Directory UMM :Journals:Journal_of_mathematics:EJQTDE:
Electronic Journal of Qualitative Theory of Differential Equations
2010, No. 52, 1-20; http://www.math.u-szeged.hu/ejqtde/
Existence of Time Periodic Solutions for
One-Dimensional Newtonian Filtration
Equation with Multiple Delays∗
Ying Yanga , Jingxue Yinb and Chunhua Jinb,c*†
a
b
Department of Mathematics, Jilin University, Changchun 130012, China
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
c
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Abstract
In this paper, we study one-dimensional Newtonian filtration equation including
unbounded sources with multiple delays. The existence of nonnegative non-trivial
time periodic solutions will be established by the Leray-Schauder fixed point theorem based on some suitable Lyapunov functionals and some a priori estimates for
all possible periodic solutions.
Keywords: Newtonian Filtration; Multiple Delays; Periodic Solution.
1
Introduction
Consider the following one-dimensional Newtonian filtration equation with multiple delays
Z t
∂ 2 um
∂u
=
+ au + f (u(x, t − τ1 ), · · · , u(x, t − τn )) + g(x, t)+γ
e−α(t−s) u(x, s)ds,
∂t
∂x2
t−τ0
x ∈ (0, 1), t ∈ R,
(1.1)
subject to the homogeneous Dirichlet boundary value condition
u(0, t) = u(1, t) = 0,
t ∈ R,
(1.2)
∗
This work is partially supported by the National Science Foundation of China, partially supported
by a Specific Foundation for Ph.D. Specialities of Educational Department of China, partially supported
by the Foundation for Post Doctor of China and partially supported by Graduate Innovation Fund of
Jilin University (No. 20101045).
†
Corresponding author. Email: jinchhua@126.com
EJQTDE, 2010 No. 52, p. 1
where m > 1, a and α are constants, γ is a positive constant, f and g are the known
functions satisfying some structure conditions.
This kind of equation arises from a variety of areas in applied mathematics, physics
and mathematical ecology. For the case m = 1, n = 1 and γ = 0 with f (r) = αr/(1 + r β ),
which appears in the blood cell production model[1]
∂u
∂2u
= 2 − au + f (u(x, t − τ )) + g(x, t).
∂t
∂x
On the other hand, for the same case, that is m = 1, n = 1, γ = 0, with different f , the
equation (1.1) also is known as the Hematopoiesis model (f (r) = e−kr (k > 0)) as well
as Nicholson’s blowflies model (f (r) = re−kr (k > 0)), see for example [2, 3]. While, it
is worth noting that all the above models are linearly diffusive, but if nonlinear diffusion
is introduced, the model will be more consistent with biologic phenomena in the real
world. However, as far as we know, only a few works are concerned with time periodic
solutions for degenerate parabolic equation with delay(s). For example, in [4], the authors
investigated the existence of time periodic solutions for p-Laplacian with multiple delays.
In [5], the authors studied the existence of periodic solutions for Nicholson’s blowflies
model with Newtonian diffusion
Z t
∂ 2 um
∂u
−au(x,t−τ )
=
− δu + pu(x, t − τ )e
+ g(x, t) + β
e−α(t−s) u(x, s)ds.
∂t
∂x2
t−τ
Nevertheless, in this paper, a more general source will be discussed, which is allowed to
be the blood cell production model or other types.
In the present paper, we pay our attention to the existence of nonnegative time periodic
solutions for (1.1). It is worth noticing that in the model of [5], the source with delay is a
typical but quite special bounded source. However, in this paper, a more general source
will be discussed, particularly, the source with delays is allowed to be unbounded, which
caused us difficulties in making the maximum norm estimates and some other a priori
estimates. On the other hand, the method used in[4] will also not work for the equation
we consider, that is the coefficient matrix associated with Lyapunov function depends on
solutions of the problem, and therefore the required estimates as did in [4] could not be
obtained. So, we must try some other methods. By constructing some suitable Lyapunov
functionals, the a priori estimates for all possible periodic solutions, and combining with
Leray-Schauder fixed point theorem, we finally establish the existence of time periodic
solutions.
The rest of this paper is organized as follows. In Section 2 we introduce some basic
assumptions, preliminary lemmas and state the main results of this paper. Section 3 is
devoted to investigating the existence of periodic solutions based on the a priori estimates
obtained in Section 2 and Leray-Schauder fixed point theorem.
EJQTDE, 2010 No. 52, p. 2
2
Preliminaries and the Main Result
Throughout this paper, we make the following assumptions:
(H1 ) 0 ≤ g ∈ C(Q), g(x, t) 6≡ 0, g(x, t + T ) = g(x, t);
(H2 ) f (0, · · · , 0) = 0, f (r1 , · · · , rn ) ≥ 0, ri ≥ 0 (i = 1, · · · , n) and
|f (a1 , · · · , an ) − f (b1 , · · · , bn )| ≤
n
X
βi |ai − bi |;
i=1
where T and βi are positive constants, Q = (0, 1) × (0, T ).
Since the equation (1.1) is degenerate parabolic and problem (1.1)–(1.2) usually admits
solutions only in some generalized sense. Hence we introduce the following definition.
Definition 2.1 A function u is said to be a weak solution of the problem (1.1)–(1.2),
m
if u ∈ {w; w ∈ L∞ , w m ∈ L∞ (0, T ; W01,2(0, 1)), ∂w∂t ∈ L2 (Q)}, and for any ϕ ∈ C ∞ (Q)
with ϕ(x, 0) = ϕ(x, T ) and ϕ(0, t) = ϕ(1, t) = 0, the following integral equality holds
Z T Z 1
∂um ∂ϕ
uϕt −
+ auϕ + f (u(x, t − τ1 ), · · · , u(x, t − τn ))ϕ + g(x, t)ϕ
∂x ∂x
0
0
Z t
−α(t−s)
+ γ
e
u(x, s)ds ϕ dxdt = 0.
t−τ
Now we state the main result of this paper.
Theorem 2.1 Suppose that (H1 ) and (H2 ) hold. Then the problem (1.1)–(1.2) admits
at least one nonnegative T -periodic solution.
To prove the existence of periodic solutions (1.1)–(1.2), let us first consider the regularized problem
∂uε
∂2
= 2 (εuε + um
ε ) + auε + f (uε (x, t − τ1 ), · · · , uε (x, t − τn ))
∂t
∂x
Z
t
−α(t−s)
+ g(x, t) + γ
e
uε ds,
(2.1)
x ∈ (0, 1), t ∈ R,
t−τ0
uε (0, t) = uε (1, t),
uε (x, t) = uε (x, t + T ),
t ∈ R,
x ∈ [0, 1], t ∈ R.
(2.2)
(2.3)
The desired solution of the problem (1.1)–(1.2) will be obtained by the limit of some
subsequence of solutions uε of the regularized problem. However, we need first to establish
the existence of solutions uε , for which, we will make use of the Leray-Schauder fixed point
theorem and our efforts center on obtaining the uniformly boundness of uε . To this end,
we prove the following lemmas.
EJQTDE, 2010 No. 52, p. 3
Lemma 2.1 Assume that (H1 ), (H2 ) hold and u is a nonnegative T-periodic solution
of the equation
ut =
∂2
m
(εu
+
u
)
+
λ
au + f (u(x, t − τ1 ), · · · , u(x, t − τn )) + g(x, t)
∂x2
Z t
+γ
e−α(t−s) u(x, s)ds ,
(2.4)
t−τ
satisfying the boundary value condition (2.2), where λ ∈ [0, 1], 0 < ε < 1 is a constant
which is arbitrary. Then for any r > 0, we have
Z TZ 1
um+r dxdt ≤ C1 (m, r),
0
0
where C1 (m, r) > 0 is a constant which depend on m and r.
Proof. Note that, multiplying Eq.(2.4) by ur and integrating over Q,
Z TZ 1
Z TZ 1
Z TZ 1 2
∂
r
m
r
ut u dxdt =
(εu + u ) · u dxdt + λ a
ur+1dxdt
2
∂x
0
0
0
0
0
0
Z TZ 1
Z TZ 1
r
+
f (u(x, t − τ1 ), · · · , u(x, t − τn ))u dxdt +
gur dxdt
0
0
0
0
Z TZ 1
Z t
r
−α(t−s)
+
uγ
e
u(x, s)dsdxdt .
0
Since u is T-periodic,
Z TZ
0
0
0
1
t−τ
1
∂u r
u dxdt =
∂t
r+1
Z
0
T
Z
0
1
∂ur+1
dxdt = 0,
∂t
it follows that
Z TZ 1
∂
∂ur
(εu + um )
dxdt
∂x
0
0 ∂x
Z TZ 1
Z TZ 1
r+1
=λa
u dxdt + λ
f (u(x, t − τ1 ), · · · , u(x, t − τn ))ur dxdt
0
0
0
0
Z TZ 1
Z TZ 1
Z t
r
r
+λ
gu dxdt + λ
uγ
e−α(t−s) u(x, s)dsdxdt
≤|a|
Z
+
Z
0
0
0
0
T Z 1
u
0
r+1
dxdt +
0
T
Z
0
1
ur γ
n
X
i=1
Z
t
βi
Z
0
T
0
t−τ
Z
1
r
u(x, t − τi )u dxdt + K
0
Z
0
T
Z
1
ur dxdt
0
e−α(t−s) u(x, s)dsdxdt
t−τ
EJQTDE, 2010 No. 52, p. 4
≤|a|ε1
T
Z
0
+
n
X
1
Z
u
m+r
0
−r/m
ε3
i=1
Z
T
0
≤ |a|ε1 +
Z
1
u
Z
0
T Z
m+r
+
1
|u(x, t − τi )|m+r dxdt +
0
βi ε2
n
X
dxdt +
βi ε2 +
−(r+1)/(m−1)
ε1
+
n
X
i=1
n
X
0
n
X
−r/m
Kε4
T
−r/m
βi ε2
ε3
+
T
Z
0
+ Kε4
Z
r
uγ
0
!Z
0
−r/m −1/(m−1)
βi ε2
ε3
1
+
T
0
1
Z
r
uγ
0
Z
t
Z
1
um+r dxdt
0
−r/m −1/(m−1)
ε3
T
βi ε2
+
T
Z
t
e−α(t−s) u(x, s)dsdxdt
t−τ
Z
1
um+r dxdt
0
−r/m
Kε4
i=1
Z
T
Z
i=1
0
i=1
+
n
X
i=1
βi ε2
+ Kε4
dxdt +
−(r+1)/(m−1)
|a|ε1
T
!
T
e−α(t−s) u(x, s)dsdxdt.
t−τ
On the other hand
Z TZ 1
Z t
r
uγ
e−α(t−s) u(x, s)dsdxdt
0
0
t−τ
Z TZ 1 Z t
Z TZ 1 Z T
|α|τ
r
|α|τ
≤e γ
u
u(x, s)dsdxdt ≤ e γ
ur
u(x, s)dsdxdt
0
0
t−τ
0
0
−τ
#
" Z Z
m+r
Z Z Z
T
1
≤e|α|τ γ ε5
0
"
Z
Z
=e|α|τ γ ε5
|α|τ
0
T
0
T
Z
−
1
um+r dxdt +
0
Z
1
m+r
T
r
um+r dxdt + ε5 m
−r
ε5 m T
1
T
m
u(x, s)ds
0
0
Z 1 Z
0
−r
ε5 m T (T
dxdt
−τ
T
u(x, s)ds
−τ
r
m
Z 1Z
T
m+r
m
#
dx
m+r
m
dsdx
+ τ)
γ ε5
u
dxdt +
u(x, s)
0
0
0
−τ
Z TZ 1
h τ i
Z 1Z T
r
m+r
r
−m
|α|τ
m+r
m
m
dsdx
≤e γ ε5
u
dxdt + ε5 T (T + τ )
+1
u(x, s)
T
0
0
0
0
Z TZ 1
Z 1Z T
h τ i
r
r
−m
|α|τ
m+r
≤e γ ε5
+ 1 ε6
u
dxdt + ε5 T (T + τ ) m
u(x, s)m+r dsdx
T
0
0
0
0
h τ i
− 1
r
r
−
+ε5 m T 2 (T + τ ) m
+ 1 ε6 m−1 .
T
≤e
Then we know that
Z TZ 1
0
0
∂
∂ur
(εu + um )
dxdt
∂x
∂x
EJQTDE, 2010 No. 52, p. 5
n
X
≤ |a|ε1 +
βi ε2 +
i=1
−(r+1)/(m−1)
+
ε1
+
n
X
i=1
n
X
−r/m
βi ε2
ε3
+ Kε4
!Z
T
0
−r/m −1/(m−1)
ε3
Z
1
um+r dxdt
0
−r/m
βi ε2
+ Kε4
i=1
+
Z
T
1
Z
0
ur γ
0
Z
t
e−α(t−s) u(x, s)dsdxdt
βi ε2 +
−r/m
γε5
T (T
−(r+1)/(m−1)
+
n
X
−r/m
βi ε2
ε3 + Kε4 + e|α|τ γε5
i=1
i=1
+e
T
t−τ
n
X
≤ |a|ε1 +
|α|τ
!
ε1
+ τ)
+
r/m
n
X
τ
([ ] + 1)ε6
T
T
Z
0
−r/m −1/(m−1)
ε3
Z
1
um+r dxdt
0
−r/m
βi ε2
+ Kε4
i=1
|α|τ
+e
r
−m
γε5 T (T + τ )
n
X
≤ |a|ε1 +
βi ε2 +
i=1
|α|τ
+e
r
m
h τ i
T
n
X
+1
−r/m
βi ε2
− 1
ε6 m−1
T
ε3 + Kε4 + e|α|τ γε5
i=1
−r/m
γε5
T (T
+ τ)
r/m
h τ i
T
+ 1 ε6
Z
T
0
Z
1
um+r dxdt + C.
0
Here and below, we use C > 0 to denote different positive constants depending only on
the known quantities. In addition, it is easy to see that
Z TZ 1
Z TZ 1
r
∂
m ∂u
(εu + u )
dxdt =
(ε + mum−1 )rur−1u2x dxdt
∂x
∂x
0
0
0
0
Z TZ 1
≥
mrum+r−2 u2x dxdt
0
0
Z T Z 1
∂ (m+r)/2 2
4mr
dxdt,
u
=
(m + r)2 0 0 ∂x
which implies
4mr
(m + r)2
≤ |a|ε1 +
Z
T
0
n
X
Z
0
1
∂ (m+r)/2 2
dxdt
u
∂x
βi ε2 +
i=1
|α|τ
+e
−r/m
γε5
T (T
n
X
−r/m
βi ε2
ε3 + Kε4 + e|α|τ γε5
i=1
+ τ)
r/m
h τ i
T
+ 1 ε6
Z
0
T
Z
1
um+r dxdt + C
0
EJQTDE, 2010 No. 52, p. 6
≤ |a|ε1 +
n
X
βi ε2 +
i=1
|α|τ
+e
−r/m
γε5
T (T
n
X
−r/m
βi ε2
ε3 + Kε4 + e|α|τ γε5
i=1
+ τ)
r/m
h τ i
T
Z
+ 1 ε6 µ
0
T
Z 1
∂ (m+r)/2 2
dxdt + C.
u
∂x
0
Thence if ε1 , ε2 , ε3 , ε4 , ε5 and ε6 are appropriately small, we can get
Z T Z 1
∂ (m+r)/2 2
dxdt ≤ C(m, r)
u
∂x
0
0
(2.5)
Using Poincar´e inequality, we see that
Z TZ 1
um+r dxdt ≤ C1 (m, r),
0
0
which completes the proof of Lemma 2.1.
Lemma 2.2 Assume that (H1 ), (H2 ) hold and u is a nonnegative T-periodic solution
of the equation (2.4) satisfying the boundary value condition (2.2). Then we have
Z T Z 1 m 2
∂u
∂x dxdt ≤ C,
0
0
where C > 0 is a constant.
Proof. In fact, choosing r = m in (2.5), we obtain
Z T Z 1 m 2
∂u
∂x dxdt ≤ C.
0
0
Lemma 2.3 Assume that (H1 ), (H2 ) hold and u is a nonnegative T-periodic solution
of the equation (2.4) satisfying the boundary value condition (2.2). Then we have
2
Z T Z 1
∂
(εu + um ) dxdt ≤ C,
∂x
0
0
where C > 0 is a constant.
Proof. The proof is a direct verification. A simple calculation shows
2
Z T Z 1
∂
m
(εu + u ) dxdt
∂x
0
0
EJQTDE, 2010 No. 52, p. 7
T
1
∂
∂
(εu + um ) (εu + um )dxdt
∂x
0
0 ∂x
Z T
Z TZ 1 2
∂
m
m−1
1
(εu + um )(εu + um )dxdt
=
(εu + u )(εux + mu
ux )|0 dt −
2
∂x
0
0
0
Z TZ 1 2
∂
=−
(εu + um )(εu + um )dxdt
2
∂x
0
0
Z TZ 1
=
(εu + um ) (−ut + aλu + λf (u(x, t − τ1 ), · · · , u(x, t − τn )) + λg(x, t)
0
Z0 t
−α(t−s)
+λγ
e
u(x, s)ds dxdt
t−τ
Z TZ 1
=
[−(εu + um )ut + aλu(εu + um ) + λf (u(x, t − τ1 ), · · · , u(x, t − τn ))(εu + um )
0
0
Z t
m
m
−α(t−s)
+λg(εu + u ) + λγ(εu + u )
e
u(x, s)ds dxdt
=
Z
Z
t−τ
≤|a|
Z
+
Z
T
Z
0
≤|a|ε
T
T
0
+
+
n
X
i=1
n
X
+γ
0
Z
m
βi
T
Z
T
0
Z
1
Z
T
1
Z
|u(x, t − τi )|(εu + um )dxdt
0
m
t
Z
g(εu + u )dxdt + γ
(εu + u )
e−α(t−s) u(x, s)dsdxdt
0
0
0
t−τ
Z 1
Z TZ 1
Z TZ 1
Z TZ 1
2
m+1
u dxdt + |a|
u
dxdt + Kε
udxdt + K
um dxdt
0
T
Z
0
βi
Z
T
0
T
n
X
i=1
1
βi ε
0
≤C.
u(εu + u )dxdt +
0
i=1
Z
m
Z
0
Z
1
Z
0
1
Z
Z
1
u2 dxdt +
0
i=1
1
0
βi ε
0
Z
|u(x, t − τi )|2 dxdt + γε
0
um
0
n
X
t
0
|u(x, t − τi )|2 dxdt +
0
n
X
i=1
Z
0
Z
0
1
T
Z
1
u
0
Z
t
βi
Z
0
0
T Z 1
u2m dxdt
0
e−α(t−s) u(x, s)dsdxdt
t−τ
e−α(t−s) u(x, s)dsdxdt
t−τ
For the convenience of further discussion, we denote
u(t) := u(x, t), u(t + θ) := u(x, t + θ),
and have the following result
Lemma 2.4 Assume that (H1 ), (H2 ) hold and u is a nonnegative T-periodic solution
of the equation (2.4) satisfying the boundary value condition (2.2). Then we have
kukL∞ (Q) ≤ C,
EJQTDE, 2010 No. 52, p. 8
where C > 0 is a constant.
Proof. Define
V (t) =
Z
2 !
Z 0 Z 1
n
X
1 2 1 ∂
2
m
βi
u2 (t + θ)dxdθ,
u + (εu + u ) dx +
2
2 ∂x
−τi 0
i=1
1
0
(2.6)
then by the Cauchy inequality and assumption (H1 ), it follows that
Z 1
n
X
∂
∂
m ∂
m
2
(εu + u )
(εu + u ) dx +
βi
[u2 − u2 (t − τi )]dx
V (t) =
uut +
∂x
∂t ∂x
0
0
i=1
Z
Z 1
n
1
X
∂2
βi2
[u2 − u2 (t − τi )]dx
=
[uut − 2 (εu + um )(ε + mum−1 )ut ]dx +
∂x
0
0
i=1
Z 1 2
∂
(εu + um ) + λau + λf (u(x, t − τ1 ), · · · , u(x, t − τn )) + λg(t, x)
=
u
2
∂x
0
Z t
−α(t−s)
+λγ
e
u(x, s)ds dx
t−τ
Z 1
+
(ε + mum−1 )ut [(−ut + λau + λf (u(x, t − τ1 ), · · · , u(x, t − τn )) + λg(t, x)
0
Z t
Z 1
n
X
−α(t−s)
2
+λγ
e
u(x, s)ds dx +
βi
[u2 − u2 (t − τi )]dx
Z
′
1
t−τ
0
i=1
n
X
"
#
∂
∂u
≤
− (εu + um )
βi u(x, t − τi )u + |u||g(x, t)| dx
+ |a|u2 +
∂x
∂x
0
i=1
#
Z 1"
n
X
˜ t ||u| + f˜
+
−f˜u2t + |a|f|u
βi u(t − τi )|ut| + f˜|ut ||g| dx
Z
1
0
+
i=1
n
X
βi2
i=1
+γ
≤
Z
1
Z
1
1
Z
u
0
[u2 − u2 (t − τi )]dx
0
Z
t
−α(t−s)
e
t−τ
−mu
Z 1 "X
n
0
+γ
Z
1
0
m−1
0
+
u(s)dsdx + γ
Z
˜ t|
f|u
!
2
n
X
∂u
+ (|a| +
βi2 )u2 dx
∂x
i=1
Z
t
e−α(t−s) u(s)dsdx
t−τ
˜ t| +
βi u(t − τi )u + ug − f˜u2t + |a|fu|u
i=1
1
u
0
Z
n
X
#
˜ t |g dxdt
βi f˜u(t − τi )|ut | + f|u
i=1
t
t−τ
e−α(t−s) u(s)dsdx + γ
Z
0
1
˜ t|
f|u
Z
t
e−α(t−s) u(s)dsdx
t−τ
EJQTDE, 2010 No. 52, p. 9
!
2
Z 1
n
X
∂u
m−1
2 2
=
−mu
βi )u dx +
F dx
∂x + (|a| +
0
0
i=1
!
Z 1 (m+1)/2 2
Z 1
Z 1
2
2
−
−
∂u
dx + ε1 (m−1)
since
u2 dx ≤ ε1
um+1 dx + ε1 (m−1) ≤ ε1 µ
∂x
0
0
0
Z 1
Z 1 (m+1)/2 2 !
Z 1
n
X
∂u
−2/(m−1)
2
2
≤ − c1
u dx +
βi + c1 )ε1
+
F dx,
dx + (|a| +
∂x
0
0
0
i=1
1
Z
where
0 0
f˜ = ε + mum−1 ,
n
n
X
X
2
˜
˜
˜
˜
F = −f ut +|a|fu|ut| +
βi f u(t − τi )|ut| + f |ut |g +
βi u(t − τi )u
i=1
+ug + γu
Z
t
i=1
e−α(t−s) u(s)ds + γ f˜|ut |
t−τ
Z
t
e−α(t−s) u(s)ds.
t−τ
Letting
c2 = (|a| +
n
X
−2/(m−1)
βi2 + c1 )ε1
,
i=1
it is obvious that
V ′ (t) ≤ −c1
Z
1
0
On the other hand
Z TZ 1
F dxdt
0
≤
Z
0
Z
Z 1 (m+1)/2 2 !
∂u
2
u dx +
dx + c2 +
∂x
0
1
F dx.
(2.7)
0
0
T
Z
+ γu
1
0
[−f˜u2t + |a|f˜u|ut| +
n
X
˜ t |g +
βi f˜u(t − τi )|ut | + f|u
i=1
Z
t
t−τ
−α(t−s)
e
u(s)ds + γ f˜ut
n
X
βi u(t − τi )u + u|g|
i=1
Z
t
e−α(t−s) u(s)ds]dxdt
t−τ
EJQTDE, 2010 No. 52, p. 10
T
Z 1"
n
n
ε2 |a| ˜ 2 |a| ˜ 2 ε3 X ˜ 2
1 X ˜ 2
ε4
≤
−f˜u2t +
βi f ut +
βi f u (t − τi ) + f˜u2t
f ut +
fu +
2
2ε2
2 i=1
2ε3 i=1
2
0
0
#
Z t
n
n
X
X
1
2
2
2
2
2
˜
βi u + u + g + γ f |ut |
f |g| +
βi u (t − τi ) +
e−α(t−s) u(s)ds dxdt
+
2ε4
t−τ
i=1
i=1
Z TZ 1
+C
um+1 dxdt + C.
Z
0
Since
Z TZ
0
1
Z t
˜
γ f |ut |
e−α(t−s) u(s)dsdxdt
0
0
t−τ
Z TZ 1
Z t
|α|τ
˜
≤e γ
f |ut|
u(s)dsdxdt
0
0
t−τ
" Z Z
#
2
Z T Z 1 Z t
T
1
ε
1
5
˜ 2 dxdt +
fu
f˜
≤e|α|τ γ
u(s)ds dxdt
t
2 0 0
2ε5 0 0
t−τ
Z TZ 1
Z TZ 1 Z t
τ
ε5
2
2
|α|τ
˜
˜
f ut dxdt +
f
u (s)dsdxdt
≤e γ
2 0 0
2ε5 0 0
t−τ
Z TZ 1
Z T Z 1Z T
Z Z
Z
τ
mτ T 1 m−1 T 2
ε5
2
2
|α|τ
˜
u
u (s)dsdxdt
f ut dxdt +
u (s)dsdxdt +
≤e γ
2 0 0
2ε5 0 0 −τ
2ε5 0 0
−τ
Z TZ 1
Z TZ 1
h τ i
Z T Z 1
ε5
mτ
τ
T
|α|τ
2
2
+1
≤e γ
u dxdt +
u2m dxdt
f˜ut dxdt +
2 0 0
2ε5 T
2ε
5 0
0
0
0
#
2
Z T Z 1 Z t
mτ
+
u2(s)ds dxdt
2ε5 0 0
−τ
Z TZ 1
Z Z
Z T Z 1
mτ T 1 2m
τ T h τ i
ε5
2
2
|α|τ
˜
+1
u dxdt +
u dxdt
f ut dxdt +
≤e γ
2 0 0
2ε5 T
2ε5 0 0
0
0
Z T Z 1
h τ i
mτ T
4
+
+1
u dxdt ,
(T + τ )
2ε5
T
0
0
choosing ε2 , ε3 , ε4 , ε5 small appropriately, we have
Z TZ T
F dxdt
0
0
ZZ "
n
n
ε4 ε5 e|α|τ ˜ 2 |a| ˜ 2
1 X ˜ 2
|a|ε2 ε3 X
βi + +
βi f u (t − τi )
fu +
+
γ)f ut +
≤
(−1 +
2
2 i=1
2
2
2ε2
2ε3 i=1
Q
n
n
X
X
1 ˜ 2
τ T h τ i
+
+ 1 u2
f |g | +
βi u2 (t − τi ) + (
βi + 1)u2 + g 2 + e|α|τ γ
2ε4
2ε5 T
i=1
i=1
EJQTDE, 2010 No. 52, p. 11
|α|τ
+e
Z TZ 1
h τ i
mτ 2m
4
|α|τ mτ T
γ
+ 1 u dxdt + C
u +e γ
(T + τ )
um+1 dxdt + C.
2ε5
2ε5
T
0
0
Applying Lemma 2.1, kukLr ≤ C, for any r > 0, yields
Z
T
0
Z
1
F dxdt ≤ C.
(2.8)
0
Since V (t) and u are T-periodic and from (2.7), (2.8), we have
Z T
0=
V ′ (t)dt
0
!
Z TZ 1
Z TZ 1
Z T Z 1 (m+1)/2 2
∂u
2
F dxdt,
≤ − c1
u dxdt +
dxdt + c2 T +
∂x
0
0
0
0
0
0
which implies
Z
T
0
Z
1
2
u dxdt ≤ C,
0
Z
T
0
Z
0
1
(m+1)/2 2
∂u
dxdt ≤ C.
∂x
Applying Lemma 2.1 and Lemma 2.3 yields
Z T
V (t)dt
0
2 !
Z TZ 0 Z 1
Z TZ 1
n
X
1 2 1 ∂
m
βi
u2 (t + θ)dxdθdt
u + (εu + u ) dxdt +
≤
2
2
∂x
0
−τi 0
0
0
i=1
≤C.
Since V is continuous, there exists a t0 ∈ [0, T ] satisfies
V (t0 ) ≤
C
≤ C.
T
Hence, if t0 ≤ t ≤ t0 + T , we obtain
Z t
V (t) =V (t0 ) +
V ′ (s) ds
t0
Z
Z "
t
≤C +
1
−c1
t0
≤C +
Z
0
T
0
Z 1"
0
c1
#
(m+1)/2 2 !
Z 1
∂u
dx + c2 +
F dx ds
u2 +
∂x
0
#
(m+1)/2 2 !
∂u
+ F dxdt + c2 T
u2 +
∂x
EJQTDE, 2010 No. 52, p. 12
≤C.
A simple calculation yields
2
Z 1
Z 1
∂
(εu + um ) dx =
|(ε + mum−1 )ux |2 dx
∂x
0
0
Z 1
Z 1 m 2
∂u
m−1
2
≥
|mu
ux | dx =
∂x dx.
0
0
By the definition of V , we see that
m 2 !
Z 1
Z 1
∂u
2
u +
dx ≤
∂x
0
0
which implies
sup
0≤t≤T
Z
0
1
2 !
∂
m
u + (εu + u ) dx ≤ 2V (t) ≤ C,
∂x
2
m 2 !
∂u
dx ≤ C.
u2 +
∂x
It follows from the definition of u that
Z x m
Z x m
m
∂u
(s,
t)
∂u
(s,
t)
m
m
ds =
ds
|u(x, t)| = |u (x, t)| = u (0, t) +
∂s
∂s
0
0
Z 1 m
∂u
≤
∂x dx
0
Z 1 m 2 !1/2
∂u
≤ sup
≤ C 1/2 ,
∂x dx
0≤t≤T 0
that is
kukL∞ (Q) ≤ C 1/(2m) := C0 .
The proof is complete.
In addition, we can also easily get the L2 boundedness of
∂um
∂t
as follows.
Lemma 2.5 Assume that (H1 ), (H2 ) hold and u is a nonnegative T -periodic solution
of the equation (2.1) satisfying the boundary value condition (2.2). Then there exists a
constant C > 0 such that
Z Z m 2
∂u
∂t dxdt ≤ C.
Q
Proof. Multiplying the equation (2.1) by
obtain
ZZ
∂u ∂
(εu + um )dxdt
∂t
∂t
Q
∂
(εu
∂t
+ um ) and integrating over Q, we
EJQTDE, 2010 No. 52, p. 13
ZZ
∂
∂2
m ∂
m
(εu + u ) (εu + u )dxdt + a
u (εu + um )dxdt
=
2
∂x
∂t
Q ∂t
ZQZ
∂
+
[f (u(x, t − τ1 ), · · · , u(x, t − τn )) + g(x, t)] (εu + um )dxdt
∂t
Q
ZZ Z t
∂
+
γ
e−α(t−s) u(s)ds (εu + um )dxdt.
∂t
Q
t−τ
ZZ
A simple calculation yields
2
Z Z
Z TZ 1 2
∂
1 T 1 ∂
∂
m ∂
m
m
(εu + u ) (εu + u )dxdt = −
(εu + u ) dxdt = 0.
2
∂t
2 0 0 ∂t ∂x
0
0 ∂x
Therefore, by Lemma 2.4, we obtain
Z TZ 1
Z TZ 1
2
ε
ut dxdt +
mum−1 u2t dxdt
0
0
0
0
Z TZ 1
Z TZ 1
n
X
∂
∂
m
≤
βi
u(t − τi ) (εu + u ) dxdt +
g(x, t) (εu + um )dxdt
∂t
∂t
0
0
0
0
i=1
Z Z Z t
∂
(εu + um )dxdt
+
γ
e−α(t−s) u(s)ds
∂t
Q
t−τ
Z TZ T
∂
m
≤C
∂t (εu + u ) dxdt
0
0
Z TZ 1
Z Z
2
ε Cε1
Cε2 T 1
C
C
2
≤
+
|ut | dxdt +
|mum−1 ut |2 dxdt +
2
2ε1
2 0 0
2ε2
0
0
Z TZ t
Z TZ 1
Cε2
mum−1 |ut |2 dxdt,
≤ε
|ut |2 dxdt + C +
2
0
0
0
0
where ε1 = 2/(Cε). Letting ε2 be appropriately small,
Z TZ 1
mum−1 u2t dxdt ≤ C,
0
0
then we can see
Z
0
T
Z TZ 1
Z 1 m 2
∂u
dxdt =
m2 u2(m−1) u2t dxdt
∂t
0
0
0
Z TZ 1
m−1
≤mC
mum−1 u2t dxdt ≤ C,
0
0
which completes the proof of Lemma 2.5.
We will show the H¨older norm estimate of solutions in the following lemma.
EJQTDE, 2010 No. 52, p. 14
Lemma 2.6 Assume that (H1 ), (H2 ) hold and u is a nonnegative T -periodic solution
of the equation (2.1) satisfying the boundary value condition (2.2). Then there exists a
constant C > 0 such that u ∈ C α,α/2 (Q) with 0 < α < 1/2.
Proof. In fact, through a similar discussion of [6] (see Chapter 2), we know that
u ∈ L∞ (0, T ; H01(0, 1)) and ∂u
∈ L2 (Q). By direct computations, for any x1 < x2 ∈ (0, 1),
∂t
we conclude that
Z x2
Z x2
∂u(x, t)
∂u(x, t)
dx ≤
|u(x2 , t) − u(x1 , t)| =
∂x dx
∂x
x1
x1
Z 1 2 !1/2
∂u
(2.9)
dx
≤
|x2 − x1 |1/2
∂x
0
≤C|x2 − x1 |1/2 .
On the other hand, to prove
|u(x, t2 ) − u(x, t1 )| ≤ C|t2 − t1 |1/4 ,
(2.10)
we need only consider the case that 0 ≤ x ≤ 21 , ∆t = t2 − t1 > 0, (∆t)α ≤ 14 , where α is
determined. Integrating (2.2) over (y, y + (∆t)α ) × (t1 , t2 ) gives
Z y+(∆t)α
[u(z, t2 ) − u(z, t1 )]dz
y
Z t2
∂
∂
α
m
α
m
=
(εu(y + (∆t) , s) + u (y + (∆t) , s)) −
(εu(y, s) + u (y, s)) ds
∂x
∂x
t1
Z t2 Z y+(∆t)α
+
[au(z, s) + f (u(z, s − τ1 ), · · · , u(z, s − τn )) + g(z, s)]dzds
+
Z
t1
t2
t1
y
y+(∆t)α
Z
γ
y
Z
s
e−α(s−σ) u(x, s)dσdzds,
s−τ0
i.e.
α
(∆t)
Z t2
1
Z
(u(y + θ(∆t)α , t2 ) − u(y + θ(∆t)α , t1 ))dθ
0
∂
∂
α
m
α
m
(εu(y + (∆t) , s) + u (y + (∆t) , s)) −
(εu(y, s) + u (y, s)) ds
=
∂x
∂x
t1
Z t2 Z y+(∆t)α
+
[au(z, s) + f (u(z, s − τ1 ), · · · , u(z, s − τn )) + g(z, s)]dzds
+
Z
t1
t2
t1
y
Z
y
y+(∆t)α
γ
Z
s
e−α(s−σ) u(x, s)dσdzds,
s−τ0
EJQTDE, 2010 No. 52, p. 15
Integrating the above equality with respect to y over (x, x + (∆t)α ), we conclude that
α
Z
x+(∆t)α
(∆t)
x
Z x+(∆t)α Z
=
t2
Z
1
(u(y + θ(∆t)α , t2 ) − u(y + θ(∆t)α , t1 ))dθdy
0
∂
∂
α
m
α
m
(εu(y + (∆t) , s) + u (y + (∆t) , s)) −
(εu(y, s) + u (y, s)) dsdy
∂x
∂x
x
t1
Z x+(∆t)α Z t2 Z y+(∆t)α
+
[au(z, s) + f (u(z, s − τ1 ), · · · , u(z, s − τn )) + g(z, s)]dzdsdy
x
+
Z
x+(∆t)α
x
≤C|(∆t)
t1
t2
Z
t1
α+1 1/2
|
y
Z
y+(∆t)α
γ
y
+ C(∆t)
Z
s
e−α(s−σ) u(x, s)dσdzdsdy,
s−τ0
2α+1
.
Hence, by a simple calculations, we have
|u(x∗ , t2 ) − u(x∗ , t1 )| ≤ C(∆t) + C(∆t)(α+1)/2−2α ≤ C|t2 − t1 |1/4 ,
where x∗ = y ∗ + θ∗ (∆t)α , y ∗ ∈ (x, x + (∆t)α ), θ∗ ∈ (0, 1), and choose α = 1/6 specially,
from which we see that (2.10) holds. Therefore, by (2.9) and (2.10), we obtain that
|u(x2 , t2 ) − u(x1 , t1 )|
≤|u(x2 , t2 ) − u(x∗ , t2 )| + |u(x∗ , t2 ) − u(x∗ , t1 )| + |u(x∗ , t1 ) − u(x1 , t1 )|
≤C(|x2 − x∗ |1/2 + |t2 − t1 |1/4 + |x∗ − x1 |1/2 )
≤C(|x2 − x1 | + |t2 − t1 |1/2 )1/2 ,
which completes the proof of Lemma 2.6.
3
Proof of the Main Result
By means of the above proved lemmas and the Leray-Schauder fixed point theorem, we
can obtain the existence of solutions uε of the regularized problem as follows.
Proposition 3.1 Assume that (H1 ) and (H2 ) hold. Then the regularized problem
(2.2)–(2.4) has a nonnegative T -periodic solution.
Proof. Denote by CT (Q) the set of all continuous functions u with the T -periodicity
in t. We study the following regularized equation
∂2
∂u
= 2 (εu + |u|m−1 u) + g(x, t)
∂t
∂x
(3.1)
EJQTDE, 2010 No. 52, p. 16
where 0 ≤ g ∈ CT (Q). We claim that, if the problem (3.1),(2.2),(2.3) has a unique T periodic solution u, then u must be nonnegative. In fact, multiplying (3.1) by u− and
integrating over Q, we obtain
ZZ
ZZ
ZZ
∂u
∂2
m−1
u− dxdt =
(εu + |u|
u)u− dxdt +
gu− dxdt,
2
Q ∂t
Q ∂x
Q
where u− = min{0, u(x, t), (x, t) ∈ Q}. Making use of integration by parts, we have
ZZ
ZZ
∂
∂u−
m−1
(εu− + |u− |
u− )
dxdt =
gu− dxdt ≤ 0.
∂x
Q ∂x
Q
Since
ZZ
Q
∂u−
∂
(εu− + |u− |m−1 u− )
dxdt =
∂x
∂x
then we get
ZZ
|u− |
m−1
Q
Therefore,
u− = 0,
ZZ
(ε + m|u− |
Q
∂u− 2
∂x dxdt ≤ 0.
m−1
∂u− 2
dxdt,
)
∂x
a.e. in Q
By the definition of u− , we see that
u ≥ 0,
a.e. in Q.
Consequently, we can rewrite the equation (3.1) as
∂2
∂u
= 2 (εu + um ) + g(x, t),
∂t
∂x
x ∈ (0, 1), t ∈ R.
(3.2)
Hence, we know that, if the problem (3.2),(2.2),(2.3) has a T -periodic solution, it must
be nonnegative.
Similarly, we have the same consequence for the equation
∂u
∂2
= 2 (εu + um ) − ζu + g(x, t),
∂t
∂x
(3.3)
with the conditions (2.2) and (2.3), where ζ ≥ 0.
On the other hand, with an argument similar to [7], we claim that for any g ∈ CT (Q),
the problem
∂2
∂u
= 2 (εu + um ) + g(x, t),
∂t
∂x
x ∈ (0, 1), t ∈ R,
EJQTDE, 2010 No. 52, p. 17
u(0, t) = u(1, t),
u(x, t) = u(x, t + T ),
t ∈ R,
x ∈ (0, 1), t ∈ R.
has a unique solution u ∈ C α,α/2 (Q). By constructing a homotopy, it is easy to obtain
that the problem (3.3),(2.2),(2.3) also admits a solution u ∈ C α,α/2 (Q) .
Next, we will obtain the existence of periodic solutions for the regularized problem
(2.1),(2.2),(2.3).
In case that a ≥ 0, we consider the periodic problem of the homotopy equation for
regularized problem
∂2
∂u
= 2 (εu + um ) + λG(x, t),
∂t
∂x
u(0, t) = u(1, t) = 0,
x ∈ (0, 1), t ∈ R,
(3.4)
t ∈ R,
(3.5)
where for any v(x, t) ∈ CT (Q),
G(x, t) = av(x, t) + f (v(x, t − τ1 ), · · · , v(x, t − τn )) + g(x, t) + γ
α,α/2
Then problem (3.4)–(3.5) admits a unique solution u ∈ CT
Z
t
e−α(t−s) v(x, s)ds.
t−τ0
(Q). Define the mapping
L : CT (Q)×[0, 1] −→ CT (Q),
(v, λ) 7−→ u.
α,α/2
Since CT
(Q) can be compactly embedded into CT (Q), L is compact. By Lemma 2.4,
we know that for any fixed point uλ of the mapping L, there is a constant C0 independent
of ε and λ, such that
kuλ kL∞ ≤ C0 .
Then in applying Leray-Schauder’s fixed point theorem, we know that the problem (2.1)–
(2.3) admits a solution uε .
In case that a < 0, we consider the periodic problem of the homotopy equation for
regularized problem
∂u
∂2
= 2 (εu + um ) + au + λG(x, t),
∂t
∂x
u(0, t) = u(1, t) = 0,
x ∈ (0, 1), t ∈ R,
(3.6)
t ∈ R,
(3.7)
where for any v(x, t) ∈ CT (Q),
G(x, t) = f (v(x, t − τ1 ), · · · , v(x, t − τn )) + g(x, t) + γ
Z
t
e−α(t−s) v(x, s)ds.
t−τ0
EJQTDE, 2010 No. 52, p. 18
α,α/2
Then problem (3.6)–(3.7) admits a unique solution u ∈ CT
(Q). The following progress
is the same as above case, then we get that the problem (2.1)–(2.3) admits a solution uε .
Now, we turn to the proof of the our main result based on the above lemmas and
Proposition 3.1
The Proof of Theorem 2.1. Let ε = 1/h (h = 1, 2, · · · ) and we note uh for the
solution of the problem (2.2)–(2.4). Clearly, according to Lemma 2.2, Lemma 2.4 and
Lemma 2.5,
kuh kL∞ (Q) ≤ C0 ,
m
∂uh
∂x
2 ≤ C1 T,
L (Q)
m
2
∂uh
∂t
2 ≤ C.
L (Q)
Hence there exists a subsequence {uh }∞
to be {uh }∞
h=1 , supposed
h=1 itself, and a function
m
u ∈ {u; u ∈ L∞ ; um ∈ L∞ (0, T ; W01,2(0, 1)); ∂u∂t ∈ L2 (Q)}, such that
uh (x, t) → u(x, t),
in L2 (Q),
∂um
∂um (x, t)
h (x, t)
→
,
in L2 (Q),
∂x
∂x
f (uh (x, t − τ1 ), · · · , uh (x, t − τn )) → f (u(x, t − τ1 ), · · · , u(x, t − τn )),
in L2 (Q).
Letting h → ∞ in
Z T Z 1
1 ∂ 2 ϕ ∂um ∂ϕ
u h ϕt + u h 2 − h
+ auh ϕ + f (uh (x, t − τ1 ), · · · , uh (x, t − τn ))ϕ
h ∂x
∂x ∂x
0
0
Z t
−α(t−s)
+g(x, t)ϕ + γ
e
uh (x, s)ds ϕ dxdt = 0,
t−τ
we have
Z
T
0
Z 1
0
∂um ∂ϕ
+ auϕ + f (u(x, t − τ1 ), · · · , u(x, t − τn ))ϕ
∂x ∂x
Z t
−α(t−s)
+g(x, t)ϕ + γ
e
u(x, s)ds ϕ dxdt = 0.
uϕt −
t−τ
That shows u satisfies the integral identity in the definition of weak solutions. Therefore,
the problem (1.1)–(1.2) admits a nonnegative T -periodic solution u ∈ {u; u ∈ L∞ ; um ∈
m
L∞ (0, T ; W01,2(0, 1)); ∂u∂t ∈ L2 (Q)}.
Since g(x, t) 6≡ 0, we see that the T -periodic solution is nontrivial. The proof of
Theorem 2.1 is complete.
EJQTDE, 2010 No. 52, p. 19
Remark. In equation (1.1), if the delays depend on time, we may construct a similar
Lyapunov-type functional
2 !
Z 1
Z 0 Z 1
n
X
1 2 1 ∂
m
2
V (t) =
u + (εu + u ) dx +
βi
u2 (t + θ)dxdθ.
2
2 ∂x
0
−τi (t) 0
i=1
Under suitable assumptions, by the similar arguments, the corresponding existence of
time periodic solution should be established for evolution equations with variable delays.
References
[1] M. Wazewska-Czyzevsia, A. Lasota, Mathematical problems of dynamics of red blood
cell system, Ann. Polish Math. Soc. Ser. 3 Appl. Math., 17(1988), 23–40.
[2] W. S. C. Gurney, S. P. Blythe, R. M. Nisbet, Nicholson’s blowflies revisited, Nature,
287(1980), 17–21.
[3] J. H. So, J. Wu, X. Zou, Structured population on two patches: modeling desperal
and delay, J. Math. Biol., 43(2001), 37–51.
[4] Ying Yang, Jingxue Yin and Chunhua Jin, Existence of time periodic solutions for
one dimensional p-Laplacian with multiple delays, Nonlinear Analysis: Real World
Applications, 11(2)(2010), 764–777.
[5] Ying Yang, Ran Zhang, Chunhua Jin and Jingxue Yin, Existence of time periodic
solutions for the Nicholson’s blowflies model with Newtonian diffusion, Mathematical
Methods in the Applied Sciences, doi: 10.1002/mma.1228.
[6] Yifu Wang, Periodic solutions of nonlinear diffusion equation, 1997.
[7] Y. Wang, J. Yin, Z. Wu, Periodic solutions of evolution p-Laplacian equations with
nonlinear sources, J. Math. Anal. Appl., 219(1998), 76–96.
(Received March 5, 2010)
EJQTDE, 2010 No. 52, p. 20
2010, No. 52, 1-20; http://www.math.u-szeged.hu/ejqtde/
Existence of Time Periodic Solutions for
One-Dimensional Newtonian Filtration
Equation with Multiple Delays∗
Ying Yanga , Jingxue Yinb and Chunhua Jinb,c*†
a
b
Department of Mathematics, Jilin University, Changchun 130012, China
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
c
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Abstract
In this paper, we study one-dimensional Newtonian filtration equation including
unbounded sources with multiple delays. The existence of nonnegative non-trivial
time periodic solutions will be established by the Leray-Schauder fixed point theorem based on some suitable Lyapunov functionals and some a priori estimates for
all possible periodic solutions.
Keywords: Newtonian Filtration; Multiple Delays; Periodic Solution.
1
Introduction
Consider the following one-dimensional Newtonian filtration equation with multiple delays
Z t
∂ 2 um
∂u
=
+ au + f (u(x, t − τ1 ), · · · , u(x, t − τn )) + g(x, t)+γ
e−α(t−s) u(x, s)ds,
∂t
∂x2
t−τ0
x ∈ (0, 1), t ∈ R,
(1.1)
subject to the homogeneous Dirichlet boundary value condition
u(0, t) = u(1, t) = 0,
t ∈ R,
(1.2)
∗
This work is partially supported by the National Science Foundation of China, partially supported
by a Specific Foundation for Ph.D. Specialities of Educational Department of China, partially supported
by the Foundation for Post Doctor of China and partially supported by Graduate Innovation Fund of
Jilin University (No. 20101045).
†
Corresponding author. Email: jinchhua@126.com
EJQTDE, 2010 No. 52, p. 1
where m > 1, a and α are constants, γ is a positive constant, f and g are the known
functions satisfying some structure conditions.
This kind of equation arises from a variety of areas in applied mathematics, physics
and mathematical ecology. For the case m = 1, n = 1 and γ = 0 with f (r) = αr/(1 + r β ),
which appears in the blood cell production model[1]
∂u
∂2u
= 2 − au + f (u(x, t − τ )) + g(x, t).
∂t
∂x
On the other hand, for the same case, that is m = 1, n = 1, γ = 0, with different f , the
equation (1.1) also is known as the Hematopoiesis model (f (r) = e−kr (k > 0)) as well
as Nicholson’s blowflies model (f (r) = re−kr (k > 0)), see for example [2, 3]. While, it
is worth noting that all the above models are linearly diffusive, but if nonlinear diffusion
is introduced, the model will be more consistent with biologic phenomena in the real
world. However, as far as we know, only a few works are concerned with time periodic
solutions for degenerate parabolic equation with delay(s). For example, in [4], the authors
investigated the existence of time periodic solutions for p-Laplacian with multiple delays.
In [5], the authors studied the existence of periodic solutions for Nicholson’s blowflies
model with Newtonian diffusion
Z t
∂ 2 um
∂u
−au(x,t−τ )
=
− δu + pu(x, t − τ )e
+ g(x, t) + β
e−α(t−s) u(x, s)ds.
∂t
∂x2
t−τ
Nevertheless, in this paper, a more general source will be discussed, which is allowed to
be the blood cell production model or other types.
In the present paper, we pay our attention to the existence of nonnegative time periodic
solutions for (1.1). It is worth noticing that in the model of [5], the source with delay is a
typical but quite special bounded source. However, in this paper, a more general source
will be discussed, particularly, the source with delays is allowed to be unbounded, which
caused us difficulties in making the maximum norm estimates and some other a priori
estimates. On the other hand, the method used in[4] will also not work for the equation
we consider, that is the coefficient matrix associated with Lyapunov function depends on
solutions of the problem, and therefore the required estimates as did in [4] could not be
obtained. So, we must try some other methods. By constructing some suitable Lyapunov
functionals, the a priori estimates for all possible periodic solutions, and combining with
Leray-Schauder fixed point theorem, we finally establish the existence of time periodic
solutions.
The rest of this paper is organized as follows. In Section 2 we introduce some basic
assumptions, preliminary lemmas and state the main results of this paper. Section 3 is
devoted to investigating the existence of periodic solutions based on the a priori estimates
obtained in Section 2 and Leray-Schauder fixed point theorem.
EJQTDE, 2010 No. 52, p. 2
2
Preliminaries and the Main Result
Throughout this paper, we make the following assumptions:
(H1 ) 0 ≤ g ∈ C(Q), g(x, t) 6≡ 0, g(x, t + T ) = g(x, t);
(H2 ) f (0, · · · , 0) = 0, f (r1 , · · · , rn ) ≥ 0, ri ≥ 0 (i = 1, · · · , n) and
|f (a1 , · · · , an ) − f (b1 , · · · , bn )| ≤
n
X
βi |ai − bi |;
i=1
where T and βi are positive constants, Q = (0, 1) × (0, T ).
Since the equation (1.1) is degenerate parabolic and problem (1.1)–(1.2) usually admits
solutions only in some generalized sense. Hence we introduce the following definition.
Definition 2.1 A function u is said to be a weak solution of the problem (1.1)–(1.2),
m
if u ∈ {w; w ∈ L∞ , w m ∈ L∞ (0, T ; W01,2(0, 1)), ∂w∂t ∈ L2 (Q)}, and for any ϕ ∈ C ∞ (Q)
with ϕ(x, 0) = ϕ(x, T ) and ϕ(0, t) = ϕ(1, t) = 0, the following integral equality holds
Z T Z 1
∂um ∂ϕ
uϕt −
+ auϕ + f (u(x, t − τ1 ), · · · , u(x, t − τn ))ϕ + g(x, t)ϕ
∂x ∂x
0
0
Z t
−α(t−s)
+ γ
e
u(x, s)ds ϕ dxdt = 0.
t−τ
Now we state the main result of this paper.
Theorem 2.1 Suppose that (H1 ) and (H2 ) hold. Then the problem (1.1)–(1.2) admits
at least one nonnegative T -periodic solution.
To prove the existence of periodic solutions (1.1)–(1.2), let us first consider the regularized problem
∂uε
∂2
= 2 (εuε + um
ε ) + auε + f (uε (x, t − τ1 ), · · · , uε (x, t − τn ))
∂t
∂x
Z
t
−α(t−s)
+ g(x, t) + γ
e
uε ds,
(2.1)
x ∈ (0, 1), t ∈ R,
t−τ0
uε (0, t) = uε (1, t),
uε (x, t) = uε (x, t + T ),
t ∈ R,
x ∈ [0, 1], t ∈ R.
(2.2)
(2.3)
The desired solution of the problem (1.1)–(1.2) will be obtained by the limit of some
subsequence of solutions uε of the regularized problem. However, we need first to establish
the existence of solutions uε , for which, we will make use of the Leray-Schauder fixed point
theorem and our efforts center on obtaining the uniformly boundness of uε . To this end,
we prove the following lemmas.
EJQTDE, 2010 No. 52, p. 3
Lemma 2.1 Assume that (H1 ), (H2 ) hold and u is a nonnegative T-periodic solution
of the equation
ut =
∂2
m
(εu
+
u
)
+
λ
au + f (u(x, t − τ1 ), · · · , u(x, t − τn )) + g(x, t)
∂x2
Z t
+γ
e−α(t−s) u(x, s)ds ,
(2.4)
t−τ
satisfying the boundary value condition (2.2), where λ ∈ [0, 1], 0 < ε < 1 is a constant
which is arbitrary. Then for any r > 0, we have
Z TZ 1
um+r dxdt ≤ C1 (m, r),
0
0
where C1 (m, r) > 0 is a constant which depend on m and r.
Proof. Note that, multiplying Eq.(2.4) by ur and integrating over Q,
Z TZ 1
Z TZ 1
Z TZ 1 2
∂
r
m
r
ut u dxdt =
(εu + u ) · u dxdt + λ a
ur+1dxdt
2
∂x
0
0
0
0
0
0
Z TZ 1
Z TZ 1
r
+
f (u(x, t − τ1 ), · · · , u(x, t − τn ))u dxdt +
gur dxdt
0
0
0
0
Z TZ 1
Z t
r
−α(t−s)
+
uγ
e
u(x, s)dsdxdt .
0
Since u is T-periodic,
Z TZ
0
0
0
1
t−τ
1
∂u r
u dxdt =
∂t
r+1
Z
0
T
Z
0
1
∂ur+1
dxdt = 0,
∂t
it follows that
Z TZ 1
∂
∂ur
(εu + um )
dxdt
∂x
0
0 ∂x
Z TZ 1
Z TZ 1
r+1
=λa
u dxdt + λ
f (u(x, t − τ1 ), · · · , u(x, t − τn ))ur dxdt
0
0
0
0
Z TZ 1
Z TZ 1
Z t
r
r
+λ
gu dxdt + λ
uγ
e−α(t−s) u(x, s)dsdxdt
≤|a|
Z
+
Z
0
0
0
0
T Z 1
u
0
r+1
dxdt +
0
T
Z
0
1
ur γ
n
X
i=1
Z
t
βi
Z
0
T
0
t−τ
Z
1
r
u(x, t − τi )u dxdt + K
0
Z
0
T
Z
1
ur dxdt
0
e−α(t−s) u(x, s)dsdxdt
t−τ
EJQTDE, 2010 No. 52, p. 4
≤|a|ε1
T
Z
0
+
n
X
1
Z
u
m+r
0
−r/m
ε3
i=1
Z
T
0
≤ |a|ε1 +
Z
1
u
Z
0
T Z
m+r
+
1
|u(x, t − τi )|m+r dxdt +
0
βi ε2
n
X
dxdt +
βi ε2 +
−(r+1)/(m−1)
ε1
+
n
X
i=1
n
X
0
n
X
−r/m
Kε4
T
−r/m
βi ε2
ε3
+
T
Z
0
+ Kε4
Z
r
uγ
0
!Z
0
−r/m −1/(m−1)
βi ε2
ε3
1
+
T
0
1
Z
r
uγ
0
Z
t
Z
1
um+r dxdt
0
−r/m −1/(m−1)
ε3
T
βi ε2
+
T
Z
t
e−α(t−s) u(x, s)dsdxdt
t−τ
Z
1
um+r dxdt
0
−r/m
Kε4
i=1
Z
T
Z
i=1
0
i=1
+
n
X
i=1
βi ε2
+ Kε4
dxdt +
−(r+1)/(m−1)
|a|ε1
T
!
T
e−α(t−s) u(x, s)dsdxdt.
t−τ
On the other hand
Z TZ 1
Z t
r
uγ
e−α(t−s) u(x, s)dsdxdt
0
0
t−τ
Z TZ 1 Z t
Z TZ 1 Z T
|α|τ
r
|α|τ
≤e γ
u
u(x, s)dsdxdt ≤ e γ
ur
u(x, s)dsdxdt
0
0
t−τ
0
0
−τ
#
" Z Z
m+r
Z Z Z
T
1
≤e|α|τ γ ε5
0
"
Z
Z
=e|α|τ γ ε5
|α|τ
0
T
0
T
Z
−
1
um+r dxdt +
0
Z
1
m+r
T
r
um+r dxdt + ε5 m
−r
ε5 m T
1
T
m
u(x, s)ds
0
0
Z 1 Z
0
−r
ε5 m T (T
dxdt
−τ
T
u(x, s)ds
−τ
r
m
Z 1Z
T
m+r
m
#
dx
m+r
m
dsdx
+ τ)
γ ε5
u
dxdt +
u(x, s)
0
0
0
−τ
Z TZ 1
h τ i
Z 1Z T
r
m+r
r
−m
|α|τ
m+r
m
m
dsdx
≤e γ ε5
u
dxdt + ε5 T (T + τ )
+1
u(x, s)
T
0
0
0
0
Z TZ 1
Z 1Z T
h τ i
r
r
−m
|α|τ
m+r
≤e γ ε5
+ 1 ε6
u
dxdt + ε5 T (T + τ ) m
u(x, s)m+r dsdx
T
0
0
0
0
h τ i
− 1
r
r
−
+ε5 m T 2 (T + τ ) m
+ 1 ε6 m−1 .
T
≤e
Then we know that
Z TZ 1
0
0
∂
∂ur
(εu + um )
dxdt
∂x
∂x
EJQTDE, 2010 No. 52, p. 5
n
X
≤ |a|ε1 +
βi ε2 +
i=1
−(r+1)/(m−1)
+
ε1
+
n
X
i=1
n
X
−r/m
βi ε2
ε3
+ Kε4
!Z
T
0
−r/m −1/(m−1)
ε3
Z
1
um+r dxdt
0
−r/m
βi ε2
+ Kε4
i=1
+
Z
T
1
Z
0
ur γ
0
Z
t
e−α(t−s) u(x, s)dsdxdt
βi ε2 +
−r/m
γε5
T (T
−(r+1)/(m−1)
+
n
X
−r/m
βi ε2
ε3 + Kε4 + e|α|τ γε5
i=1
i=1
+e
T
t−τ
n
X
≤ |a|ε1 +
|α|τ
!
ε1
+ τ)
+
r/m
n
X
τ
([ ] + 1)ε6
T
T
Z
0
−r/m −1/(m−1)
ε3
Z
1
um+r dxdt
0
−r/m
βi ε2
+ Kε4
i=1
|α|τ
+e
r
−m
γε5 T (T + τ )
n
X
≤ |a|ε1 +
βi ε2 +
i=1
|α|τ
+e
r
m
h τ i
T
n
X
+1
−r/m
βi ε2
− 1
ε6 m−1
T
ε3 + Kε4 + e|α|τ γε5
i=1
−r/m
γε5
T (T
+ τ)
r/m
h τ i
T
+ 1 ε6
Z
T
0
Z
1
um+r dxdt + C.
0
Here and below, we use C > 0 to denote different positive constants depending only on
the known quantities. In addition, it is easy to see that
Z TZ 1
Z TZ 1
r
∂
m ∂u
(εu + u )
dxdt =
(ε + mum−1 )rur−1u2x dxdt
∂x
∂x
0
0
0
0
Z TZ 1
≥
mrum+r−2 u2x dxdt
0
0
Z T Z 1
∂ (m+r)/2 2
4mr
dxdt,
u
=
(m + r)2 0 0 ∂x
which implies
4mr
(m + r)2
≤ |a|ε1 +
Z
T
0
n
X
Z
0
1
∂ (m+r)/2 2
dxdt
u
∂x
βi ε2 +
i=1
|α|τ
+e
−r/m
γε5
T (T
n
X
−r/m
βi ε2
ε3 + Kε4 + e|α|τ γε5
i=1
+ τ)
r/m
h τ i
T
+ 1 ε6
Z
0
T
Z
1
um+r dxdt + C
0
EJQTDE, 2010 No. 52, p. 6
≤ |a|ε1 +
n
X
βi ε2 +
i=1
|α|τ
+e
−r/m
γε5
T (T
n
X
−r/m
βi ε2
ε3 + Kε4 + e|α|τ γε5
i=1
+ τ)
r/m
h τ i
T
Z
+ 1 ε6 µ
0
T
Z 1
∂ (m+r)/2 2
dxdt + C.
u
∂x
0
Thence if ε1 , ε2 , ε3 , ε4 , ε5 and ε6 are appropriately small, we can get
Z T Z 1
∂ (m+r)/2 2
dxdt ≤ C(m, r)
u
∂x
0
0
(2.5)
Using Poincar´e inequality, we see that
Z TZ 1
um+r dxdt ≤ C1 (m, r),
0
0
which completes the proof of Lemma 2.1.
Lemma 2.2 Assume that (H1 ), (H2 ) hold and u is a nonnegative T-periodic solution
of the equation (2.4) satisfying the boundary value condition (2.2). Then we have
Z T Z 1 m 2
∂u
∂x dxdt ≤ C,
0
0
where C > 0 is a constant.
Proof. In fact, choosing r = m in (2.5), we obtain
Z T Z 1 m 2
∂u
∂x dxdt ≤ C.
0
0
Lemma 2.3 Assume that (H1 ), (H2 ) hold and u is a nonnegative T-periodic solution
of the equation (2.4) satisfying the boundary value condition (2.2). Then we have
2
Z T Z 1
∂
(εu + um ) dxdt ≤ C,
∂x
0
0
where C > 0 is a constant.
Proof. The proof is a direct verification. A simple calculation shows
2
Z T Z 1
∂
m
(εu + u ) dxdt
∂x
0
0
EJQTDE, 2010 No. 52, p. 7
T
1
∂
∂
(εu + um ) (εu + um )dxdt
∂x
0
0 ∂x
Z T
Z TZ 1 2
∂
m
m−1
1
(εu + um )(εu + um )dxdt
=
(εu + u )(εux + mu
ux )|0 dt −
2
∂x
0
0
0
Z TZ 1 2
∂
=−
(εu + um )(εu + um )dxdt
2
∂x
0
0
Z TZ 1
=
(εu + um ) (−ut + aλu + λf (u(x, t − τ1 ), · · · , u(x, t − τn )) + λg(x, t)
0
Z0 t
−α(t−s)
+λγ
e
u(x, s)ds dxdt
t−τ
Z TZ 1
=
[−(εu + um )ut + aλu(εu + um ) + λf (u(x, t − τ1 ), · · · , u(x, t − τn ))(εu + um )
0
0
Z t
m
m
−α(t−s)
+λg(εu + u ) + λγ(εu + u )
e
u(x, s)ds dxdt
=
Z
Z
t−τ
≤|a|
Z
+
Z
T
Z
0
≤|a|ε
T
T
0
+
+
n
X
i=1
n
X
+γ
0
Z
m
βi
T
Z
T
0
Z
1
Z
T
1
Z
|u(x, t − τi )|(εu + um )dxdt
0
m
t
Z
g(εu + u )dxdt + γ
(εu + u )
e−α(t−s) u(x, s)dsdxdt
0
0
0
t−τ
Z 1
Z TZ 1
Z TZ 1
Z TZ 1
2
m+1
u dxdt + |a|
u
dxdt + Kε
udxdt + K
um dxdt
0
T
Z
0
βi
Z
T
0
T
n
X
i=1
1
βi ε
0
≤C.
u(εu + u )dxdt +
0
i=1
Z
m
Z
0
Z
1
Z
0
1
Z
Z
1
u2 dxdt +
0
i=1
1
0
βi ε
0
Z
|u(x, t − τi )|2 dxdt + γε
0
um
0
n
X
t
0
|u(x, t − τi )|2 dxdt +
0
n
X
i=1
Z
0
Z
0
1
T
Z
1
u
0
Z
t
βi
Z
0
0
T Z 1
u2m dxdt
0
e−α(t−s) u(x, s)dsdxdt
t−τ
e−α(t−s) u(x, s)dsdxdt
t−τ
For the convenience of further discussion, we denote
u(t) := u(x, t), u(t + θ) := u(x, t + θ),
and have the following result
Lemma 2.4 Assume that (H1 ), (H2 ) hold and u is a nonnegative T-periodic solution
of the equation (2.4) satisfying the boundary value condition (2.2). Then we have
kukL∞ (Q) ≤ C,
EJQTDE, 2010 No. 52, p. 8
where C > 0 is a constant.
Proof. Define
V (t) =
Z
2 !
Z 0 Z 1
n
X
1 2 1 ∂
2
m
βi
u2 (t + θ)dxdθ,
u + (εu + u ) dx +
2
2 ∂x
−τi 0
i=1
1
0
(2.6)
then by the Cauchy inequality and assumption (H1 ), it follows that
Z 1
n
X
∂
∂
m ∂
m
2
(εu + u )
(εu + u ) dx +
βi
[u2 − u2 (t − τi )]dx
V (t) =
uut +
∂x
∂t ∂x
0
0
i=1
Z
Z 1
n
1
X
∂2
βi2
[u2 − u2 (t − τi )]dx
=
[uut − 2 (εu + um )(ε + mum−1 )ut ]dx +
∂x
0
0
i=1
Z 1 2
∂
(εu + um ) + λau + λf (u(x, t − τ1 ), · · · , u(x, t − τn )) + λg(t, x)
=
u
2
∂x
0
Z t
−α(t−s)
+λγ
e
u(x, s)ds dx
t−τ
Z 1
+
(ε + mum−1 )ut [(−ut + λau + λf (u(x, t − τ1 ), · · · , u(x, t − τn )) + λg(t, x)
0
Z t
Z 1
n
X
−α(t−s)
2
+λγ
e
u(x, s)ds dx +
βi
[u2 − u2 (t − τi )]dx
Z
′
1
t−τ
0
i=1
n
X
"
#
∂
∂u
≤
− (εu + um )
βi u(x, t − τi )u + |u||g(x, t)| dx
+ |a|u2 +
∂x
∂x
0
i=1
#
Z 1"
n
X
˜ t ||u| + f˜
+
−f˜u2t + |a|f|u
βi u(t − τi )|ut| + f˜|ut ||g| dx
Z
1
0
+
i=1
n
X
βi2
i=1
+γ
≤
Z
1
Z
1
1
Z
u
0
[u2 − u2 (t − τi )]dx
0
Z
t
−α(t−s)
e
t−τ
−mu
Z 1 "X
n
0
+γ
Z
1
0
m−1
0
+
u(s)dsdx + γ
Z
˜ t|
f|u
!
2
n
X
∂u
+ (|a| +
βi2 )u2 dx
∂x
i=1
Z
t
e−α(t−s) u(s)dsdx
t−τ
˜ t| +
βi u(t − τi )u + ug − f˜u2t + |a|fu|u
i=1
1
u
0
Z
n
X
#
˜ t |g dxdt
βi f˜u(t − τi )|ut | + f|u
i=1
t
t−τ
e−α(t−s) u(s)dsdx + γ
Z
0
1
˜ t|
f|u
Z
t
e−α(t−s) u(s)dsdx
t−τ
EJQTDE, 2010 No. 52, p. 9
!
2
Z 1
n
X
∂u
m−1
2 2
=
−mu
βi )u dx +
F dx
∂x + (|a| +
0
0
i=1
!
Z 1 (m+1)/2 2
Z 1
Z 1
2
2
−
−
∂u
dx + ε1 (m−1)
since
u2 dx ≤ ε1
um+1 dx + ε1 (m−1) ≤ ε1 µ
∂x
0
0
0
Z 1
Z 1 (m+1)/2 2 !
Z 1
n
X
∂u
−2/(m−1)
2
2
≤ − c1
u dx +
βi + c1 )ε1
+
F dx,
dx + (|a| +
∂x
0
0
0
i=1
1
Z
where
0 0
f˜ = ε + mum−1 ,
n
n
X
X
2
˜
˜
˜
˜
F = −f ut +|a|fu|ut| +
βi f u(t − τi )|ut| + f |ut |g +
βi u(t − τi )u
i=1
+ug + γu
Z
t
i=1
e−α(t−s) u(s)ds + γ f˜|ut |
t−τ
Z
t
e−α(t−s) u(s)ds.
t−τ
Letting
c2 = (|a| +
n
X
−2/(m−1)
βi2 + c1 )ε1
,
i=1
it is obvious that
V ′ (t) ≤ −c1
Z
1
0
On the other hand
Z TZ 1
F dxdt
0
≤
Z
0
Z
Z 1 (m+1)/2 2 !
∂u
2
u dx +
dx + c2 +
∂x
0
1
F dx.
(2.7)
0
0
T
Z
+ γu
1
0
[−f˜u2t + |a|f˜u|ut| +
n
X
˜ t |g +
βi f˜u(t − τi )|ut | + f|u
i=1
Z
t
t−τ
−α(t−s)
e
u(s)ds + γ f˜ut
n
X
βi u(t − τi )u + u|g|
i=1
Z
t
e−α(t−s) u(s)ds]dxdt
t−τ
EJQTDE, 2010 No. 52, p. 10
T
Z 1"
n
n
ε2 |a| ˜ 2 |a| ˜ 2 ε3 X ˜ 2
1 X ˜ 2
ε4
≤
−f˜u2t +
βi f ut +
βi f u (t − τi ) + f˜u2t
f ut +
fu +
2
2ε2
2 i=1
2ε3 i=1
2
0
0
#
Z t
n
n
X
X
1
2
2
2
2
2
˜
βi u + u + g + γ f |ut |
f |g| +
βi u (t − τi ) +
e−α(t−s) u(s)ds dxdt
+
2ε4
t−τ
i=1
i=1
Z TZ 1
+C
um+1 dxdt + C.
Z
0
Since
Z TZ
0
1
Z t
˜
γ f |ut |
e−α(t−s) u(s)dsdxdt
0
0
t−τ
Z TZ 1
Z t
|α|τ
˜
≤e γ
f |ut|
u(s)dsdxdt
0
0
t−τ
" Z Z
#
2
Z T Z 1 Z t
T
1
ε
1
5
˜ 2 dxdt +
fu
f˜
≤e|α|τ γ
u(s)ds dxdt
t
2 0 0
2ε5 0 0
t−τ
Z TZ 1
Z TZ 1 Z t
τ
ε5
2
2
|α|τ
˜
˜
f ut dxdt +
f
u (s)dsdxdt
≤e γ
2 0 0
2ε5 0 0
t−τ
Z TZ 1
Z T Z 1Z T
Z Z
Z
τ
mτ T 1 m−1 T 2
ε5
2
2
|α|τ
˜
u
u (s)dsdxdt
f ut dxdt +
u (s)dsdxdt +
≤e γ
2 0 0
2ε5 0 0 −τ
2ε5 0 0
−τ
Z TZ 1
Z TZ 1
h τ i
Z T Z 1
ε5
mτ
τ
T
|α|τ
2
2
+1
≤e γ
u dxdt +
u2m dxdt
f˜ut dxdt +
2 0 0
2ε5 T
2ε
5 0
0
0
0
#
2
Z T Z 1 Z t
mτ
+
u2(s)ds dxdt
2ε5 0 0
−τ
Z TZ 1
Z Z
Z T Z 1
mτ T 1 2m
τ T h τ i
ε5
2
2
|α|τ
˜
+1
u dxdt +
u dxdt
f ut dxdt +
≤e γ
2 0 0
2ε5 T
2ε5 0 0
0
0
Z T Z 1
h τ i
mτ T
4
+
+1
u dxdt ,
(T + τ )
2ε5
T
0
0
choosing ε2 , ε3 , ε4 , ε5 small appropriately, we have
Z TZ T
F dxdt
0
0
ZZ "
n
n
ε4 ε5 e|α|τ ˜ 2 |a| ˜ 2
1 X ˜ 2
|a|ε2 ε3 X
βi + +
βi f u (t − τi )
fu +
+
γ)f ut +
≤
(−1 +
2
2 i=1
2
2
2ε2
2ε3 i=1
Q
n
n
X
X
1 ˜ 2
τ T h τ i
+
+ 1 u2
f |g | +
βi u2 (t − τi ) + (
βi + 1)u2 + g 2 + e|α|τ γ
2ε4
2ε5 T
i=1
i=1
EJQTDE, 2010 No. 52, p. 11
|α|τ
+e
Z TZ 1
h τ i
mτ 2m
4
|α|τ mτ T
γ
+ 1 u dxdt + C
u +e γ
(T + τ )
um+1 dxdt + C.
2ε5
2ε5
T
0
0
Applying Lemma 2.1, kukLr ≤ C, for any r > 0, yields
Z
T
0
Z
1
F dxdt ≤ C.
(2.8)
0
Since V (t) and u are T-periodic and from (2.7), (2.8), we have
Z T
0=
V ′ (t)dt
0
!
Z TZ 1
Z TZ 1
Z T Z 1 (m+1)/2 2
∂u
2
F dxdt,
≤ − c1
u dxdt +
dxdt + c2 T +
∂x
0
0
0
0
0
0
which implies
Z
T
0
Z
1
2
u dxdt ≤ C,
0
Z
T
0
Z
0
1
(m+1)/2 2
∂u
dxdt ≤ C.
∂x
Applying Lemma 2.1 and Lemma 2.3 yields
Z T
V (t)dt
0
2 !
Z TZ 0 Z 1
Z TZ 1
n
X
1 2 1 ∂
m
βi
u2 (t + θ)dxdθdt
u + (εu + u ) dxdt +
≤
2
2
∂x
0
−τi 0
0
0
i=1
≤C.
Since V is continuous, there exists a t0 ∈ [0, T ] satisfies
V (t0 ) ≤
C
≤ C.
T
Hence, if t0 ≤ t ≤ t0 + T , we obtain
Z t
V (t) =V (t0 ) +
V ′ (s) ds
t0
Z
Z "
t
≤C +
1
−c1
t0
≤C +
Z
0
T
0
Z 1"
0
c1
#
(m+1)/2 2 !
Z 1
∂u
dx + c2 +
F dx ds
u2 +
∂x
0
#
(m+1)/2 2 !
∂u
+ F dxdt + c2 T
u2 +
∂x
EJQTDE, 2010 No. 52, p. 12
≤C.
A simple calculation yields
2
Z 1
Z 1
∂
(εu + um ) dx =
|(ε + mum−1 )ux |2 dx
∂x
0
0
Z 1
Z 1 m 2
∂u
m−1
2
≥
|mu
ux | dx =
∂x dx.
0
0
By the definition of V , we see that
m 2 !
Z 1
Z 1
∂u
2
u +
dx ≤
∂x
0
0
which implies
sup
0≤t≤T
Z
0
1
2 !
∂
m
u + (εu + u ) dx ≤ 2V (t) ≤ C,
∂x
2
m 2 !
∂u
dx ≤ C.
u2 +
∂x
It follows from the definition of u that
Z x m
Z x m
m
∂u
(s,
t)
∂u
(s,
t)
m
m
ds =
ds
|u(x, t)| = |u (x, t)| = u (0, t) +
∂s
∂s
0
0
Z 1 m
∂u
≤
∂x dx
0
Z 1 m 2 !1/2
∂u
≤ sup
≤ C 1/2 ,
∂x dx
0≤t≤T 0
that is
kukL∞ (Q) ≤ C 1/(2m) := C0 .
The proof is complete.
In addition, we can also easily get the L2 boundedness of
∂um
∂t
as follows.
Lemma 2.5 Assume that (H1 ), (H2 ) hold and u is a nonnegative T -periodic solution
of the equation (2.1) satisfying the boundary value condition (2.2). Then there exists a
constant C > 0 such that
Z Z m 2
∂u
∂t dxdt ≤ C.
Q
Proof. Multiplying the equation (2.1) by
obtain
ZZ
∂u ∂
(εu + um )dxdt
∂t
∂t
Q
∂
(εu
∂t
+ um ) and integrating over Q, we
EJQTDE, 2010 No. 52, p. 13
ZZ
∂
∂2
m ∂
m
(εu + u ) (εu + u )dxdt + a
u (εu + um )dxdt
=
2
∂x
∂t
Q ∂t
ZQZ
∂
+
[f (u(x, t − τ1 ), · · · , u(x, t − τn )) + g(x, t)] (εu + um )dxdt
∂t
Q
ZZ Z t
∂
+
γ
e−α(t−s) u(s)ds (εu + um )dxdt.
∂t
Q
t−τ
ZZ
A simple calculation yields
2
Z Z
Z TZ 1 2
∂
1 T 1 ∂
∂
m ∂
m
m
(εu + u ) (εu + u )dxdt = −
(εu + u ) dxdt = 0.
2
∂t
2 0 0 ∂t ∂x
0
0 ∂x
Therefore, by Lemma 2.4, we obtain
Z TZ 1
Z TZ 1
2
ε
ut dxdt +
mum−1 u2t dxdt
0
0
0
0
Z TZ 1
Z TZ 1
n
X
∂
∂
m
≤
βi
u(t − τi ) (εu + u ) dxdt +
g(x, t) (εu + um )dxdt
∂t
∂t
0
0
0
0
i=1
Z Z Z t
∂
(εu + um )dxdt
+
γ
e−α(t−s) u(s)ds
∂t
Q
t−τ
Z TZ T
∂
m
≤C
∂t (εu + u ) dxdt
0
0
Z TZ 1
Z Z
2
ε Cε1
Cε2 T 1
C
C
2
≤
+
|ut | dxdt +
|mum−1 ut |2 dxdt +
2
2ε1
2 0 0
2ε2
0
0
Z TZ t
Z TZ 1
Cε2
mum−1 |ut |2 dxdt,
≤ε
|ut |2 dxdt + C +
2
0
0
0
0
where ε1 = 2/(Cε). Letting ε2 be appropriately small,
Z TZ 1
mum−1 u2t dxdt ≤ C,
0
0
then we can see
Z
0
T
Z TZ 1
Z 1 m 2
∂u
dxdt =
m2 u2(m−1) u2t dxdt
∂t
0
0
0
Z TZ 1
m−1
≤mC
mum−1 u2t dxdt ≤ C,
0
0
which completes the proof of Lemma 2.5.
We will show the H¨older norm estimate of solutions in the following lemma.
EJQTDE, 2010 No. 52, p. 14
Lemma 2.6 Assume that (H1 ), (H2 ) hold and u is a nonnegative T -periodic solution
of the equation (2.1) satisfying the boundary value condition (2.2). Then there exists a
constant C > 0 such that u ∈ C α,α/2 (Q) with 0 < α < 1/2.
Proof. In fact, through a similar discussion of [6] (see Chapter 2), we know that
u ∈ L∞ (0, T ; H01(0, 1)) and ∂u
∈ L2 (Q). By direct computations, for any x1 < x2 ∈ (0, 1),
∂t
we conclude that
Z x2
Z x2
∂u(x, t)
∂u(x, t)
dx ≤
|u(x2 , t) − u(x1 , t)| =
∂x dx
∂x
x1
x1
Z 1 2 !1/2
∂u
(2.9)
dx
≤
|x2 − x1 |1/2
∂x
0
≤C|x2 − x1 |1/2 .
On the other hand, to prove
|u(x, t2 ) − u(x, t1 )| ≤ C|t2 − t1 |1/4 ,
(2.10)
we need only consider the case that 0 ≤ x ≤ 21 , ∆t = t2 − t1 > 0, (∆t)α ≤ 14 , where α is
determined. Integrating (2.2) over (y, y + (∆t)α ) × (t1 , t2 ) gives
Z y+(∆t)α
[u(z, t2 ) − u(z, t1 )]dz
y
Z t2
∂
∂
α
m
α
m
=
(εu(y + (∆t) , s) + u (y + (∆t) , s)) −
(εu(y, s) + u (y, s)) ds
∂x
∂x
t1
Z t2 Z y+(∆t)α
+
[au(z, s) + f (u(z, s − τ1 ), · · · , u(z, s − τn )) + g(z, s)]dzds
+
Z
t1
t2
t1
y
y+(∆t)α
Z
γ
y
Z
s
e−α(s−σ) u(x, s)dσdzds,
s−τ0
i.e.
α
(∆t)
Z t2
1
Z
(u(y + θ(∆t)α , t2 ) − u(y + θ(∆t)α , t1 ))dθ
0
∂
∂
α
m
α
m
(εu(y + (∆t) , s) + u (y + (∆t) , s)) −
(εu(y, s) + u (y, s)) ds
=
∂x
∂x
t1
Z t2 Z y+(∆t)α
+
[au(z, s) + f (u(z, s − τ1 ), · · · , u(z, s − τn )) + g(z, s)]dzds
+
Z
t1
t2
t1
y
Z
y
y+(∆t)α
γ
Z
s
e−α(s−σ) u(x, s)dσdzds,
s−τ0
EJQTDE, 2010 No. 52, p. 15
Integrating the above equality with respect to y over (x, x + (∆t)α ), we conclude that
α
Z
x+(∆t)α
(∆t)
x
Z x+(∆t)α Z
=
t2
Z
1
(u(y + θ(∆t)α , t2 ) − u(y + θ(∆t)α , t1 ))dθdy
0
∂
∂
α
m
α
m
(εu(y + (∆t) , s) + u (y + (∆t) , s)) −
(εu(y, s) + u (y, s)) dsdy
∂x
∂x
x
t1
Z x+(∆t)α Z t2 Z y+(∆t)α
+
[au(z, s) + f (u(z, s − τ1 ), · · · , u(z, s − τn )) + g(z, s)]dzdsdy
x
+
Z
x+(∆t)α
x
≤C|(∆t)
t1
t2
Z
t1
α+1 1/2
|
y
Z
y+(∆t)α
γ
y
+ C(∆t)
Z
s
e−α(s−σ) u(x, s)dσdzdsdy,
s−τ0
2α+1
.
Hence, by a simple calculations, we have
|u(x∗ , t2 ) − u(x∗ , t1 )| ≤ C(∆t) + C(∆t)(α+1)/2−2α ≤ C|t2 − t1 |1/4 ,
where x∗ = y ∗ + θ∗ (∆t)α , y ∗ ∈ (x, x + (∆t)α ), θ∗ ∈ (0, 1), and choose α = 1/6 specially,
from which we see that (2.10) holds. Therefore, by (2.9) and (2.10), we obtain that
|u(x2 , t2 ) − u(x1 , t1 )|
≤|u(x2 , t2 ) − u(x∗ , t2 )| + |u(x∗ , t2 ) − u(x∗ , t1 )| + |u(x∗ , t1 ) − u(x1 , t1 )|
≤C(|x2 − x∗ |1/2 + |t2 − t1 |1/4 + |x∗ − x1 |1/2 )
≤C(|x2 − x1 | + |t2 − t1 |1/2 )1/2 ,
which completes the proof of Lemma 2.6.
3
Proof of the Main Result
By means of the above proved lemmas and the Leray-Schauder fixed point theorem, we
can obtain the existence of solutions uε of the regularized problem as follows.
Proposition 3.1 Assume that (H1 ) and (H2 ) hold. Then the regularized problem
(2.2)–(2.4) has a nonnegative T -periodic solution.
Proof. Denote by CT (Q) the set of all continuous functions u with the T -periodicity
in t. We study the following regularized equation
∂2
∂u
= 2 (εu + |u|m−1 u) + g(x, t)
∂t
∂x
(3.1)
EJQTDE, 2010 No. 52, p. 16
where 0 ≤ g ∈ CT (Q). We claim that, if the problem (3.1),(2.2),(2.3) has a unique T periodic solution u, then u must be nonnegative. In fact, multiplying (3.1) by u− and
integrating over Q, we obtain
ZZ
ZZ
ZZ
∂u
∂2
m−1
u− dxdt =
(εu + |u|
u)u− dxdt +
gu− dxdt,
2
Q ∂t
Q ∂x
Q
where u− = min{0, u(x, t), (x, t) ∈ Q}. Making use of integration by parts, we have
ZZ
ZZ
∂
∂u−
m−1
(εu− + |u− |
u− )
dxdt =
gu− dxdt ≤ 0.
∂x
Q ∂x
Q
Since
ZZ
Q
∂u−
∂
(εu− + |u− |m−1 u− )
dxdt =
∂x
∂x
then we get
ZZ
|u− |
m−1
Q
Therefore,
u− = 0,
ZZ
(ε + m|u− |
Q
∂u− 2
∂x dxdt ≤ 0.
m−1
∂u− 2
dxdt,
)
∂x
a.e. in Q
By the definition of u− , we see that
u ≥ 0,
a.e. in Q.
Consequently, we can rewrite the equation (3.1) as
∂2
∂u
= 2 (εu + um ) + g(x, t),
∂t
∂x
x ∈ (0, 1), t ∈ R.
(3.2)
Hence, we know that, if the problem (3.2),(2.2),(2.3) has a T -periodic solution, it must
be nonnegative.
Similarly, we have the same consequence for the equation
∂u
∂2
= 2 (εu + um ) − ζu + g(x, t),
∂t
∂x
(3.3)
with the conditions (2.2) and (2.3), where ζ ≥ 0.
On the other hand, with an argument similar to [7], we claim that for any g ∈ CT (Q),
the problem
∂2
∂u
= 2 (εu + um ) + g(x, t),
∂t
∂x
x ∈ (0, 1), t ∈ R,
EJQTDE, 2010 No. 52, p. 17
u(0, t) = u(1, t),
u(x, t) = u(x, t + T ),
t ∈ R,
x ∈ (0, 1), t ∈ R.
has a unique solution u ∈ C α,α/2 (Q). By constructing a homotopy, it is easy to obtain
that the problem (3.3),(2.2),(2.3) also admits a solution u ∈ C α,α/2 (Q) .
Next, we will obtain the existence of periodic solutions for the regularized problem
(2.1),(2.2),(2.3).
In case that a ≥ 0, we consider the periodic problem of the homotopy equation for
regularized problem
∂2
∂u
= 2 (εu + um ) + λG(x, t),
∂t
∂x
u(0, t) = u(1, t) = 0,
x ∈ (0, 1), t ∈ R,
(3.4)
t ∈ R,
(3.5)
where for any v(x, t) ∈ CT (Q),
G(x, t) = av(x, t) + f (v(x, t − τ1 ), · · · , v(x, t − τn )) + g(x, t) + γ
α,α/2
Then problem (3.4)–(3.5) admits a unique solution u ∈ CT
Z
t
e−α(t−s) v(x, s)ds.
t−τ0
(Q). Define the mapping
L : CT (Q)×[0, 1] −→ CT (Q),
(v, λ) 7−→ u.
α,α/2
Since CT
(Q) can be compactly embedded into CT (Q), L is compact. By Lemma 2.4,
we know that for any fixed point uλ of the mapping L, there is a constant C0 independent
of ε and λ, such that
kuλ kL∞ ≤ C0 .
Then in applying Leray-Schauder’s fixed point theorem, we know that the problem (2.1)–
(2.3) admits a solution uε .
In case that a < 0, we consider the periodic problem of the homotopy equation for
regularized problem
∂u
∂2
= 2 (εu + um ) + au + λG(x, t),
∂t
∂x
u(0, t) = u(1, t) = 0,
x ∈ (0, 1), t ∈ R,
(3.6)
t ∈ R,
(3.7)
where for any v(x, t) ∈ CT (Q),
G(x, t) = f (v(x, t − τ1 ), · · · , v(x, t − τn )) + g(x, t) + γ
Z
t
e−α(t−s) v(x, s)ds.
t−τ0
EJQTDE, 2010 No. 52, p. 18
α,α/2
Then problem (3.6)–(3.7) admits a unique solution u ∈ CT
(Q). The following progress
is the same as above case, then we get that the problem (2.1)–(2.3) admits a solution uε .
Now, we turn to the proof of the our main result based on the above lemmas and
Proposition 3.1
The Proof of Theorem 2.1. Let ε = 1/h (h = 1, 2, · · · ) and we note uh for the
solution of the problem (2.2)–(2.4). Clearly, according to Lemma 2.2, Lemma 2.4 and
Lemma 2.5,
kuh kL∞ (Q) ≤ C0 ,
m
∂uh
∂x
2 ≤ C1 T,
L (Q)
m
2
∂uh
∂t
2 ≤ C.
L (Q)
Hence there exists a subsequence {uh }∞
to be {uh }∞
h=1 , supposed
h=1 itself, and a function
m
u ∈ {u; u ∈ L∞ ; um ∈ L∞ (0, T ; W01,2(0, 1)); ∂u∂t ∈ L2 (Q)}, such that
uh (x, t) → u(x, t),
in L2 (Q),
∂um
∂um (x, t)
h (x, t)
→
,
in L2 (Q),
∂x
∂x
f (uh (x, t − τ1 ), · · · , uh (x, t − τn )) → f (u(x, t − τ1 ), · · · , u(x, t − τn )),
in L2 (Q).
Letting h → ∞ in
Z T Z 1
1 ∂ 2 ϕ ∂um ∂ϕ
u h ϕt + u h 2 − h
+ auh ϕ + f (uh (x, t − τ1 ), · · · , uh (x, t − τn ))ϕ
h ∂x
∂x ∂x
0
0
Z t
−α(t−s)
+g(x, t)ϕ + γ
e
uh (x, s)ds ϕ dxdt = 0,
t−τ
we have
Z
T
0
Z 1
0
∂um ∂ϕ
+ auϕ + f (u(x, t − τ1 ), · · · , u(x, t − τn ))ϕ
∂x ∂x
Z t
−α(t−s)
+g(x, t)ϕ + γ
e
u(x, s)ds ϕ dxdt = 0.
uϕt −
t−τ
That shows u satisfies the integral identity in the definition of weak solutions. Therefore,
the problem (1.1)–(1.2) admits a nonnegative T -periodic solution u ∈ {u; u ∈ L∞ ; um ∈
m
L∞ (0, T ; W01,2(0, 1)); ∂u∂t ∈ L2 (Q)}.
Since g(x, t) 6≡ 0, we see that the T -periodic solution is nontrivial. The proof of
Theorem 2.1 is complete.
EJQTDE, 2010 No. 52, p. 19
Remark. In equation (1.1), if the delays depend on time, we may construct a similar
Lyapunov-type functional
2 !
Z 1
Z 0 Z 1
n
X
1 2 1 ∂
m
2
V (t) =
u + (εu + u ) dx +
βi
u2 (t + θ)dxdθ.
2
2 ∂x
0
−τi (t) 0
i=1
Under suitable assumptions, by the similar arguments, the corresponding existence of
time periodic solution should be established for evolution equations with variable delays.
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(Received March 5, 2010)
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