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]
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AbstractIn this work, we study the existence of periodic solutions for the evolution of p-Laplacian system and we show that these periodic solutions be-long toL∞
(ω, W1,∞
(Ω)) and give a bound ofk∇ui(t)k∞under certain geometric conditions on∂Ω.
Keywords: Periodic solutions; nonlinear parabolic systems; p-Laplacian; existence of solutions; gradients estimates.
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1.
Introduction
We consider the following nonlinear system(S)of the form
(S)
∂u1
∂t − △p1u1+a1(u1).∇u1=f1(t)u α1
1 +h1(x, u2) inQ,
∂u2
∂t − △p2u2+a2(u2).∇u2=f2(t)uα22 +h2(x, u1) inQ,
u1(x, t) =u2(x, t) = 0 onS,
((u1(x, t+ω),(u2(x, t+ω)) = (u1(x, t), u2(x, t)) in Q,
where △pu= div(|∇u|p
−2
∇u) is the so-called p−Laplacian operator, Ω is a bounded convex subset inRN, Q := Ω×(0, ω), S := ∂Ω×(0, ω), ∂Ω is the boundary of Ω andω >0.Precise conditions onai, fiandhi 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
∂t − △pu+b(u).∇u=f(t)u
α+h(x, t),
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 letE=Cω(Q), the set of all functions inC(Ω×R)
which are periodic int with period w. k.kq and k.ks,q denote Lq =Lq(Ω) and
Ws,q=Ws,q(Ω) norms, respectively, 1≤q≤ ∞.Since , Ω⊂RN is a bounded
convex domain, we take the equivalent norm inW01,q(Ω) to be
k∇ukq =
N X
j=1
Z
Ω
∂u ∂xj
q
1
q
f or any u∈W01,q(Ω).
In the sequel, the same symbolcwill be used to indicate some positive con-stants, 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 likeMi, i= 1,2...,instead.
We make the following assumptions:
(H1)
a
i(s) = (ai1, ai2, ..., aIN) is anRN−valued function onR,
|ai(s)| ≤ki|s|βi, for some 0≤βi< pi−2,(i= 1,2).
(H2)
fi(t)∈L∞(0, ω) is periodic int (i= 1,2), with periodω.
(H3)
hi(x, t)∈Cω(Q)∩L
∞
(0, ω;W1,∞
0 (Ω)), hi(x,0) = 0
hi(x, t)>0 , for all (x, t)∈Ω×R, (i= 1,2).
(H4) {0≤αi< pi−1 and N > pi≥2.
We make the following definition of solutions.
Definition 2.1 A functionu= (u1, u2) is called a solution of system(S) if
ui∈Lpi(0, ω;W01,pi(Ω))∩Cω(Q)
andusatisfies
Z Z
Q
{−u1ϕ1t+|∇u1|p1−2∇u1.∇ϕ1−A1(u1).∇ϕ1−f1(t)uα11 ϕ1−h1(x, u2)ϕ1}dxdt = 0,
Z Z
Q
{−u2ϕ2t+|∇u2|p2−2∇u2.∇ϕ2−A2(u2).∇ϕ2−f2(t)uα22 ϕ2−h2(x, u1)ϕ2}dxdt = 0,
for any (ϕ1, ϕ2)∈C01(0, ω;C01(Ω)) withϕi(x,0) = ϕi(x, ω), whereAi(u) = Ru
Let (H1) to (H4) be satisfied. Then there exists a solution (u1, u2) of problem (S)such that fori= 1,2, we have
ui(t)∈L∞(0, ω;W01,pi(Ω))∩Cω(Q),
∂ui
∂t ∈L
2(Q).
Theorem 1.2. Let (H1) to (H4) be satisfied. Then there exists a solution (u1, u2) of problem(S)such that fori= 1,2, we have
ui(t)∈L∞(0, ω;W1,
∞ 0 (Ω)), sup
t
k∇ui(t)k∞≤Mi <∞.
For the proof of the theorems we use the following elementary lemmas. Lemma 2.1. (Gagliardo-Nirenberg, cf[4])
letβ ≥0, N > p≥1, β+ 1≤q, and 1≤r≤q≤(β+ 1)N p/(N−p),then forvsuch that|v|βv∈W1,p(Ω),
kvkq ≤Cβ+11 kvk1−θ r
|v|
β
v
θ/(β+1) 1,p ,
withθ= (β+ 1)(r−1−q−1)/
N−1−p−1+ (β+ 1)r−1 ,whereCis a constant independent ofq, r, β, and θ.
Lemma 2.2. (cf [13])
Lety(t)∈C1(R) be a nonnegativeωperiodic function satisfying the differ-ential inequality
y′
(t) +Ayα+1(t)≤By(t) +C, t∈R, with someα, A >0, B≥0, andC≥0.Then
y(t)≤maxn1,(A−1(B+C))1
α
o
.
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
△εpu=div(
|∇u|2+ε p−2
2
∇u).
To prove theorem 1.1, we consider the following. By theorem 1 in [3],we can chooseu0
iε such that :
∂u0
iε
∂t − △
ε
u0iε= 0 onS,
u0
iε(x, ω) =u0iε(x, t) in Q,
and we construct two sequences of functions (un
1ε) and (un2ε),such that :
∂un
1ε
∂t − △
ε
p1un1ε+a1(un1ε).∇un1ε=f1(t)(un1ε)α1+h1(x, un
−1
2ε ) inQ , (1.1)
un1ε= 0 onS, (1.2)
un
1ε(x, ω) =un1ε(x, t) in Q. (1.3)
∂un
2ε
∂t − △
ε
p2un2ε+a2(un2ε).∇un2ε=f2(t)(un2ε)α2+h2(x, un
−1
1ε ) inQ (1.6)
un2ε= 0 onS, (1.5)
un
2ε(x, ω) =un2ε(x, t) in Q. (1.3)
It is clear that for eachn= 1,2,3, ..., the above systems consist of two nonde-generated and uncoupled initial boundary-value problems. By theorem 1, [3] for fixednandε, the problem (1.1)-(1.3) and (1.4)-(1.6) has a solution (un
1ε, 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 constantMi independent ofε andn,
such that
kuniε(t)k∞≤Mi. (1.7) Proof. Forn= 0,(1.7) is proved in [11], so suppose (1.7) for (n−1).
Multiplying (1.1) by|un1ε| k
un1ε,kinteger, and integrating over Ω to obtain :
1
k+ 2
Z
Ω
|un1ε| k+2
dx+ε(k+ 1)( p1
k+p1 )p1
∇(u n
1ε)
(k+p1)/p1 p1
p1
+
Z
Ω
a1(un1ε).∇un1ε|un1ε| k
un1εdx=
Z
Ω
f1(t)(un1ε)α1+1|un1ε| k
dx+
Z
Ω
h1(x, un−1
2ε )un1ε|un1ε| k
dx. (1.8)
We note that
Z
Ω
a1(un1ε).∇un1ε|un1ε| k
un1εdx= Z
Ω
N X
j=1
a1j(un1ε)|un1ε| k
un1ε
∂un
1ε
∂xj
dx
=
N X
j=1
Z
Ω
Z un1ε
0
a1j(s)|s|ksds !
=
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 andσ >0 is a constant independent ofk.
If 0< α1<1,by H¨older’s inequality and Young’s inequality, we have
kun1εk It follows from (1.8) - (1.13) that
d The same holds also forun
2ε.
Lemma 2.4. un
iε satisfies the following:
Z ω
Proof of lemma 2.4. Multiplying (1.1) byun
1ε,we get :
c
Hence, by (H1), (1.24) and Young’s inequality, we have
Z ω
Proof of theorem 1.2
Multiplying (1.1) by −div(|∇un
By hypothesis and Young’s inequality, we get
≤ck∇un1ε(t)k
By integration by parts, we have
Z
x∈∂Ω nonpositive with respect to the ouward normal. We have from (1.26)-(1.30) that
By lemma 2.1 and Young’s inequality, we have
k∇un1ε(t)k
Therefore, (1.31) can be rewritten as
Then, setting
k1=p1−2, kl= 2kl−1−p1+ 2, θl= 2N
N+ 2(1−
pl−1
pl
), l= 2,3, ...
Again by lemma 2.1, we have
k∇un1ε(t)kpl≤c
2
pl+p1−2k∇un
1εk
1−θl
pl−1 |∇u
n
1ε|
pl+p1−2 2
2θl/(pl+p1−2)
1,2 . (1.34) Therefore, from (1.33) and (1.34), we obtain
d dtk∇u
n
1ε(t)k pl
pl+cc
−2/θlk|∇un
1ε|k
(pl+p1−2)(θl−1)/θl
1,2 k∇u
n
1ε(t)k
(pl+p1−2)/θl
pl )
≤ckl3(1 +k∇un1ε(t)k kl
kl) +cklk∇u
n
1εk kl−1
kl . (1.35)
Applying Lemma 2.2, by the same argument as in lemma 2.3, we can obtain (1.20).
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