Directory UMM :Journals:Journal_of_mathematics:EJQTDE:
Electronic Journal of Qualitative Theory of Differential Equations
2009, No. 68, 1-10; http://www.math.u-szeged.hu/ejqtde/
Periodic Solutions of p-Laplacian Systems
with a Nonlinear Convection Term
Hamid El Ouardi
Equipe Architures des Systemes
Universit´e Hassan II Ain Chock
ENSEM, BP. 9118, Oasis Casablanca,
Morocco, Fax : +212.05.22.23.12.99
e-mail : [email protected] / h.elou [email protected]
———————————————————————————————–
Abstract In this work, we study the existence of periodic solutions for the
evolution of p-Laplacian system and we show that these periodic solutions belong to L∞ (ω, W 1,∞ (Ω)) and give a bound of k∇ui (t)k∞ under certain geometric
conditions on ∂Ω.
Keywords: Periodic solutions; nonlinear parabolic systems; p-Laplacian;
existence of solutions; gradients estimates.
———————————————————————————————–
1. Introduction
We consider the following nonlinear system (S) of the form
∂u
α1
1
in Q,
∂t − △p1 u1 + a1 (u1 ).∇u1 = f1 (t)u1 + h1 (x, u2 )
α2
2
in Q,
∂u
∂t − △p2 u2 + a2 (u2 ).∇u2 = f2 (t)u2 + h2 (x, u1 )
(S)
on S,
u1 (x, t) = u2 (x, t) = 0
((u1 (x, t + ω), (u2 (x, t + ω)) = (u1 (x, t), u2 (x, t))
in Q,
where △p u = div(|∇u|p−2 ∇u) is the so-called p−Laplacian operator, Ω is
a bounded convex subset in RN , Q := Ω × (0, ω), S := ∂Ω × (0, ω), ∂Ω is the
boundary of Ω and ω > 0. Precise conditions on ai , fi and hi will be given later.
The system of form (S) is a class of degenerate parabolic systems and
appears in the theory of non-Newtonian fluids perturbed by nonlinear terms
and forced by rather irregular period in time excitations, see [1, 5] . Periodic
parabolic single equations have been the subject of numbers of extentive works
see [3, 5, 11, 12, 14, 15, 16]. In particular, Lui [11], has considered the following
single equation:
∂u
− △p u + b(u).∇u = f (t)uα + h(x, t),
∂t
with Dirichlet boundary condition and has obtained the existence and gradient
estimates of a periodic solution.The basic tools that have been used are a priori
estimates, Leray-Shauder fixed point theorem, topological degree theory and
EJQTDE, 2009 No. 68, p. 1
Moser method. We first consider the parabolic regularisation of (S) and we
employ Moser’s technique as in [4] and we make some devices as in [7, 13] to
obtain the existence of periodic solutions and we derive estimates of ∇ui (t).
2. Main results
For convenience, hereafter let E = Cω (Q), the set of all functions in C(Ω×R)
which are periodic in t with period w. k.kq and k.ks,q denote Lq = Lq (Ω) and
W s,q = W s,q (Ω) norms, respectively, 1 ≤ q ≤ ∞. Since , Ω ⊂ RN is a bounded
convex domain, we take the equivalent norm in W01,q (Ω) to be
1
q
N Z
X
∂u q
f or any u ∈ W 1,q (Ω).
k∇ukq =
0
∂xj
j=1 Ω
In the sequel, the same symbol c will be used to indicate some positive constants, possibly different from each other, appearing in the various hypotheses
and computations and depending only on data. When we need to fix the precise
value of one constant, we shall use a notation like Mi , i = 1, 2....., instead.
We make the following assumptions:
ai (s) = (ai1 , ai2 , ....., aIN ) is an RN − valued function on R,
|a (s)| ≤ ki |s|βi , for some 0 ≤ βi < pi − 2, (i = 1, 2) .
i
(H2) fi (t) ∈ L ∞ (0, ω) is periodic in t (i = 1, 2) , with period ω.
hi (x, t) ∈ Cω (Q)∩L ∞ (0, ω; W01,∞ (Ω)), hi (x, 0) = 0
(H3)
hi (x, t) > 0 , for all (x, t) ∈ Ω × R, (i = 1, 2) .
(H4) {0 ≤ αi < pi − 1 and N > pi ≥ 2.
(H1)
We make the following definition of solutions.
Definition 2.1 A function u = (u1 , u2 ) is called a solution of system (S) if
ui ∈ Lpi (0, ω; W01,pi (Ω)) ∩ Cω (Q)
and u satisfies
Z Z
1
{−u1 ϕ1t + |∇u1 |p1−2 ∇u1 .∇ϕ1 − A1 (u1 ).∇ϕ1 − f1 (t)uα
1 ϕ1 − h1 (x, u2 )ϕ1 } dxdt
=
0,
=
0,
Q
Z Z
p2−2
{−u2 ϕ2t + |∇u2 |
Q
2
∇u2 .∇ϕ2 − A2 (u2 ).∇ϕ2 − f2 (t)uα
2 ϕ2 − h2 (x, u1 )ϕ2 } dxdt
1
1
R u for any (ϕ1 , ϕ2 ) ∈ C0 (0, ω; C0 (Ω)) with ϕi (x, 0) = ϕi (x, ω), where Ai (u) =
a (s)ds is set.
0 i
Theorem 1.1.
EJQTDE, 2009 No. 68, p. 2
Let (H1) to (H4) be satisfied. Then there exists a solution (u1 , u2 ) of problem
(S) such that for i = 1, 2, we have
ui (t) ∈ L∞ (0, ω; W01,pi (Ω)) ∩ Cω (Q),
∂ui
∈ L2 (Q).
∂t
Theorem 1.2. Let (H1) to (H4) be satisfied. Then there exists a solution
(u1 , u2 ) of problem (S) such that for i = 1, 2, we have
ui (t) ∈ L∞ (0, ω; W01,∞ (Ω)),
sup k∇ui (t)k∞ ≤ Mi < ∞.
t
For the proof of the theorems we use the following elementary lemmas.
Lemma 2.1. (Gagliardo-Nirenberg, cf[4])
let β ≥ 0, N > p ≥ 1, β + 1 ≤ q, and 1 ≤ r ≤ q ≤ (β + 1)N p/(N − p), then
for v such that |v|β v ∈ W 1,p (Ω),
1
1−θ
kvkq ≤ C β+1 kvkr
β
θ/(β+1)
,
|v| v
1,p
with θ = (β + 1)(r−1 − q −1 )/ N −1 − p−1 + (β + 1)r−1 , where C is a constant
independent of q, r, β, and θ.
Lemma 2.2. (cf [13])
Let y(t) ∈ C 1 (R) be a nonnegative ω periodic function satisfying the differential inequality
y ′ (t) + Ay α+1 (t) ≤ By(t) + C,
t ∈ R,
with some α, A > 0, B ≥ 0, and C ≥ 0. Then
o
n
1
y(t) ≤ max 1, (A−1 (B + C)) α .
3. Proofs of Theorem 1.1 and Theorem 1.2.
In this section we derive a priori estimates of solutions (u1 , u2 ) of problem
(S). We first replace the p-Laplacian by the regularized one
p−2
2
2
∇u).
△εp u = div( |∇u| + ε
To prove theorem 1.1, we consider the following.
By theorem 1 in [3] , we can choose u0iε such that :
∂u0iε
− △εp1 u0iε ) + ai (u0iε ).∇u0iε = fi (t)(uniε )αi
∂t
in Q,
EJQTDE, 2009 No. 68, p. 3
u0iε = 0
on S,
u0iε (x, ω) = u0iε (x, t)
in Q,
and we construct two sequences of functions (un1ε ) and (un2ε ), such that :
∂un1ε
n−1
− △εp1 un1ε + a1 (un1ε ).∇un1ε = f1 (t)(un1ε )α1 + h1 (x, u2ε
) in Q ,
∂t
un1ε = 0
on S,
n
n
u1ε (x, ω) = u1ε (x, t)
in Q.
∂un2ε
n−1
)
− △εp2 un2ε + a2 (un2ε ).∇un2ε = f2 (t)(un2ε )α2 + h2 (x, u1ε
∂t
n
u2ε = 0
un2ε (x, ω) = un2ε (x, t)
(1.1)
(1.2)
(1.3)
in Q
(1.6)
on S,
(1.5)
in Q. (1.3)
It is clear that for each n = 1, 2, 3, ....., the above systems consist of two nondegenerated and uncoupled initial boundary-value problems. By theorem 1, [3] for
fixed n and ε, the problem (1.1)-(1.3) and (1.4)-(1.6) has a solution (un1ε , un2ε ).
We need lemma 2.3 and lemma 2.4 below to complete the proof of theorem
1.1.
Lemma 2.3. There exists a positive constant Mi independent of ε and n,
such that
kuniε (t)k∞ ≤ Mi .
(1.7)
Proof. For n = 0, (1.7) is proved in [11], so suppose (1.7) for (n − 1).
k
Multiplying (1.1) by |un1ε | un1ε , k integer, and integrating over Ω to obtain :
Z
p
1
p1 p1
(k+p1 )/p1
1
k+2
)
∇ (un1ε )
|un1ε |
dx + ε(k + 1)(
k+2 Ω
k + p1
p1
Z
k
+
a1 (un1ε ).∇un1ε |un1ε | un1ε dx =
Ω
Z
Ω
k
f1 (t)(un1ε )α1 +1 |un1ε | dx +
We note that
Z
Ω
a1 (un1ε ).∇un1ε |un1ε |k un1ε dx =
=
N Z
X
j=1
Ω
Z
0
un
1ε
Z
Ω
k
n−1 n
h1 (x, u2ε
)u1ε |un1ε | dx.
Z X
N
Ω j=1
a1j (un1ε ) |un1ε |k un1ε
k
a1j (s) |s| sds
!
(1.8)
∂un1ε
dx
∂xj
dx
xj
EJQTDE, 2009 No. 68, p. 4
=
N Z
X
j=1
Z
∂Ω
Z
un
1ε
a1j (s) |s| sds cos(n, xj )ds = 0,
0
k
Ω
!
k
f1 (t)(un1ε )α1 +1 |un1ε | dx +
Z
Ω
(1.9)
k
n−1 n
h1 (x, u2ε
)u1ε |un1ε | dx
h
i
k+α +1
k+1
≤ C f (t) kun1ε kk+α11 +1 + khkk+2 kun1ε kk+2 .
(1.10)
If 1 ≤ α1 < pi − 1, by H¨
older’s inequality and Lemma 2.1, we have
k+α +1
θ
θ
1
kun1ε kk+α11 +1 ≤ C kun1ε kk+2
kun1ε kq2
θ p /(k+p1 )
θ1
(k+p1 )/p1
2 1
≤ C kun1ε kk+2
∇ (un1ε )
pi
p
ε
p1 p1
(k+p1 )/p1
1
≤
)
∇ (un1ε )
(k + 1)(
2M
k + p1
p1
r/θ1
θ1
+C kun1ε kk+2
(k + 2)σ ,
(1.11)
where we set
q = (k + p1 )N/(N − p1 ), θ1 = (k + 2) [q − (p1 + α1 + 1)] /(q − k − 2)
, θ2 = q(α1 − 1)/(q − k − 2), r < k + 2 and σ > 0 is a constant independent
of k.
If 0 < α1 < 1, by H¨
older’s inequality and Young’s inequality, we have
k+α1 +1
n k+1
kun1ε kk+α
≤ kun1ε kk+2
k+2 + C ku1ε kk+2 .
1 +1
If α1 = 0, by H¨
older’s inequality , we have
1
k+1
k+α +1
kun1ε kk+α11 +1 ≤ max 1, |Ω| 2 kun1ε kk+2 .
(1.12)
(1.13)
It follows from (1.8) - (1.13) that
p
d
k+2
(k+p1 )/p1
1
kun1ε kk+2 + c1 (k + 2)−(p1 −2)
∇ (un1ε )
dt
p1
h
i
k+2
≤ c′ (k + 2)σ+1 kun1ε kk+2 + 1 .
Setting
k1 = p1 , kl = p1 kl−1 − p1 + 2, δl =
θl =
N ((p1 −1)kl −p1 +2)
kl ((p1 −1)N +p1 ) .
(kl +p1 −2)
θl
(1.14)
− kl (> p1 − 2),
By lemma 2.1, we have
θ p /(kl +p1 −2)
1−θ
(k +p )/p
l 1
kun1ε kkl ≤ c kun1ε kkl−1l
∇ (un1ε ) l 1 1
.
p1
(1.15)
Set k = kl in (1.14) and by (1.15), we have
EJQTDE, 2009 No. 68, p. 5
−1
d
(kl +p1 −2)(θl −1)/θl
(k +p −2)/θl
k
kun k l +c1 c−(kl +p1 −2)θl kl−p1 +2 . kun1ε (t)kkl−1
kun1ε (t)kkl l 1
dt 1ε kl
h
i
≤ c (kl )σ+1 kun1ε (t)kkkll + 1 .
(1.16)
Therefore,
−1
d
(p −1−δl )
δ +1
kun1ε (t)kkll
kun k + c1 c−(kl +p1 −2)θl kl−p1 +1 kun1ε (t)kkl 1
dt 1ε kl
Let λl =
sup kun1ε kkl
t
≤ c (kl )σ kun1ε kkl + 1 .
(1.17)
, by lemma 2.2, we obtain
n
o1
−1
(p1 −1−δl ) δl
R = c′ c(kl +p1 −2)θl klp1 −1+σ .λl−1
.
λl ≤ max(1, R),
By [7], we have {λl } bounded and we set without loss of generality that R > 1,
which implies
sup kun1ε k∞ ≤ c0 .
(1.18)
t
un2ε .
The same holds also for
Lemma 2.4. uniε satisfies the following:
Z ω
p
k∇uniε kpii dt ≤ ci < ∞,
(1.19)
0
ω
Z
0
n
2
∂uiε
∂t
dt ≤ ci < ∞.
2
(1.20)
Proof of lemma 2.4. Multiplying (1.1) by un1ε , we get :
Z ωZ
Z ωZ
Z ωZ
∂un
p
un1ε 1ε dxdt +
|∇un1ε | 1 dxdt +
a1 (un1ε ).∇un1ε un1ε dxdt
∂t
0
Ω
0
Ω
0
Ω
Z ωZ
Z ωZ
n−1
n α1 +1
=
f (t)(u1ε )
dxdt +
h1 (t, u2ε
)dxdt.
(1.21)
0
0
Ω
Ω
By the periodicity, H¨
older’s inequality, Poincare’s inequality and the L∞ norm
n
bounded of uiε , we have
Z
0
ω
k∇un1ε kpp11 dt
≤
≤
Z
ω
0
Z
1 +1
dt +
kf (t)kr∗ kun1ε kα
p1
ω
0
+
Z
∗
kf (t)krr∗
0
ω
dt
1/r∗ Z
p
kh(., t)kp11
0
dt
ω
Z
ω
0
kh(., t)kp1 kun1ε kpp11 −1 dt,
kun1ε kpp11
1/p1 Z
0
ω
dt
(α1 +1)/p1
p
kun1ε kp11
dt
(p1 −1)/p1
EJQTDE, 2009 No. 68, p. 6
c
"Z
ω
0
k∇un1ε kpp11
dt
(α1 +1)/p1
+
ω
Z
0
k∇un1ε kpp11
dt
(p1 −1)/p1 #
,
(1.22)
in which r∗ = p1 /(p1 − 1 − α1 ). Thus, we have
Z ω
p
k∇un1 kp11 dt ≤ c < ∞,
(1.23)
0
∂un
Multiplying (1.1) by ∂t1ε and integrating over [0, ω] × Ω, we get :
Z ω
n
Z ωZ
n
∂u1ε
2
n
n ∂u1ε
(t)
dxdt
dt
+
a
(u
).∇u
1
1ε
1ε
∂t
∂t
0
≤
Z
ω
f (t)dt
Ω
0
0
2
Z
(un1ε )α1
∂un1ε
dx +
∂t
Ω
Z
ω
0
Z
n
h1 (t, un−1 ) ∂u1ε dxdt
2ε
∂t
(1.24)
Ω
Hence, by (H1), (1.24) and Young’s inequality, we have
Z ω
n
2
∂u1ε
∂t
dt ≤ c < ∞.
0
2
(1.25)
Whence lemma 2.4. Passing to the limit in ε → 0 and n → +∞ is almost the
same in [9].
Proof of theorem 1.2
k−2
Multiplying (1.1) by −div(|∇un1ε |
∇un1ε ) (k > p1 ), and integrating over
Ω, we have
#
"
Z
p12−2
1 d
k−2
2
k
∇un1ε ) div(|∇un1ε |
∇un1ε )dx
div |∇un1ε | + ε
k∇un1ε (t)kk +
k dt
Ω
=
Z
Ω
a1 (un1ε ).∇un1ε
k−2
div(|∇un1ε |
−
Z
Ω
∇un1ε )dx−
Z
k−2
Ω
f1 (t)(un1ε )α1 +1 div(|∇un1ε |
k−2
n−1
h1 (x, u2ε
) div(|∇un1ε |
∇un1ε )dx.
∇un1ε )dx
(1.26)
By hypothesis and Young’s inequality, we get
Z
k−2
a1 (un1ε ).∇un1ε div(|∇un1ε |
∇un1ε )dx
Ω
≤
Z
Ω
k+p1 −4
|∇un1ε |
≤
−
Z
Ω
Z
Ω
2 n 2
D u1ε dx +
Z
2
Ω
k−p1 −2
k 2 |a1 (un1ε )| |∇un1ε |
2
|∇un1ε |k+p1 −4 D2 un1ε dx + ck 2 (1 + k∇un1ε (t)kkk ),
f1 (t)(un1ε )α1 +1
k−2
div(|∇un1ε |
∇un1ε )dx
=−
Z
Ω
k−2
∇f1 . |∇un1ε |
dx
(1.27)
∇un1ε )dx
EJQTDE, 2009 No. 68, p. 7
k−1
k
≤ c k∇un1ε (t)kk + k∇un1ε (t)kk ,
−
Z
Ω
=−
k−2
n−1
h1 (x, u2ε
) div(|∇un1ε |
(1.28)
∇un1ε )dx
Z
∂h1 (x, s)
k−2
n−1
n−1
∇x h1 (x, u2ε
)+
. |∇un1ε |
∇un1ε )dx
∇u2ε
∂s
Ω
k−1
≤ c k∇un1ε (t)kk
.
(1.29)
By integration by parts, we have
Z
Ω
"
#
Z
pi2−2
2
n 2
n
n k−2
n
|∇un1ε |k+p1 −4 D2 un1ε dx
div |∇u1ε | + ε
∇u1ε ) div(|∇u1ε |
∇u1ε )dx ≥
≥
Z
Ω
Ω
k+p −4
|∇un1ε | 1
2 n 2
D u1ε dx + p1 − 2
4
−c(N − 1)
Z
∂Ω
Z
Ω
k+p1 −6
|∇un1ε |
|∇un1ε |k+p1 −2 H(x)dS,
2
2
∇ |∇un1ε | dx
(1.30)
∂v
on ∂Ω and H(x) = (N −1)−1 ( ∂v −1 (△v− ∂ 2 v2 )
where we note that |∇v| = ∂n
∂n
∂n
is the mean curvature on ∂Ω. Using that Ω is convex so ∂Ω is of C 2 and H(x) at
x ∈ ∂Ω nonpositive with respect to the ouward normal. We have from (1.26)(1.30) that
k+p1 −2 2
c
1 d
k∇un1ε (t)kkk +
|∇un1ε | 2
≤ ck 2 (1 + k∇un1ε (t)kkk ) + c k∇un1ε (t)kkk−1
k dt
k
1,2
c
+
k
Z
k+p1 −2
∂Ω
|∇un1ε |
dx.
(1.31)
By lemma 2.1 and Young’s inequality, we have
k+p −2
k∇un1ε (t)kk+p11 −2 ≤
k+p1 −2 2
1
k−1
p −1
|∇un1ε | 2
+ c k∇un1ε kp11 k∇un1ε kk .
2
1,2
(1.32)
Therefore, (1.31) can be rewritten as
k+p1 −2 2
d
k
k−1
k
k∇un1ε (t)kk + c
|∇un1ε | 2
≤ ck 3 (1 + k∇un1ε (t)kk ) + ck k∇un1ε kk .
dt
1,2
(1.33)
EJQTDE, 2009 No. 68, p. 8
Then, setting
k1 = p1 − 2, kl = 2kl−1 − p1 + 2, θl =
2N
pl−1
), l = 2, 3, ........
(1 −
N +2
pl
Again by lemma 2.1, we have
pl +p1 −2 2θl /(pl +p1 −2)
2
1−θ
k∇un1ε (t)kpl ≤ c pl +p1 −2 k∇un1ε kpl−1l
|∇un1ε | 2
.
1,2
(1.34)
Therefore, from (1.33) and (1.34), we obtain
d
l +p1 −2)(θl −1)/θl
k∇un1ε (t)kppll + cc−2/θl k|∇un1ε |k(p
k∇un1ε (t)kp(pl l +p1 −2)/θl )
1,2
dt
k
k −1
≤ ckl3 (1 + k∇un1ε (t)kkll ) + ckl k∇un1ε kkll
.
(1.35)
Applying Lemma 2.2, by the same argument as in lemma 2.3, we can obtain
(1.20).
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non lin´eaires. Dunod, Paris, 1969.
[11] W. Liu: Periodic solutions of evolution m-Laplacian equations with a nonlinear convection term. International Journal of Mathematical Sciences. Vo.
2007, doi:10.1115/2007/27368.
[12] M. Nakao: Periodic solutions of some nonlinear degenerate parabolic equations. Journal of Mathematical Analysis and Applications, vol.104, No. 2,
pp. 554-567, 1984.
[13] M. Nakao, Y. Ohara: Gradient estimates of periodic solutions for some
quasilinear parabolic equations. Journal of Mathematical Analysis and Applications, vol. 204, No. 3, pp. 868-883, 1996.
[14] J. Sun, Y. Ke, C. Jin, J. Yin: Existence of positive periodic solutions for
the p-Laplacian system. Applied Mathematics letters, (20), pp. 696-701,
2007.
[15] J. Wang, W. Gao: Existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with
nonlocal terms. Journal of Mathematical Analysis and Applications,
doi:10.1016/j.jmaa.2006.08.059. 2006.
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domains with periodically moving boundaries. Nagoya Math. J. 70, pp.
111-123, 1978.
(Received March 15, 2009)
EJQTDE, 2009 No. 68, p. 10
2009, No. 68, 1-10; http://www.math.u-szeged.hu/ejqtde/
Periodic Solutions of p-Laplacian Systems
with a Nonlinear Convection Term
Hamid El Ouardi
Equipe Architures des Systemes
Universit´e Hassan II Ain Chock
ENSEM, BP. 9118, Oasis Casablanca,
Morocco, Fax : +212.05.22.23.12.99
e-mail : [email protected] / h.elou [email protected]
———————————————————————————————–
Abstract In this work, we study the existence of periodic solutions for the
evolution of p-Laplacian system and we show that these periodic solutions belong to L∞ (ω, W 1,∞ (Ω)) and give a bound of k∇ui (t)k∞ under certain geometric
conditions on ∂Ω.
Keywords: Periodic solutions; nonlinear parabolic systems; p-Laplacian;
existence of solutions; gradients estimates.
———————————————————————————————–
1. Introduction
We consider the following nonlinear system (S) of the form
∂u
α1
1
in Q,
∂t − △p1 u1 + a1 (u1 ).∇u1 = f1 (t)u1 + h1 (x, u2 )
α2
2
in Q,
∂u
∂t − △p2 u2 + a2 (u2 ).∇u2 = f2 (t)u2 + h2 (x, u1 )
(S)
on S,
u1 (x, t) = u2 (x, t) = 0
((u1 (x, t + ω), (u2 (x, t + ω)) = (u1 (x, t), u2 (x, t))
in Q,
where △p u = div(|∇u|p−2 ∇u) is the so-called p−Laplacian operator, Ω is
a bounded convex subset in RN , Q := Ω × (0, ω), S := ∂Ω × (0, ω), ∂Ω is the
boundary of Ω and ω > 0. Precise conditions on ai , fi and hi will be given later.
The system of form (S) is a class of degenerate parabolic systems and
appears in the theory of non-Newtonian fluids perturbed by nonlinear terms
and forced by rather irregular period in time excitations, see [1, 5] . Periodic
parabolic single equations have been the subject of numbers of extentive works
see [3, 5, 11, 12, 14, 15, 16]. In particular, Lui [11], has considered the following
single equation:
∂u
− △p u + b(u).∇u = f (t)uα + h(x, t),
∂t
with Dirichlet boundary condition and has obtained the existence and gradient
estimates of a periodic solution.The basic tools that have been used are a priori
estimates, Leray-Shauder fixed point theorem, topological degree theory and
EJQTDE, 2009 No. 68, p. 1
Moser method. We first consider the parabolic regularisation of (S) and we
employ Moser’s technique as in [4] and we make some devices as in [7, 13] to
obtain the existence of periodic solutions and we derive estimates of ∇ui (t).
2. Main results
For convenience, hereafter let E = Cω (Q), the set of all functions in C(Ω×R)
which are periodic in t with period w. k.kq and k.ks,q denote Lq = Lq (Ω) and
W s,q = W s,q (Ω) norms, respectively, 1 ≤ q ≤ ∞. Since , Ω ⊂ RN is a bounded
convex domain, we take the equivalent norm in W01,q (Ω) to be
1
q
N Z
X
∂u q
f or any u ∈ W 1,q (Ω).
k∇ukq =
0
∂xj
j=1 Ω
In the sequel, the same symbol c will be used to indicate some positive constants, possibly different from each other, appearing in the various hypotheses
and computations and depending only on data. When we need to fix the precise
value of one constant, we shall use a notation like Mi , i = 1, 2....., instead.
We make the following assumptions:
ai (s) = (ai1 , ai2 , ....., aIN ) is an RN − valued function on R,
|a (s)| ≤ ki |s|βi , for some 0 ≤ βi < pi − 2, (i = 1, 2) .
i
(H2) fi (t) ∈ L ∞ (0, ω) is periodic in t (i = 1, 2) , with period ω.
hi (x, t) ∈ Cω (Q)∩L ∞ (0, ω; W01,∞ (Ω)), hi (x, 0) = 0
(H3)
hi (x, t) > 0 , for all (x, t) ∈ Ω × R, (i = 1, 2) .
(H4) {0 ≤ αi < pi − 1 and N > pi ≥ 2.
(H1)
We make the following definition of solutions.
Definition 2.1 A function u = (u1 , u2 ) is called a solution of system (S) if
ui ∈ Lpi (0, ω; W01,pi (Ω)) ∩ Cω (Q)
and u satisfies
Z Z
1
{−u1 ϕ1t + |∇u1 |p1−2 ∇u1 .∇ϕ1 − A1 (u1 ).∇ϕ1 − f1 (t)uα
1 ϕ1 − h1 (x, u2 )ϕ1 } dxdt
=
0,
=
0,
Q
Z Z
p2−2
{−u2 ϕ2t + |∇u2 |
Q
2
∇u2 .∇ϕ2 − A2 (u2 ).∇ϕ2 − f2 (t)uα
2 ϕ2 − h2 (x, u1 )ϕ2 } dxdt
1
1
R u for any (ϕ1 , ϕ2 ) ∈ C0 (0, ω; C0 (Ω)) with ϕi (x, 0) = ϕi (x, ω), where Ai (u) =
a (s)ds is set.
0 i
Theorem 1.1.
EJQTDE, 2009 No. 68, p. 2
Let (H1) to (H4) be satisfied. Then there exists a solution (u1 , u2 ) of problem
(S) such that for i = 1, 2, we have
ui (t) ∈ L∞ (0, ω; W01,pi (Ω)) ∩ Cω (Q),
∂ui
∈ L2 (Q).
∂t
Theorem 1.2. Let (H1) to (H4) be satisfied. Then there exists a solution
(u1 , u2 ) of problem (S) such that for i = 1, 2, we have
ui (t) ∈ L∞ (0, ω; W01,∞ (Ω)),
sup k∇ui (t)k∞ ≤ Mi < ∞.
t
For the proof of the theorems we use the following elementary lemmas.
Lemma 2.1. (Gagliardo-Nirenberg, cf[4])
let β ≥ 0, N > p ≥ 1, β + 1 ≤ q, and 1 ≤ r ≤ q ≤ (β + 1)N p/(N − p), then
for v such that |v|β v ∈ W 1,p (Ω),
1
1−θ
kvkq ≤ C β+1 kvkr
β
θ/(β+1)
,
|v| v
1,p
with θ = (β + 1)(r−1 − q −1 )/ N −1 − p−1 + (β + 1)r−1 , where C is a constant
independent of q, r, β, and θ.
Lemma 2.2. (cf [13])
Let y(t) ∈ C 1 (R) be a nonnegative ω periodic function satisfying the differential inequality
y ′ (t) + Ay α+1 (t) ≤ By(t) + C,
t ∈ R,
with some α, A > 0, B ≥ 0, and C ≥ 0. Then
o
n
1
y(t) ≤ max 1, (A−1 (B + C)) α .
3. Proofs of Theorem 1.1 and Theorem 1.2.
In this section we derive a priori estimates of solutions (u1 , u2 ) of problem
(S). We first replace the p-Laplacian by the regularized one
p−2
2
2
∇u).
△εp u = div( |∇u| + ε
To prove theorem 1.1, we consider the following.
By theorem 1 in [3] , we can choose u0iε such that :
∂u0iε
− △εp1 u0iε ) + ai (u0iε ).∇u0iε = fi (t)(uniε )αi
∂t
in Q,
EJQTDE, 2009 No. 68, p. 3
u0iε = 0
on S,
u0iε (x, ω) = u0iε (x, t)
in Q,
and we construct two sequences of functions (un1ε ) and (un2ε ), such that :
∂un1ε
n−1
− △εp1 un1ε + a1 (un1ε ).∇un1ε = f1 (t)(un1ε )α1 + h1 (x, u2ε
) in Q ,
∂t
un1ε = 0
on S,
n
n
u1ε (x, ω) = u1ε (x, t)
in Q.
∂un2ε
n−1
)
− △εp2 un2ε + a2 (un2ε ).∇un2ε = f2 (t)(un2ε )α2 + h2 (x, u1ε
∂t
n
u2ε = 0
un2ε (x, ω) = un2ε (x, t)
(1.1)
(1.2)
(1.3)
in Q
(1.6)
on S,
(1.5)
in Q. (1.3)
It is clear that for each n = 1, 2, 3, ....., the above systems consist of two nondegenerated and uncoupled initial boundary-value problems. By theorem 1, [3] for
fixed n and ε, the problem (1.1)-(1.3) and (1.4)-(1.6) has a solution (un1ε , un2ε ).
We need lemma 2.3 and lemma 2.4 below to complete the proof of theorem
1.1.
Lemma 2.3. There exists a positive constant Mi independent of ε and n,
such that
kuniε (t)k∞ ≤ Mi .
(1.7)
Proof. For n = 0, (1.7) is proved in [11], so suppose (1.7) for (n − 1).
k
Multiplying (1.1) by |un1ε | un1ε , k integer, and integrating over Ω to obtain :
Z
p
1
p1 p1
(k+p1 )/p1
1
k+2
)
∇ (un1ε )
|un1ε |
dx + ε(k + 1)(
k+2 Ω
k + p1
p1
Z
k
+
a1 (un1ε ).∇un1ε |un1ε | un1ε dx =
Ω
Z
Ω
k
f1 (t)(un1ε )α1 +1 |un1ε | dx +
We note that
Z
Ω
a1 (un1ε ).∇un1ε |un1ε |k un1ε dx =
=
N Z
X
j=1
Ω
Z
0
un
1ε
Z
Ω
k
n−1 n
h1 (x, u2ε
)u1ε |un1ε | dx.
Z X
N
Ω j=1
a1j (un1ε ) |un1ε |k un1ε
k
a1j (s) |s| sds
!
(1.8)
∂un1ε
dx
∂xj
dx
xj
EJQTDE, 2009 No. 68, p. 4
=
N Z
X
j=1
Z
∂Ω
Z
un
1ε
a1j (s) |s| sds cos(n, xj )ds = 0,
0
k
Ω
!
k
f1 (t)(un1ε )α1 +1 |un1ε | dx +
Z
Ω
(1.9)
k
n−1 n
h1 (x, u2ε
)u1ε |un1ε | dx
h
i
k+α +1
k+1
≤ C f (t) kun1ε kk+α11 +1 + khkk+2 kun1ε kk+2 .
(1.10)
If 1 ≤ α1 < pi − 1, by H¨
older’s inequality and Lemma 2.1, we have
k+α +1
θ
θ
1
kun1ε kk+α11 +1 ≤ C kun1ε kk+2
kun1ε kq2
θ p /(k+p1 )
θ1
(k+p1 )/p1
2 1
≤ C kun1ε kk+2
∇ (un1ε )
pi
p
ε
p1 p1
(k+p1 )/p1
1
≤
)
∇ (un1ε )
(k + 1)(
2M
k + p1
p1
r/θ1
θ1
+C kun1ε kk+2
(k + 2)σ ,
(1.11)
where we set
q = (k + p1 )N/(N − p1 ), θ1 = (k + 2) [q − (p1 + α1 + 1)] /(q − k − 2)
, θ2 = q(α1 − 1)/(q − k − 2), r < k + 2 and σ > 0 is a constant independent
of k.
If 0 < α1 < 1, by H¨
older’s inequality and Young’s inequality, we have
k+α1 +1
n k+1
kun1ε kk+α
≤ kun1ε kk+2
k+2 + C ku1ε kk+2 .
1 +1
If α1 = 0, by H¨
older’s inequality , we have
1
k+1
k+α +1
kun1ε kk+α11 +1 ≤ max 1, |Ω| 2 kun1ε kk+2 .
(1.12)
(1.13)
It follows from (1.8) - (1.13) that
p
d
k+2
(k+p1 )/p1
1
kun1ε kk+2 + c1 (k + 2)−(p1 −2)
∇ (un1ε )
dt
p1
h
i
k+2
≤ c′ (k + 2)σ+1 kun1ε kk+2 + 1 .
Setting
k1 = p1 , kl = p1 kl−1 − p1 + 2, δl =
θl =
N ((p1 −1)kl −p1 +2)
kl ((p1 −1)N +p1 ) .
(kl +p1 −2)
θl
(1.14)
− kl (> p1 − 2),
By lemma 2.1, we have
θ p /(kl +p1 −2)
1−θ
(k +p )/p
l 1
kun1ε kkl ≤ c kun1ε kkl−1l
∇ (un1ε ) l 1 1
.
p1
(1.15)
Set k = kl in (1.14) and by (1.15), we have
EJQTDE, 2009 No. 68, p. 5
−1
d
(kl +p1 −2)(θl −1)/θl
(k +p −2)/θl
k
kun k l +c1 c−(kl +p1 −2)θl kl−p1 +2 . kun1ε (t)kkl−1
kun1ε (t)kkl l 1
dt 1ε kl
h
i
≤ c (kl )σ+1 kun1ε (t)kkkll + 1 .
(1.16)
Therefore,
−1
d
(p −1−δl )
δ +1
kun1ε (t)kkll
kun k + c1 c−(kl +p1 −2)θl kl−p1 +1 kun1ε (t)kkl 1
dt 1ε kl
Let λl =
sup kun1ε kkl
t
≤ c (kl )σ kun1ε kkl + 1 .
(1.17)
, by lemma 2.2, we obtain
n
o1
−1
(p1 −1−δl ) δl
R = c′ c(kl +p1 −2)θl klp1 −1+σ .λl−1
.
λl ≤ max(1, R),
By [7], we have {λl } bounded and we set without loss of generality that R > 1,
which implies
sup kun1ε k∞ ≤ c0 .
(1.18)
t
un2ε .
The same holds also for
Lemma 2.4. uniε satisfies the following:
Z ω
p
k∇uniε kpii dt ≤ ci < ∞,
(1.19)
0
ω
Z
0
n
2
∂uiε
∂t
dt ≤ ci < ∞.
2
(1.20)
Proof of lemma 2.4. Multiplying (1.1) by un1ε , we get :
Z ωZ
Z ωZ
Z ωZ
∂un
p
un1ε 1ε dxdt +
|∇un1ε | 1 dxdt +
a1 (un1ε ).∇un1ε un1ε dxdt
∂t
0
Ω
0
Ω
0
Ω
Z ωZ
Z ωZ
n−1
n α1 +1
=
f (t)(u1ε )
dxdt +
h1 (t, u2ε
)dxdt.
(1.21)
0
0
Ω
Ω
By the periodicity, H¨
older’s inequality, Poincare’s inequality and the L∞ norm
n
bounded of uiε , we have
Z
0
ω
k∇un1ε kpp11 dt
≤
≤
Z
ω
0
Z
1 +1
dt +
kf (t)kr∗ kun1ε kα
p1
ω
0
+
Z
∗
kf (t)krr∗
0
ω
dt
1/r∗ Z
p
kh(., t)kp11
0
dt
ω
Z
ω
0
kh(., t)kp1 kun1ε kpp11 −1 dt,
kun1ε kpp11
1/p1 Z
0
ω
dt
(α1 +1)/p1
p
kun1ε kp11
dt
(p1 −1)/p1
EJQTDE, 2009 No. 68, p. 6
c
"Z
ω
0
k∇un1ε kpp11
dt
(α1 +1)/p1
+
ω
Z
0
k∇un1ε kpp11
dt
(p1 −1)/p1 #
,
(1.22)
in which r∗ = p1 /(p1 − 1 − α1 ). Thus, we have
Z ω
p
k∇un1 kp11 dt ≤ c < ∞,
(1.23)
0
∂un
Multiplying (1.1) by ∂t1ε and integrating over [0, ω] × Ω, we get :
Z ω
n
Z ωZ
n
∂u1ε
2
n
n ∂u1ε
(t)
dxdt
dt
+
a
(u
).∇u
1
1ε
1ε
∂t
∂t
0
≤
Z
ω
f (t)dt
Ω
0
0
2
Z
(un1ε )α1
∂un1ε
dx +
∂t
Ω
Z
ω
0
Z
n
h1 (t, un−1 ) ∂u1ε dxdt
2ε
∂t
(1.24)
Ω
Hence, by (H1), (1.24) and Young’s inequality, we have
Z ω
n
2
∂u1ε
∂t
dt ≤ c < ∞.
0
2
(1.25)
Whence lemma 2.4. Passing to the limit in ε → 0 and n → +∞ is almost the
same in [9].
Proof of theorem 1.2
k−2
Multiplying (1.1) by −div(|∇un1ε |
∇un1ε ) (k > p1 ), and integrating over
Ω, we have
#
"
Z
p12−2
1 d
k−2
2
k
∇un1ε ) div(|∇un1ε |
∇un1ε )dx
div |∇un1ε | + ε
k∇un1ε (t)kk +
k dt
Ω
=
Z
Ω
a1 (un1ε ).∇un1ε
k−2
div(|∇un1ε |
−
Z
Ω
∇un1ε )dx−
Z
k−2
Ω
f1 (t)(un1ε )α1 +1 div(|∇un1ε |
k−2
n−1
h1 (x, u2ε
) div(|∇un1ε |
∇un1ε )dx.
∇un1ε )dx
(1.26)
By hypothesis and Young’s inequality, we get
Z
k−2
a1 (un1ε ).∇un1ε div(|∇un1ε |
∇un1ε )dx
Ω
≤
Z
Ω
k+p1 −4
|∇un1ε |
≤
−
Z
Ω
Z
Ω
2 n 2
D u1ε dx +
Z
2
Ω
k−p1 −2
k 2 |a1 (un1ε )| |∇un1ε |
2
|∇un1ε |k+p1 −4 D2 un1ε dx + ck 2 (1 + k∇un1ε (t)kkk ),
f1 (t)(un1ε )α1 +1
k−2
div(|∇un1ε |
∇un1ε )dx
=−
Z
Ω
k−2
∇f1 . |∇un1ε |
dx
(1.27)
∇un1ε )dx
EJQTDE, 2009 No. 68, p. 7
k−1
k
≤ c k∇un1ε (t)kk + k∇un1ε (t)kk ,
−
Z
Ω
=−
k−2
n−1
h1 (x, u2ε
) div(|∇un1ε |
(1.28)
∇un1ε )dx
Z
∂h1 (x, s)
k−2
n−1
n−1
∇x h1 (x, u2ε
)+
. |∇un1ε |
∇un1ε )dx
∇u2ε
∂s
Ω
k−1
≤ c k∇un1ε (t)kk
.
(1.29)
By integration by parts, we have
Z
Ω
"
#
Z
pi2−2
2
n 2
n
n k−2
n
|∇un1ε |k+p1 −4 D2 un1ε dx
div |∇u1ε | + ε
∇u1ε ) div(|∇u1ε |
∇u1ε )dx ≥
≥
Z
Ω
Ω
k+p −4
|∇un1ε | 1
2 n 2
D u1ε dx + p1 − 2
4
−c(N − 1)
Z
∂Ω
Z
Ω
k+p1 −6
|∇un1ε |
|∇un1ε |k+p1 −2 H(x)dS,
2
2
∇ |∇un1ε | dx
(1.30)
∂v
on ∂Ω and H(x) = (N −1)−1 ( ∂v −1 (△v− ∂ 2 v2 )
where we note that |∇v| = ∂n
∂n
∂n
is the mean curvature on ∂Ω. Using that Ω is convex so ∂Ω is of C 2 and H(x) at
x ∈ ∂Ω nonpositive with respect to the ouward normal. We have from (1.26)(1.30) that
k+p1 −2 2
c
1 d
k∇un1ε (t)kkk +
|∇un1ε | 2
≤ ck 2 (1 + k∇un1ε (t)kkk ) + c k∇un1ε (t)kkk−1
k dt
k
1,2
c
+
k
Z
k+p1 −2
∂Ω
|∇un1ε |
dx.
(1.31)
By lemma 2.1 and Young’s inequality, we have
k+p −2
k∇un1ε (t)kk+p11 −2 ≤
k+p1 −2 2
1
k−1
p −1
|∇un1ε | 2
+ c k∇un1ε kp11 k∇un1ε kk .
2
1,2
(1.32)
Therefore, (1.31) can be rewritten as
k+p1 −2 2
d
k
k−1
k
k∇un1ε (t)kk + c
|∇un1ε | 2
≤ ck 3 (1 + k∇un1ε (t)kk ) + ck k∇un1ε kk .
dt
1,2
(1.33)
EJQTDE, 2009 No. 68, p. 8
Then, setting
k1 = p1 − 2, kl = 2kl−1 − p1 + 2, θl =
2N
pl−1
), l = 2, 3, ........
(1 −
N +2
pl
Again by lemma 2.1, we have
pl +p1 −2 2θl /(pl +p1 −2)
2
1−θ
k∇un1ε (t)kpl ≤ c pl +p1 −2 k∇un1ε kpl−1l
|∇un1ε | 2
.
1,2
(1.34)
Therefore, from (1.33) and (1.34), we obtain
d
l +p1 −2)(θl −1)/θl
k∇un1ε (t)kppll + cc−2/θl k|∇un1ε |k(p
k∇un1ε (t)kp(pl l +p1 −2)/θl )
1,2
dt
k
k −1
≤ ckl3 (1 + k∇un1ε (t)kkll ) + ckl k∇un1ε kkll
.
(1.35)
Applying Lemma 2.2, by the same argument as in lemma 2.3, we can obtain
(1.20).
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(Received March 15, 2009)
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