Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 37, pp. 1–15.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
DYNAMICS OF THE p-LAPLACIAN EQUATIONS WITH
NONLINEAR DYNAMIC BOUNDARY CONDITIONS
XIYOU CHENG, LEI WEI
Abstract. In this article, we study the long-time behavior of the p-Laplacian
equation with nonlinear dynamic boundary conditions for both autonomous
and non-autonomous cases. For the autonomous case, some asymptotic regularity of solutions is proved. For the non-autonomous case, we obtain the
existence and structure of a compact uniform attractor in Lr1 (Ω) × Lr (Γ)
(r = min(r1 , r2 )).
1. Introduction
In this article, we consider the asymptotic behavior of solutions of the following
p-Laplacian equations with nonlinear dynamic boundary conditions:
ut − ∆p u + f (u) = h(x, t),
ut + |∇u|
p−2
in Ω,
∂n u + g(u) = 0,
on Γ,
(1.1)
where Ω is a bounded domain in RN (N > 3) with a smooth boundary Γ, ∆p denotes
the p-Laplacian operator, which is defined as ∆p u = div(|∇u|p−2 ∇u), p > 2, and
about the external forcing h(x, t), we consider two cases: the autonomous case
0
h(x, t) = h(x) ∈ Lr1 (Ω), where r10 is conjugate to r1 , and the non-autonomous case
h(x, t), which will be given later in Sections 3 and 4 respectively. The functions f
and g ∈ C 1 (R, R), satisfy the following conditions:
C1 |s|r1 − k1 ≤ f (s)s ≤ C2 |s|r1 + k2 ,
r2
C3 |s|
− k3 ≤ g(s)s ≤ C4 |s|
0
0
f (s) ≥ −l,
r2
+ k4 ,
g (s) ≥ −m,
r1 ≥ p,
(1.2)
r2 ≥ 2,
(1.3)
(1.4)
here l, m > 0, Ci , ki > 0, i = 1, 2, 3, 4.
In the case p = 2, the problem (1.1) is a general reaction-diffusion equation,
the dynamical behavior have been studied in [3, 4, 8, 22, 25, 26, 27, 31] for the
Dirichlet boundary conditions and [10, 11, 14, 15, 28, 29] for the dynamic boundary
conditions.
The long-time behavior of the solutions of (1.1) has been considered by many
researchers, e.g., see [3, 4, 8, 27] and the references therein.
2000 Mathematics Subject Classification. 37L05, 35B40, 35B41.
Key words and phrases. p-Laplacian equation; boundary condition; asymptotic regularity;
attractor.
c
2015
Texas State University - San Marcos.
Submitted May 6, 2014. Published February 10, 2015.
1
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X. CHENG, L. WEI
EJDE-2015/37
For the autonomous systems; i.e., h(x, t) = h(x), in the Dirichlet boundary case,
the nonlinear eigenvalue problem for the p-Laplacian operator was considered in [18]
by using the Ljusternik-Schnirelman principle. In [3], Babin & Vishik established
the existence of a (L2 (Ω), (W01, p (Ω) ∩ Lq (Ω))w )-global attractor. In [27], a special
case of f = ku was discussed by Temam. In [5], Carvalho, Cholewa and Dlotko
considered the existence of global attractors for problems with monotone operators,
and as an application, they proved the existence of (L2 (Ω), L2 (Ω))-global attractor
for p-Laplacian equation, see also Cholewa & Dlotko [8]. In [6], Carvalho & Gentile
obtained that the corresponding semigroup has a (L2 (Ω), W01, p (Ω))-global attractor
under some additional conditions. In [30], Yang, Sun and Zhong obtained the
existence of a (L2 (Ω), W01, p (Ω) ∩ Lr1 (Ω))-global attractor, which holds only under
the assumptions (1.2) and (1.4). Some asymptotic regularity of the solutions was
proved by Liu, Yang and Zhong in [20]. In the dynamic boundary case, recently,
Gal et al [16, 17] presented firstly the general result for the problem (1.1), the
well-posedness and the asymptotic behavior of the solutions were studied.
Inspired by the ideas of [20, 26, 29], we obtain the asymptotic regularity of
the solutions of equation (1.1), where we only assume the external forcing h(x) ∈
0
Lr1 (Ω), r10 is conjugate to r1 . As a direct application of the asymptotic regularity
results, we can obtain the existence of a global attractor in (W 1,p (Ω) ∩ Lr1 (Ω)) ×
(W 1−1/p,p (Γ) ∩ Lr2 (Γ)) immediately. Moreover, we also can show further that the
global attractor attracts every L2 (Ω) × L2 (Γ)-bounded subset with (W 1,p (Ω) ∩
Lr1 +δ (Ω)) × (W 1−1/p,p (Γ) ∩ Lr2 +γ (Γ))-norm for any δ, γ ∈ [0, ∞).
For the non-autonomous systems, in the Dirichlet boundary case, the existence of
the (L2 (Ω), W01,p (Ω) ∩ Lr1 (Ω))-uniform attractor was proved by Chen and Zhong in
[7]. However, for the nonlinear dynamic boundary conditions, the non-autonomous
p-Laplacian equation is less considered. In this article, we obtain the existence and
structure of a compactly uniform attractor in Lr1 (Ω) × Lr (Γ) (r = min(r1 , r2 )),
which holds only under the assumptions (1.2)–(1.4), and no any restrictions on
p, r1 , r2 and N .
The main results of this article are Theorem 3.4 (asymptotic regularity), Theorem 3.5 (global attractor) and Theorem 4.5 (uniform attractor and its structure).
Hereafter, we assume that
2 < p < N.
For the case p = 2, system (1.1) is a reaction-diffusion equation and we refer
the reader to [15, 28]; and if p > N , then embeddings W 1,p (Ω) ,→ Ls1 (Ω) and
W 1,p (Ω) ,→ Ls2 (Γ) hold for any s1 , s2 ∈ [1, ∞), which make the nonlinear terms
f (·) and g(·) to be trivial terms.
For convenience, hereafter k · k and k · kΓ stand for the norm in L2 (Ω) and L2 (Γ),
h·, ·i and h·, ·iΓ stand for the inner product in L2 (Ω) and L2 (Γ), respectively. C,
Ci denote general positive constants, i = 1, . . . , which will be different in different
estimates.
This article is organized as follows: in Section 2, we introduce some preliminary
results; in Section 3, for the autonomous cases, i.e., h(x, t) = h(x), we prove some asymptotic regularity of the solution; in Section 4, for the non-autonomous cases, the
existence and structure of a uniform attractor in Lr1 (Ω) × Lr (Γ) (r = min(r1 , r2 ))
is obtained.
EJDE-2015/37
DYNAMICS OF THE p-LAPLACIAN EQUATIONS
3
2. Preliminaries
In this section, we give some auxiliary results which will be used later. We first
introduce the spaces of time-dependent external forcing h(x, t) to be considered in
this article (see[4]).
Definition 2.1 ([4]). A function ϕ is said to be translation bounded in L2loc (R; X),
if
Z t+1
kϕk2b = sup
kϕk2X ds < +∞.
t∈R
Denote by
L2b (R; X)
t
the set of all translation bounded functions in L2loc (R; X).
We now introduce a class of functions that was defined first in [21].
Definition 2.2 ([21]). A function ϕ ∈ L2loc (R; X) is said to be normal if for any
ε > 0, there exists η > 0 such that
Z t+η
sup
kϕk2X ds ≤ ε.
t∈R
t
Denote by L2n (R; X) the set of all normal functions in L2loc (R; X).
Lemma 2.3 ([21]). If ϕ0 ∈ L2n (R; X), then for any τ ∈ R,
Z t
lim sup
e−γ(t−s) kϕ(s)k2X ds = 0,
γ→∞ t≥τ
τ
uniformly (with respect to ϕ ∈ H(ϕ0 )), where H(ϕ0 ) = {ϕ0 (t + h) | h ∈ R} .
The next result is an estimate of the p-Laplacian operator; see [9] for the proof.
Lemma 2.4. Let p > 2. Then there exists constant K > 0 such that for any
a, b ∈ RN ,
h|a|p−2 a − |b|p−2 b, a − bi > K|a − b|p ,
(2.1)
where K depends only on p and N ; h·, ·i denotes the inner product of RN .
3. Autonomous cases: h(x, t) = h(x)
In this section, we consider the autonomous case of (1.1); that is,
ut − ∆p u + f (u) = h(x),
ut + |∇u|
p−2
∂n u + g(u) = 0,
in Ω,
on Γ,
(3.1)
u(x, 0) = u0 (x),
0
where h(x) ∈ Lr1 (Ω), r10 is conjugate to r1 .
3.1. Mathematical setting. At first, following [17], it is more convenient to introduce the unknown function v(t) := u(t)|Γ , defined on the boundary Γ, so we
think of our problem as a coupled system of two parabolic equations, one in the
bulk Ω and the other on the boundary Γ. Thus, the function u(t) tracks diffusion
in the bulk, while v(t) records the information on the boundary. Throughout the
remainder of this section, we formulate the problem as following:
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X. CHENG, L. WEI
EJDE-2015/37
Problem (P). Let Ω ⊂ RN (N ≥ 3) be a bounded domain with a smooth boundary
Γ := ∂Ω (e.g., of class C 2 ). The nonlinearities f and g satisfy (1.2)–(1.4). For any
given pair of initial data (u0 , v0 ) ∈ L2 (Ω) × L2 (Γ), find (u(t), v(t)) with
(u, v) ∈ C([0, +∞); L2 (Ω) × L2 (Γ)) ∩ L∞ ((0, +∞); W 1,p (Ω) × W 1−1/p,p (Γ)),
1,2
(u, v) ∈ Wloc
((0, ∞); L2 (Ω) × L2 (Γ)),
u ∈ Lploc ([0, +∞); W 1,p (Ω)),
v ∈ Lploc ([0, +∞); W 1−1/p,p (Γ))
(3.2)
such that (u(0), v(0)) = (u0 , v0 ), and for almost all t ≥ 0, (u(t), v(t)) satisfies
u(t)|Γ = v(t) a.e. for t ∈ (0, ∞), and the following partial differential equations:
∂t u − div(|∇u|p−2 ∇u) + f (u) = h(x),
∂t v + |∇u|
p−2
∂n u + g(v) = 0,
in Ω × (0, +∞),
on Γ × (0, +∞).
(3.3)
Secondly, we give the following existence and uniqueness results, where we use
the definition of weak solution as in [17, Definition 2.3]. For more details we refer
the reader to [17].
Theorem 3.1 ([17]). Let Ω be a bounded smooth domain in RN (N > 3), f and
0
g satisfy (1.2)–(1.4), h(x) ∈ Lr1 (Ω). Then for any initial data (u0 , v0 ) ∈ L2 (Ω) ×
2
L (Γ) and any T > 0, the problem (P) has a unique weak solution (u(t), v(t)) ∈
C([0, T ]; L2 (Ω) × L2 (Γ)). In addition to the regularity stated in (3.2), we also have
that
u(t) ∈ Lr1 (0, T ; Lr1 (Ω)), v(t) ∈ Lr2 (0, T ; Lr2 (Γ)).
Furthermore, (u0 , v0 ) 7→ (u(t), v(t)) is continuous on L2 (Ω) × L2 (Γ).
By Theorem 2.3, we can define the operator semigroup {S(t)}t>0 on the phase
space L2 (Ω) × L2 (Γ) as follows:
S(t) : L2 (Ω) × L2 (Γ) → L2 (Ω) × L2 (Γ),
S(t)(u0 , v0 ) = (u(t), v(t)),
(3.4)
which is continuous in L2 (Ω) × L2 (Γ).
Next, exactly as in [17], we have the following dissipative results.
Lemma 3.2 ([17]). Under the assumption of Theorem 2.3, {S(t)}t>0 has a positively invariant (L2 (Ω) × L2 (Γ), W 1,p (Ω) ∩ Lr1 (Ω) × W 1−1/p,p (Γ) ∩ Lr2 (Γ))-bounded
absorbing set; that is, there is a positive constant M , such that for any bounded
subset B ⊂ L2 (Ω) × L2 (Γ), there exists a positive constant T which depends only
on the L2 (Ω) × L2 (Γ)-norm of B such that
Z
Z
Z
|∇u(t)|p dx+ |u(t)|r1 dx+ |v(t)|r2 dS 6 M for all t > T and (u0 , v0 ) ∈ B.
Ω
Ω
Γ
Lemma 3.3 ([17]). Under the assumption of Theorem 2.3, for any bounded subset
B ⊂ L2 (Ω) × L2 (Γ), there exists a positive constant T1 which depends only on the
L2 (Ω) × L2 (Γ)-norm of B such that
Z
Z
|ut (s)|2 dx + |vt (s)|2 dS 6 M 0 for all s > T1 and (u0 , v0 ) ∈ B,
(3.5)
Ω
Γ
0
where M is a positive constant which depends on M .
EJDE-2015/37
DYNAMICS OF THE p-LAPLACIAN EQUATIONS
5
Hereafter, from Lemma 3.2, we denote one of the positively invariant absorbing
set by B0 with
B0 ⊂ {(u(t), v(t)) : ku(t)kW 1,p (Ω)∩Lr1 (Ω) + kv(t)kW 1−1/p,p (Γ)∩Lr2 (Γ) 6 M },
note that here the positive invariance means S(t)B0 ⊂ B0 for any t > 0.
3.2. Asymptotic regularity. In this subsection, we consider the asymptotic regularity of solutions of systems (3.1), which excel the regularity allowed by the
corresponding elliptic equation.
At first, we consider the elliptic equation
− div(|∇φ|p−2 ∇φ) + f (φ) = h(x)
|∇φ|
p−2
∂n φ + g(φ) = 0
in Ω,
(3.6)
on Γ.
Due to the assumptions (1.2)–(1.4), from the classical results about elliptic equations, we know that (3.6) at least has one solution φ(x) with
φ(x) ∈ W 1,p (Ω) ∩ Lr1 (Ω).
(3.7)
For the rest of this article, we assume that φ(x) denotes a fixed solution of (3.6).
Then, for the solution (u(x, t), v(x, t)) of (3.1), we decompose (u(x, t), v(x, t)) as
follows
(u(x, t), v(x, t)) = (φ(x) + w(x, t), φ(x) + w(x,
e t))
(3.8)
with u0 (x) = φ(x) + w(x, 0), v0 (x) = φ(x) + w(x,
e 0), where (w(x, t), w(x,
e t)) solves
the equation
wt − div(|∇u|p−2 ∇u) + div(|∇φ|p−2 ∇φ) + f (u) − f (φ) = 0
w
et + |∇u|p−2 ∂n u − |∇φ|p−2 ∂n φ + g(v) − g(φ) = 0,
in Ω,
on Γ,
w(x,
e t) := w(x, t)|Γ ,
(3.9)
w(x, 0) = u0 (x) − φ(x),
w(x,
e 0) = v0 (x) − φ(x).
It is easy to see that this equation is also globally well posed. Moreover, thanks
to Lemma 3.2, without loss of generality, hereafter we assume (u0 , v0 ) ∈ B0 and so
(w(x, 0), w(x,
e 0)) ∈ (W 1,p (Ω) ∩ Lr1 (Ω)) × (W 1−1/p,p (Γ) ∩ Lr2 (Γ)).
At the same time, from the positive invariance of B0 and (3.7) we have that
kw(x, t)kW 1,p (Ω)∩Lr1 (Ω) + kw(x,
e t)kW 1−1/p,p (Γ)∩Lr2 (Γ) 6 M1
(3.10)
for all t > 0, with some positive constant M1 .
The main result of this section reads as follows.
Theorem 3.4. Let Ω be a bounded smooth domain in RN (N > 3), f and g satisfy
0
(1.2)–(1.4), h(x) ∈ Lr1 (Ω), and suppose that {S(t)}t≥0 is the semigroup generated
by the solutions of equation (3.1) with initial data (u0 , v0 ) ∈ L2 (Ω) × L2 (Γ). Then,
for any δ, γ ∈ [0, ∞), there exists a bounded subset Bδ,γ satisfying the following
properties:
n
Bδ,γ = (w, w)
e : kwkW 1,p (Ω)∩Lr1 +δ (Ω)
o
+ kwk
e W 1−1/p,p (Γ)∩Lr2 +γ (Γ) 6 Λp,r1 ,r2 ,N,δ,γ < ∞ ,
and for any bounded subset B ⊂ L2 (Ω) × L2 (Γ), there exists a
T = T (kBkL2 (Ω) , kBkL2 (Γ) , δ, γ)
6
X. CHENG, L. WEI
EJDE-2015/37
such that
S(t)B ⊂ φ(x) + Bδ,γ
for all t > T,
(3.11)
where φ(x) is a fixed solution of (3.6), (w(x, t), w(x,
e t)) satisfies (3.9); the constant
Λp,r1 ,r2 ,N,δ,γ depends only on p, r1 , r2 , N, δ, γ.
Proof. We use the Moser-Alikakos iteration technique [2] to prove the following
induction estimates about the solution of (3.9). For clarity, we separate our proof
into two steps.
Step 1 : We first claim that
For each k = 0, 1, 2, . . . , there exist two positive constants Tk and Mk , which
depend only on k, p, r1 , r2 , N and kB0 kW 1,p (Ω)∩Lr1 (Ω)×W 1−1/p,p (Γ)∩Lr2 (Γ) , such that
for any (u0 , v0 ) ∈ B0 and t > Tk , we have
Z
Z
|w(t)|σk dx + |w(t)|
e σk dS 6 Mk ,
(Ak )
Ω
and
Z
t+1
t
Z
|w(s)|σk+1 dx
Γ
−p
N
N −1
Z
t
Ω
t+1
Z
ds +
Γ
σk+1
|w(s)|
e
dS
−p
N
N −1
ds 6 Mk .
(Bk )
where (w(t), w(t))
e
is the solution of equation (3.9), and
σk = 2(
k
X
N −1 i
N −1 k
) + (p − 2)
(
) −1 ,
N −p
N
−
p
i=0
k = 0, 1, 2, . . . .
(3.12)
(i) Initialization of the induction (k = 0). From (3.10), we can deduce (A0 ) immediately. To prove (B0 ), we multiply (3.9) by w and w,
e and integrate over Ω, then
we obtain
Z
Z
Z
1 d
1 d
|w|2 dx +
|w|
e 2 dS + h|∇u|p−2 ∇u − |∇φ|p−2 ∇φ, ∇wi dx
2 dt Ω
2 dt Γ
Ω
Z
Z
(3.13)
+ (f (u) − f (φ))w dx + (g(v) − g(φ))w
e dS = 0.
Ω
Γ
By (1.4), we have
Z
Z
(f (u) − f (φ))w dx > −l
Ω
(3.14)
|w|
e 2 dS.
(3.15)
Ω
Z
Z
Γ
|w|2 dx,
(g(v) − g(φ))w
e dS > −m
Γ
Then applying Lemma 2.4, we have
Z
Z
h|∇u|p−2 ∇u − |∇φ|p−2 ∇φ, ∇wi dx > K
|∇w|p dx.
Ω
(3.16)
Ω
Inserting (3.14)–(3.16) into (3.13), we obtain
Z
Z
Z
1 d
1 d
|w|2 dx +
|w|
e 2 dS + K
|∇w|p dx
2 dt Ω
2 dt Γ
Ω
Z
Z
2
2
6l
|w| dx + m |w|
e dS
Ω
Z
ZΓ
6C
|w|2 dx + |w|
e 2 dS .
Ω
Γ
(3.17)
EJDE-2015/37
DYNAMICS OF THE p-LAPLACIAN EQUATIONS
7
Then, for any t > 0, integrating the above inequality over [t, t + 1] and using (3.10),
we deduce that
Z t+1 Z
|∇w(x, s)|p dx ds 6 CK,M,M1 for all t > 0.
(3.18)
t
Ω
By the Sobolev embeddings (e.g., see Adams and Fourier [1])
W 1,p (Ω) ,→ L
p(N −1)
N −p
W 1,p (Ω) ,→ L
(Ω),
p(N −1)
N −p
(Γ),
from (3.18), for all t > 0, we have
Z t+1 Z
N −p
p(N −1)
|w(x, s)| N −p dx N −1 ds
t
Ω
t+1
Z
p
|∇w(x, s)| dx ds 6 CK,M,M1 ,N ,
6 C1
Ω
t
Z
(3.19)
Z
t+1
Z
|w(x,
e s)|
t
Z
Γ
t+1
p(N −1)
N −p
dS
−p
N
N −1
ds
(3.20)
Z
p
|∇w(x, s)| dx ds 6 CK,M,M1 ,N ,
6 C2
t
Ω
where C1 , C2 are constants of embeddings W 1,p (Ω) ,→ L
p(N −1)
N −p
(Ω) and W 1,p (Ω) ,→
p(N −1)
N −p
L
(Γ), note that here C1 , C2 depend only on N . This implies (B0 ) holds.
(ii) The induction argument. We now assume that (Ak ) and (Bk ) hold for k > 1,
and we need only to prove that (Ak+1 ) and (Bk+1 ) hold. Multiplying (3.9) by
e and integrating over Ω, we obtain
|w|σk+1 −2 w and |w|
e σk+1 −2 w,
Z
Z
1 d
σk+1
|w|
dx + |w|
e σk+1 dS
σk+1 dt Ω
Γ
Z
p−2
+ (σk+1 − 1) h|∇u| ∇u − |∇φ|p−2 ∇φ, ∇wi|w|σk+1 −2 dx
(3.21)
Ω
Z
Z
σ −2
+
f (u) − f (φ) |w|σk+1 −2 w dx +
g(v) − g(φ) |w|
e k+1 w
e dS = 0.
Ω
Γ
Similar to (3.14)–(3.16), we have
Z
Z
f (u) − f (φ) |w|σk+1 −2 w dx > −l
|w|σk+1 dx,
ZΩ
ZΩ
σk+1 −2
(g(v) − g(φ))|w|
e
w
e dS > −m |w|
e σk+1 dS,
Γ
Γ
Z
(σk+1 − 1) h|∇u|p−2 ∇u − |∇φ|p−2 ∇φ, ∇wi|w|σk+1 −2 dx
Ω
Z
> K(σk+1 − 1)
|∇w|p |w|σk+1 −2 dx,
(3.22)
(3.23)
(3.24)
Ω
so we have
Z
Z
Z
1 d
e σk+1 dS + K(σk+1 − 1)
|∇w|p |w|σk+1 −2 dx
|w|σk+1 dx + |w|
σk+1 dt Ω
Γ
Ω
Z
Z
Z
Z
σk+1
σk+1
σk+1
6l
|w|
dx + m |w|
e
dS 6 C
|w|
dx + |w|
e σk+1 dS .
Ω
Γ
Ω
Γ
(3.25)
8
X. CHENG, L. WEI
EJDE-2015/37
Then, combining with (Bk ) and application of the uniform Gronwall lemma to
(3.25) we can get (Ak+1 ) immediately. For (Bk+1 ), we integrate the above inequality
over [t, t + 1] and use (Ak+1 ), we have
Z t+1 Z
|∇w|p |w|σk+1 −2 dx ds 6 Mk+1 for all t > 0,
(3.26)
t
Ω
where Mk+1 depends on k, p, r1 , r2 , N, M, M1 . By the embeddings W 1,p (Ω) ,→
L
p(N −1)
N −p
p(N −1)
(Ω) and W 1,p (Ω) ,→ L N −p (Γ) again, we have
−p
N
Z
N −1
N −1
|w|(σk+1 −2+p) N −p dx
Ω
Z
p
p
6 C1 ·
|w|σk+1 −2 |∇w|p dx,
σk+1 − 2 + p
Ω
−p
N
Z
N −1
N −1
|w|
e (σk+1 −2+p) N −p dS
Γ
Z
p
p
6 C2 ·
|w|σk+1 −2 |∇w|p dx,
σk+1 − 2 + p
Ω
(3.27)
(3.28)
and from the definition of σk , we have
(σk+1 − 2 + p)
N −1
= σk+2 .
N −p
(3.29)
Combining (3.26)–(3.29), we deduce (Bk+1 ) immediately.
Step 2 : Based on Step 1, since N > 3, from the definition of σk given in (3.12),
it is easy to see that σk → ∞ as k → ∞.
Hence, for any δ, γ ∈ [0, ∞), we can take k so large that r1 + δ 6 σk , r2 + γ 6 σk .
Consequently, we can define Bδ,γ as
n
Bδ,γ := (z, z̃) : kz + φkpW 1,p (Ω) + kzkrL1r+δ
1 +δ (Ω)
o
p
r2 +γ
+ kz̃ + φkW 1−1/p,p (Γ) + kz̃kLr2 +γ (Γ) 6 M + Mk ,
where z(t)|Γ = z̃(t), and recall that φ(x) is a fixed solution of (3.6).
Hence, from Theorem 3.4, using the interpolation inequality, we can obtain immediately the following results.
Theorem 3.5. Under the assumptions of Theorem 3.4, the semigroup {S(t)}t>0
has a (L2 (Ω) × L2 (Γ), W 1,p (Ω) ∩ Lr1 (Ω) × W 1−1/p,p (Γ) ∩ Lr2 (Γ))-global attractor
A . Moreover, A attracts every L2 (Ω) × L2 (Γ)-bounded subset with (W 1,p (Ω) ∩
Lr1 +δ (Ω)) × (W 1−1/p,p (Γ) ∩ Lr2 +γ (Γ))-norm for any δ, γ ∈ [0, ∞); and A allows
the decomposition A = φ(x) + A0 with A0 is bounded in (W 1,p (Ω) ∩ Lr1 +δ (Ω)) ×
(W 1−1/p,p (Γ)∩Lr2 +γ (Γ)) for any δ, γ ∈ [0, ∞), and φ(x) is a fixed solution of (3.6).
Proof. From Theorem 3.4, combining with the (L2 (Ω) × L2 (Γ), L2 (Ω) × L2 (Γ))asymptotic compactness (obtained in [17]) and the interpolation inequality, it is
easily to verify that {S(t)}t>0 is asymptotically compact in Lr1 (Ω) × Lr2 (Γ), then
it is sufficient to verify that {S(t)}t>0 is asymptotically compact in W 1,p (Ω) ×
W 1−1/p,p (Γ).
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DYNAMICS OF THE p-LAPLACIAN EQUATIONS
9
Let B0 be a (W 1,p (Ω) ∩ Lr1 (Ω)) × (W 1−1/p,p (Γ) ∩ Lr2 (Γ))-bounded absorbing
set obtained in Lemma 3.2, then we need only to show that
for any {(u0n , v0n )} ⊂ B0 and tn → ∞, {(un (tn ), vn (tn ))}∞
n=1 is precompact in W 1,p (Ω) × W 1−1/p,p (Γ),
(3.30)
where un (tn ) = S(tn )u0n , vn (tn ) = S(tn )v0n .
2
2
In fact, we know that {(un (tn ), vn (tn ))}∞
n=1 is precompact in L (Ω) × L (Γ) and
r1
r2
in L (Ω) × L (Γ).
Without loss of generality, we assume that {(unk (tnk ), vnk (tnk ))}∞
n=1 is a Cauchy
sequence in L2 (Ω) × L2 (Γ) and Lr1 (Ω) × Lr2 (Γ).
1,p
Now, we prove that {(unk (tnk ), vnk (tnk ))}∞
(Ω)×
n=1 is a Cauchy sequence in W
1−1/p,p
W
(Γ). From Lemma 2.4, and after standard transformations, we know that
there exists a constant K > 0, such that
Kk∇(unk (tnk ) − unj (tnj ))kpLp (Ω)
d
d
≤ − unk (tnk ) − f (unk (tnk )) + unj (tnj ) + f (unj (tnj )), unk (tnk ) − unj (tnj )
dt
dt
d
d
+ − vnk (tnk ) − g(vnk (tnk )) + vnj (tnj ) + g(vnj (tnj )), vnk (tnk ) − vnj (tnj ) Γ
dt
dt
Z
d
d
un (tn ) − un (tn )|un (tn ) − un (tn )|
≤
j
j
k
k
k
k
dt j j
Ω dt
Z
+
|f (unk (tnk )) − f (unj (tnj ))||unk (tnk ) − unj (tnj )|
ZΩ
d
d
+ vnk (tnk ) − vnj (tnj )|vnk (tnk ) − vnj (tnj )|
dt
dt
ZΓ
+ |g(vnk (tnk )) − g(vnj (tnj ))||vnk (tnk ) − vnj (tnj )|,
Γ
so we have
Kk∇(unk (tnk ) − unj (tnj ))kpLp (Ω)
d
d
≤ unk (tnk ) − unj (tnj ) kunk (tnk ) − unj (tnj )k
dt
dt
d
d
vnk (tnk ) − vnj (tnj )Γ kvnk (tnk ) − vnj (tnj )kΓ
+
dt
dt
+ C 1 + kunk (tnk )krL1r1 (Ω) + kunj (tnj )krL1r1 (Ω) unk (tnk ) − unj (tnj )Lr1 (Ω)
r2
e 1 + kvn (tn )kr2r
vn (tn ) − vnj (tnj ) r
+C
.
k
k
k
k
L 2 (Γ) + kvnj (tnj )kLr2 (Γ)
L 2 (Γ)
(3.31)
Combining Lemma 3.2, Lemma 3.3 and the compactness of Lr1 (Ω) × Lr2 (Γ), and
since W 1,p (Ω) ,→ W 1−1/p,p (Γ), we know that the norms on W 1,p (Ω) × W 1−1/p,p (Γ)
and W 1,p (Ω) are equivalent, (3.31) yields (3.30) immediately.
4. Non-autonomous case
In this section, we discuss the non-autonomous case of (1.1); that is,
ut − ∆p u + f (u) = h(x, t),
ut + |∇u|
p−2
∂n u + g(u) = 0,
u(x, τ ) = uτ (x),
in Ω̄,
in Ω,
on Γ,
(4.1)
10
X. CHENG, L. WEI
EJDE-2015/37
where h(x, t) ∈ L2b (R; L2 (Ω)).
4.1. Mathematical setting. Similar to the autonomous cases (e.g., Problem (p)
and Theorem 2.3), for each h ∈ Σ, we can also easily obtain the following wellposedness result and the time-dependent terms make no essential complications.
Theorem 4.1 ([17]). Let Ω be a bounded smooth domain in RN (N > 3), f and
g satisfy (1.2)–(1.4), h(x, t) ∈ L2b (R; L2 (Ω)). Then for any initial data (uτ , vτ ) ∈
L2 (Ω) × L2 (Γ), and any τ, T ∈ R, T > τ , the solution (u(t), v(t)) of problem (4.1)
is globally defined and satisfies
u(t) ∈ C([τ, T ]; L2 (Ω)) ∩ Lploc (τ, T ; W 1,p (Ω)) ∩ Lr1 (τ, T ; Lr1 (Ω)),
v(t) ∈ C([τ, T ]; L2 (Γ)) ∩ Lploc (τ, T ; W 1−1/p,p (Γ)) ∩ Lr2 (τ, T ; Lr2 (Γ)),
where v(t) := u(t)|Γ . Furthermore, (uτ , vτ ) 7→ (u(t), v(t)) is continuous on L2 (Ω) ×
L2 (Γ).
We now define the symbol space Σ for (4.1). Taking a fixed symbol σ0 (s) =
2
2
2
h0 (s), h0 (s) ∈ L2b (R; L2 (Ω)). We denote by L2,w
loc (R; L (Ω)) the space Lloc (R; L (Ω))
endowed with local weak convergence topology. Set
Σ0 = {h0 (s + h)| h ∈ R},
(4.2)
and let
2
Σ be the closure of Σ0 in L2,w
loc (R; L (Ω)).
Systems (4.1) can be rewritten in the operator form
∂t y = Aσ(t) (y),
y|t=τ = yτ ,
(4.3)
(4.4)
where σ(t) = h(t) is the symbol of equation (4.4). Thus, from Theorem 4.1, we
know that problem (4.1) is well posed for all σ(s) ∈ Σ and generates a family of
processes {Uσ (t, τ )}, σ ∈ Σ given by the formula Uσ (t, τ )yτ = y(t), and the y(t) is
the solution of (4.1).
4.2. Existence of a bounded uniformly (w. r. t. σ ∈ Σ) absorbing set
in (W 1,p (Ω) ∩ Lr1 (Ω)) × (W 1−1/p,p (Γ) ∩ Lr2 (Γ)). In this subsection, (W 1,p (Ω) ∩
Lr1 (Ω) × W 1−1/p,p (Γ) ∩ Lr2 (Γ))-bounded uniformly (with respect to σ ∈ Σ) absorbing set is obtained. The proof is similar to [17] (autonomous case).
Theorem 4.2. Let Ω be a bounded smooth domain in RN (N > 3), f and g satisfy
(1.2)–(1.4), h(x, t) ∈ L2b (R; L2 (Ω)). Then the family of processes {Uσ (t, τ )}, σ ∈ Σ
corresponding to (4.1) has a bounded uniformly (with respect to σ ∈ Σ) absorbing
set B0 in (W 1,p (Ω) ∩ Lr1 (Ω)) × (W 1−1/p,p (Γ) ∩ Lr2 (Γ)), that is, there is a positive
constant M , such that for any τ ∈ R and any bounded subset B, there exists a
positive constant T = T (B, τ ) ≥ τ such that
Z
Z
Z
|∇u(t)|p dx +
|u(t)|r1 dx + |v(t)|r2 dS 6 M
Ω
Ω
Γ
for all t > T , (uτ , vτ ) ∈ B, σ ∈ Σ.
Proof. Multiplying (4.1) by u and v, and integrating by parts, we obtain
Z
Z
Z
Z
Z
1 d
1 d
|u|2 dx +
|v|2 dS +
|∇u|p dx +
f (u)u dx + g(v)v dS
2 dt Ω
2 dt Γ
Ω
Ω
Γ
Z
(4.5)
=
h0 (t)u dx,
Ω
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DYNAMICS OF THE p-LAPLACIAN EQUATIONS
11
combining with assumptions (1.2)–(1.4), Young’s inequality and Poincaré inequality, we obtain
Z
Z
Z
Z
d
d
2
2
2
|u| dx +
|v| dS + C( |u| dx + |v|2 dS)
dt Ω
dt Γ
(4.6)
Ω
Γ
≤ C|Ω|,S(Γ) + Ckh0 k2 .
Applying the suitable version of Gronwall’s inequality to (4.6), we can find T0 > 0
and ρ0 > 0, such that
ku(t)k2 + kv(t)k2Γ ≤ ρ20 , for any t ≥ T0 .
(4.7)
Rs
Rs
Let F (s) = 0 f (τ )dτ , G(s) = 0 g(τ )dτ , by assumptions (1.2)–(1.3) again, from
(4.5), we obtain
Z
Z
Z
Z
Z
d
d
|u|2 dx +
|v|2 dS +
|∇u|p dx + C1
F (u) dx + C2 G(v) dS
dt Ω
dt Γ
Ω
Ω
Γ
≤ C|Ω|,S(Γ) + Ckh0 k2 .
Integrating this inequality above from t to t + 1, and combining (4.7), it follows
that for any t ≥ T0 ,
Z t+1 Z
Z
Z
p
( |∇u| dx + C1
F (u) dx + C2 G(v) dS)ds
t
Ω
Ω
Z
Γ
t+1
(4.8)
kh0 k2 ds
≤ C|Ω|,S(Γ),ρ0 + C
t
≤ C|Ω|,S(Γ),ρ0 ,kh0 k2b .
On the other hand, multiplying (1.1) by ut and vt , we have
Z
Z
Z
Z
Z
d
1 d
p
2
2
|∇u| dx +
F (u) dx + G(v) dS
|ut | dx + |vt | dS +
p
dt
dt
Ω
Ω
Γ
Ω
Γ
Z
Z
(4.9)
1
1
2
2
≤
|h0 | dx +
|ut | dx,
2 Ω
2 Ω
so we obtain
Z
Z
Z
d
p
( |∇u| dx + p
F (u) dx + p G(v) dS) ≤ Ckh0 k2 .
dt Ω
Ω
Γ
(4.10)
Combining (4.8) and (4.10), by the uniformly Gronwall lemma, we have that for
any t ≥ T0 + 1, σ ∈ Σ,
Z
Z
Z
p
|∇u| dx +
F (u) dx + G(v) dS ≤ C|Ω|,S(Γ),ρ0 ,khk2b ,
(4.11)
Ω
Ω
Γ
which implies that for any t ≥ T0 + 1, σ ∈ Σ,
Z
Z
Z
|∇u|p dx +
|u|r1 dx + |v|r2 dS ≤ M,
Ω
Ω
where M depends on |Ω|, S(Γ), ρ0 , khk2b .
(4.12)
Γ
As a direct result of Theorem 4.2, we have the existence of a uniform attractor
in L2 (Ω) × L2 (Γ):
12
X. CHENG, L. WEI
EJDE-2015/37
Corollary 4.3. Under the assumptions of Theorem 4.2, the family of processes
{Uσ (t, τ )}, σ ∈ Σ corresponding to (4.1) has a uniform attractor AΣ in L2 (Ω) ×
L2 (Γ) , which is compact in L2 (Ω)×L2 (Γ) and attracts every L2 (Ω)×L2 (Γ)-bounded
subset with L2 (Ω) × L2 (Γ)-norm. Moreover,
AΣ = ω0,Σ (B0 ) = ∪σ∈Σ Kσ (s),
∀ s ∈ R,
where Kσ (s) is the section at t = s of the kernel Kσ of the process {Uσ (t, τ )} with
symbol σ.
Proof. Theorem 4.2 and the Sobolev compactness imbedding theorem imply the
existence of a uniform attractor AΣ in L2 (Ω) × L2 (Γ) immediately.
4.3. Existence of a uniform attractor in Lr1 (Ω) × Lr (Γ) (r = min(r1 , r2 )).
First, we give some a priori estimates for the solution of (4.1) to verify the uniformly
asymptotic compactness in Lr1 (Ω) × Lr1 (Γ). The idea of the proof comes from [31].
Theorem 4.4. Assume that h(t) is normal in L2loc (R; L2 (Ω)), f and g satisfy (1.2)–
(1.3). Then for any ε > 0, τ ∈ R and any bounded subset B ⊂ L2 (Ω) × L2 (Γ),
there exist two positive constants T = T (B, ε, τ ) and M = M (ε), such that
Z
r1
|Uσ (t, τ )uτ |
Z
|Uσ (t, τ )vτ |r1 ≤ ε,
+
Ω(|Uσ (t,τ )uτ |≥M )
Γ(|Uσ (t,τ )vτ |≥M )
for all t ≥ T , (uτ , vτ ) ∈ B, σ ∈ Σ.
r1 −1
Proof. We multiply (4.1) by (u − M )r+1 −1 and (v − M )+
, and integrate over Ω,
then we have
Z
Z
1 d
1 d
r1
|u − M | dx +
|v − M |r1 dS
r1 dt Ω(u≥M )
r1 dt Γ(v≥M )
Z
Z
r1 −2
p
+ (r1 − 1)
(u − M )
|∇u| dx +
f (u)(u − M )r1 −1 dx
Ω(u≥M )
Ω(u≥M )
Z
(4.13)
+
g(v)(v − M )r1 −1 dS
Γ(v≥M )
Z
=
h0 (t)(u − M )r1 −1 dx,
Ω(u≥M )
where (u − M )+ denotes the positive part of (u − M ); that is,
(
(u − M )+ =
u − M, u ≥ M,
0,
u ≤ M.
From conditions (1.2)–(1.3), we can take M large enough such that
C3 |v|r2 −1 ≤ g(v),
r1 −1
C4 |u|
≤ f (u),
in Γ(v(t) ≥ M ),
in Ω(u(t) ≥ M ).
EJDE-2015/37
DYNAMICS OF THE p-LAPLACIAN EQUATIONS
13
Let Ω1 = Ω(u(t) ≥ M ), Γ1 = Γ(v(t) ≥ M ), using Young’s inequality and the
inequalities above, we obtain
Z
Z
1 d
1 d
|u − M |r1 dx +
|v − M |r1 dS
r1 dt Ω1
r1 dt Γ1
Z
+ (r1 − 1)
(u − M )r1 −2 |∇u|p dx
Ω1
Z
Z
(4.14)
+ C4
|u|r1 −1 (u − M )r1 −1 dx + C3
|v|r2 −1 (v − M )r1 −1 dS
Γ1
ZΩ1
Z
1
C4
|u − M |2r1 −2 dx +
|h0 (t)|2 dx,
≤
2 Ω1
2C4 Ω1
so we have
Z
1 d
|u − M |r1 dx +
|v − M |r1 dS
r
dt
1
Ω1
Γ1
Z
r1 −2
p
+ (r1 − 1)
(u − M )
|∇u| dx
1 d
r1 dt
Z
Ω1
r1 −2
Z
Z
C4 M
|u − M |r1 dx + C3 M r2 −2
|v − M |r1 dS
2
Ω1
Γ1
Z
1
2
|h0 (t)| dx.
≤
2C4 Ω1
+
By using the Gronwall lemma and together with the Lemma 2.3, we can choose M
large enough, such that
Z
Z
|u − M |r1 dx +
|v − M |r1 dS ≤ ε.
(4.15)
Ω1
Γ1
Noting that
Z
Z
1
r1
|u|
dx
≤
|u − M |r1 dx,
2r1 Ω(u≥2M )
Ω(u≥M )
Z
Z
1
r1
|v|
dS
≤
|v − M |r1 dS,
2r1 Γ(v≥2M )
Γ(v≥M )
combining (4.15)–(4.17), we obtain
Z
Z
|u(t)|r1 dx +
Ω(u≥2M )
|v(t)|r1 dS ≤ 2r1 ε.
(4.16)
(4.17)
(4.18)
Γ(v≥2M )
r1 −1
r1 −1
Repeating the same steps above, just taking (u + M )−
instead of (u − M )+
,
r1 −1
(v + M )r−1 −1 instead of (v − M )+
, we deduce that
Z
Z
|u(t)|r1 dx +
|v(t)|r1 dS ≤ 2r1 ε.
(4.19)
Ω(u≤−2M )
Γ(v≤−2M )
Combining (4.18)–(4.19), we obtain
Z
Z
r1
|u(t)| dx +
Ω(|u(t)|≥2M )
|v(t)|r1 dS ≤ 2r1 ε.
(4.20)
Γ(|v(t)|≥2M )
Now we state the existence and structure of a uniform attractor in Lr1 (Ω)×Lr (Γ)
(r = min(r1 , r2 )).
14
X. CHENG, L. WEI
EJDE-2015/37
Theorem 4.5. Assume that h(t) is normal in L2loc (R; L2 (Ω)), f and g satisfy
(1.2)–(1.4). Then the family of processes {Uσ (t, τ )}, σ ∈ Σ corresponding to (4.1)
has a compact uniform (with respect to σ ∈ Σ) attractor AΣ in Lr1 (Ω) × Lr (Γ)
(r = min(r1 , r2 )) and AΣ satisfies
AΣ = ω0,Σ (B0 ) = ∪σ∈Σ Kσ (s),
∀s ∈ R,
where Kσ (s) is the section at t = s of the kernel Kσ of the process {Uσ (t, τ )} with
symbol σ.
Proof. From Corollary 4.3 and Theorem 4.4, it is easy to verify that {Uσ (t, τ )}, σ ∈
Σ has uniformly asymptotic compactness in Lr1 (Ω) × Lr1 (Γ), which combining
with Theorem 4.2, we can obtain the existence of a compactly uniform attractor
in Lr1 (Ω) × Lr (Γ) (r = min(r1 , r2 )). Then, similar to [24, 28], we can obtain the
structure of AΣ , see more details in [24, 28].
Acknowledgments. This work is partly supported by the NNSF of China
(11101404, 11201204, 11471148) and by the State Scholarship Fund of China Scholarship Council (201308620021).
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Differential Equations, 223 (2006), 367-399.
Xiyou Cheng
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China.
Key Laboratory of Applied Mathematics and Complex Systems, Gansu Province, China
E-mail address: chengxy@lzu.edu.cn
Lei Wei (corresponding author)
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
E-mail address: wlxznu@163.com