x ; q
namely, u nŽ t, x . satisfies u n Ž t , x . s n Ž 0, hy1 Ž h Ž x . eyt . . = exp y q
t
y1
H0 g ⬘ Ž h
ž
t
t
Ž h Ž x . eyŽ ty . . . d
y1
H0 exp yH g ⬘ Ž h
ž
/
Ž h Ž x . eyŽ ty . . . d
/
Ž 6.2.
= f Ž u n Ž y n , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d , tG0, xg w 0, 1 x , u Ž t , x . s n Ž t , x . ,
tg w yn , 0 x , xg w 0, 1 x ,
First, we estimate u n . For t g w0, yn x, x g w0, 1x, from Ž6.2., we have u n Ž t , x . s n Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
/
= f Ž n Ž y n , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d ,
Ž 6.3.
478
MENGXING AND RONGGUI
or t
u n Ž t , x . s n Ž 0, hy1 Ž h Ž x . eyt . . exp y q
t
t
y1
H0 exp yH g ⬘ Ž h
ž
y1
H0 g ⬘ Ž h
ž
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
/
= f Ž n Ž y n , ␣ hy1 Ž h Ž x . eyŽyty . . . . yf Ž 0 . q f Ž 0 . d .
Ž 6.4.
It follows that < u n < F eyI t q q
t yIŽ ty .
L n Ž y n ␣ hy1 Ž h Ž x . eyŽ ty . . . d
H0 e
t yI Ž ty . <
H0 e
F eyI t q
½
L I
F eyI t 1 y
f Ž 0. < d
Ž 1 y eyI t . q I
ž
< f Ž 0. <
< f Ž 0. <
Lq
I
/
q
Ž 1 y eyI t .
I
ž
Lq
< f Ž 0. <
⬘
/5
F , where 0 - - IrŽ L q < f Ž0.
for t G yn ,
x g w 0, 1 x ,
n s 0, 1, 2, . . . . Ž 6.5.
Thus, the sequence u n4 is uniformly bounded on t G 0, x g w0, 1x. We now estimate ⭸ ur⭸ n. Differentiating Ž6.3. with respect to x, we have
⭸ un ⭸x
s
⭸ ⭸x
n Ž 0, hy1 Ž h Ž x . eyt . . hy1 Ž h Ž x . eyt .
=h⬘ Ž x . eyt exp y
ž
t
y1
H0 g ⬘ Ž h
qn Ž 0, hy1 Ž h Ž x . eyt . .
X x
Ž h Ž x . eyŽ ty . . . d
/
479
SEMILINEAR TRANSPORT EQUATION t
= y
y1
H0 g ⬙ Ž h
½
Ž h Ž x . eyŽ ty . . .
hy1 Ž h Ž x . eyŽ ty . .
X x
=h⬘ Ž x . eyt d t
=exp y
y1
H0 g ⬘ Ž h
ž
t
q
H0
½
t
y
Ž h Ž x . eyŽ ty . . . d y1
H g ⬙ Ž h
/
Ž h Ž x . eyŽ ty . . .
hy1 Ž h Ž x . eyŽ ty . . =h⬘ Ž x . eyt d
t
=exp y
y1
H g ⬘ Ž h
ž
5
Ž h Ž x . eyŽ ty . . . d
X x
5
/
= f Ž n Ž y n , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d t
q
t
y1
H0 exp yH0 g ⬘ Ž h
= =
⭸f ⭸x
⭸ ⭸x
ž
Ž h Ž x . eyŽ ty . . . d
/
Ž n Ž y n , ␣ hy1 Ž h Ž x . eyŽ ty . . . .
n Ž y n , ␣ hy1 Ž h Ž x . eyŽyty . . .
=␣ hy1 Ž h Ž x . eyŽ ty . .
X x
h⬘ Ž x . eyŽ ty . d
s I1 q I2 q I3 q I4 .
Ž 6.6.
X Notice that , xX , g ⬙, and w hy1 Ž hŽ x . eyt .x x h⬘Ž x . eyt are bounded, Ž nŽ t y n , ␣ hy1 Ž hŽ x . eyt . eyŽ ty . .., ⭸⭸ xf Ž nŽ t y n , ␣ hy1 Ž hŽ x . eyt . eyŽ ty . .. are uniformly bounded, and then there exist constants ⬘ ) 0, L⬘ ) 0, L⬙ ) 0 such that
< I1 < q < I2 < F ⬘eyI t , < I3 < F < I4 < F
t yIŽ ty .
H0 e
L⬙
t yI Ž ty .
H0 e
⭸n ⭸x
L⬘ d ;
Ž y n ␣ hy1 Ž h Ž x . eyŽ ty . . .
d .
480
MENGXING AND RONGGUI
It follows that
⭸n ⭸x
F ⬘eyI t q q
t yIŽ ty .
L⬘ d
H0 e
t yIŽ ty .
L⬙
H0 e
F ⬘eyI t q
L⬘ I
½
F eyI t 1 y
I
⭸n ⭸x
Ž y n ␣ hy1 Ž h Ž x . eyŽ ty . . .
Ž 1 y eyI t . q L⬙ q
ž
L⬘
⬘
/
L⬙ I
q
I
d
Ž 1 y eyI t .
ž
L⬙ q
L⬘
⬘
/5
F ⬘ for t g w0, n x, x g w0, 1x. Using the step method and induction, we obtain
⭸ un ⭸x
F ⬘
x g w 0, 1 x .
for t G 0,
Ž 6.7.
From Ž6.1., Ž6.5., and Ž6.7., it is not difficult to see that
⭸ un ⭸x
F ⬙
x g w 0, 1 x ,
for t G 0,
Ž 6.8.
where ⬙ is a positive constant which is independent of n. Equation Ž6.1. and estimates Ž6.5., Ž6.7., Ž6.8. show that the sequence u n4 is uniformly bounded and equicontinuous on an arbitrary compact region w0, T x = w0, 1x. Then, we can choose a subsequence u n j 4 which is uniformly convergent on w0, T x = w0, 1x. Let lim jª⬁ u n j s uŽ t, x ., and it is easily verified that uŽ t, x . satisfies u Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . exp y
ž
q
t
t
y1
H0 exp yH0 g ⬘ Ž h
ž
t
y1
H0 g ⬘ Ž h
Ž h Ž x . eyŽ ty . . . d
Ž h Ž x . eyŽ ty . . . d
/
/
= f Ž u Ž , ␣ hy1 Ž h Ž x . eyŽyty . . . . d ,
tG0,
xg w 0, 1 x .
Ž 6.9. Namely, uŽ t, x . is a mild solution of Ž4.4.. Again, since the solution of Ž6.9. is unique, it follows that the sequence u n4 uniformly converges to uŽ t, x ..
481
SEMILINEAR TRANSPORT EQUATION
Summarizing the above discussions, we obtain THEOREM 6.1. Let the hypothesis of Theorem 4.1 be satisfied and suppose 0 - - IrŽ L q < f Ž0.<., Ž0, x . s Ž x .. Then the mild solution of Ž4.1. con¨ erges to the mild solution of Ž4.4. as the time lag n ª 0 Ž n ª q⬁.. 7. GENERALIZATION OF PROBLEM We only discuss two cases. 7.1. Multidimensional Case We consider
⭸u ⭸t
q
m
⭸
Ý
⭸ xi
is1
t ) 0, x i g w 0, 1 x ,
Ž gi Ž x i . u. s f Ž uŽ t y , ␣ x . . ,
u Ž t , x . s Ž x, t . ,
t g w y x , x i g w 0, 1 x , i s 1, 2, . . . , m,
Ž 7.1.
where x s Ž x 1 , . . . , x m ., ␣ s Ž ␣ 1 , . . . , ␣ m .T , 0 - ␣ i - 1. For f, g i , and , we suppose that ŽH. f, are continuous, and g i is continuously differentiable and satisfies g i Ž 0 . s 0,
g i Ž x i . ) 0, 0 - x i - 1,
1
H0
dx i gi Ž xi .
s ⬁.
As we have seen above, the characteristic equations of Ž7.1. are dx i dt du dt
s gi Ž xi . , m
sy Ý
Ž 7.2. g Xi
Ž x i . u q f Ž uŽ t y , ␣ x . . .
is1
Define h i Ž x i . s expŽ H1x d irg i Ž i .., and then h i Ž1. s 1, h i Ž0. s 0, and h i is a monotone increasing function and satisfies dh i hi
s dt.
Ž 7.3.
482
MENGXING AND RONGGUI
Consider the initial conditions h i Ž xUi . < ts0 s h i Ž x i0 . ,
0 F x i0 F 1,
u Ž t , x . < ts0 s Ž 0, x 0 . ,
Ž 7.4.
0 F x i0 F 1,
It is known from Ž7.3. and Ž7.4. that h i Ž x i Ž t .. s h i Ž x i0 . e t. So t x i Ž t . s hy1 i Ž h Ž x i0 . e . ,
Ž 7.5.
yt x i0 s hy1 ., i Ž hŽ xi . e
where i s 1, 2, . . . , m. Consider the initial condition Ž7.4. and the second equation of Ž7.2. to obtain the relation m
t X
u Ž t , x . s Ž 0, x 0 . exp Ý y is1 t
q
H0
m
exp y Ý
ž
is1
H0 g Ž x Ž s . . ds Ž 0, x
ž
i
i
t X
H0 g Ž x Ž s . . ds i
i
0
.
/
f u Ž s y , ␣ x Ž s . . . ds. Ž 7.6.
/Ž
Then, substituting Ž7.4., Ž7.5. into Ž7.6., we obtain u Ž t , x . s Ž 0, hy1 Ž h Ž x . eyt . . m
= exp y Ý
ž
q
is1
t H0 g Ž h Ž h Ž x . e
m
t
H0 exp
ž
yÝ is1
X i
y1 i
i
yŽ ty .
i
. . d
t H g Ž h Ž h Ž x . e X i
y1 i
i
i
yŽ ty .
/
. . d
/
Ž 7.7.
= f Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d , t ) 0, uŽ t , x . s Ž t , x . ,
t g w y , 0 x ,
x i g w 0, 1 x ,
x g w 0, 1 x ,
Ž h1Ž x 1 . eyt ., . . . , hy1 Ž Ž . yt ... where hy1 Ž hŽ x . eyt . s Ž hy1 1 m hm xm e 7.2. System of Equation Case We consider
⭸ ui ⭸t
q
⭸ ⭸ xi
Ž g i Ž x i . u i . s fi Ž uŽ t y , ␣ x . . ,
u i Ž t , x . s i Ž x, t . ,
t g w y x ,
t ) 0, x g w 0, 1 x ,
Ž 7.8.
x g w 0, 1 x ,
where i s 1, 2, . . . , n, u s Ž u1 , . . . , u n ., x s Ž x 1 , . . . , x n ., ␣ s Ž ␣ 1 , . . . , ␣ n .T .
483
SEMILINEAR TRANSPORT EQUATION
It is well known that the characteristic equation is dx i dt du i dt
s gi Ž xi . , s
yg Xi u
Ž 7.9.
q
f iX
Ž uŽ t y , ␣ x . . ,
i s 1, 2, . . . , n,
where f, , and g satisfy again condition ŽH.. By means of the above method, we can transform Ž7.8. into the integral equation u i Ž t , x . s i Ž 0, hy1 Ž h Ž x . eyt . . t H0 g Ž h Ž h Ž x . e
= exp y
ž
X i
y1 i
i
yŽ ty .
i
t t H0 exp yH g Ž h Ž h Ž x . e
q
ž
X i
y1 i
i
i
. . d
yŽ ty .
/
. . d
/
Ž 7.10.
=f i Ž u Ž y , ␣ hy1 Ž h Ž x . eyŽy ty . . . . d , t ) 0, u i Ž t , x . s i Ž t , x . ,
t g w y , 0 x ,
x i g w 0, 1 x , x i g w 0, 1 x ,
For Ž7.1. and Ž7.8., we can prove, under suitable conditions, that corresponding results hold. Because the methods are similar, we omit them. ACKNOWLEDGMENT The author appreciates the anonymous referee for his valuable comments and helpful suggestions on this paper.
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