On quasilinear parabolic equations in the Orlicz spaces

Nonlinear Analysis: Real World Applications 22 (2015) 307–318

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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

On quasilinear parabolic equations in the Orlicz spaces Fei Fang a , Chao Ji b,∗ a

School of Mathematical Sciences, Peking University, Beijing, 100871, China

b

Department of Mathematics, East China University of Science and Technology, Shanghai, 20037, China

article

abstract

info

Article history: Received 16 April 2014 Received in revised form 19 September 2014 Accepted 24 September 2014

In this work, we consider a class of quasilinear parabolic equations in the Orlicz–Sobolev spaces. The existence and the asymptotic behavior of solutions are obtained via operator theory and nonlinear semigroups theory. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Orlicz–Sobolev spaces Quasilinear parabolic equations Attractor Nonlinear semigroups

1. Introduction In [1–5], the following quasilinear elliptic equation is considered in the Orlicz space



−div(a(|∇ u|)∇ u) = f (x, u), u = 0,

in Ω , on ∂ Ω .

(1.1)

These references are concerned with existence and multiplicity of solutions to (1.1) via variational methods. The principal project of this paper is to investigate existence of solutions and the asymptotic behavior to the corresponding parabolic version

 ut − div(a(|∇ u|)∇ u) = g (u), (x, t ) ∈ Ω × (0, T ), (P) u(x, t ) = 0, (x, t ) ∈ ∂ Ω × (0, T ), u(0) = u ∈ M := L2 (Ω ), 0

(1.2)

where Ω ∈ RN is a bounded domain with smooth boundary ∂ Ω . When a(|s|) = |s|p−2 , the corresponding operator is the well-known p-Laplacian operator −div(|∇·|p−2 ∇·). This operator has been widely considered in the literature of PDE (see [6–10] and the literature cited therein). Obviously, our problem (P) possesses more complicated operator, for example, it is inhomogeneous, so in the discussions, some special techniques will be needed. The inhomogeneous operator has important physical background, e.g., (a) nonlinear elasticity: P (t ) = (1 + t 2 )γ − 1, γ > 12 , (b) plasticity: P (t ) = t α (log(1 + t ))β , α ≥ 1, β > 0, t (c) generalized Newtonian fluids: P (t ) = 0 s1−α (sinh−1 s)β ds, 0 ≤ α ≤ 1, β > 0.

∗

Corresponding author. E-mail address: [email protected] (C. Ji).

http://dx.doi.org/10.1016/j.nonrwa.2014.09.005 1468-1218/© 2014 Elsevier Ltd. All rights reserved.

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F. Fang, C. Ji / Nonlinear Analysis: Real World Applications 22 (2015) 307–318

The paper is organized as follows. In Section 2, we present some preliminary knowledge on the Orlicz–Sobolev spaces. In Section 3, we define an operator with respect to problem (P) and prove this operator is maximal monotone in L2 (Ω ) and its domain is dense in L2 (Ω ) if p+ − p− < 1 and p− > 2. Using these properties of the operator, existence results of problem (P) are obtained. Section 4 is devoted to proving the existence of the global B-attractor for problem (P). 2. Preliminaries As in [1,4,3,5], we can also construct an Orlicz–Sobolev space setting for problem (P). Let the function p(t ) = ta(t ) t p ( s ) ds (such functions are called Young 0 or N-functions). If we set

(p(0) = 0) be odd increasing homeomorphism from R onto itself and let P (t ) = P˜ (t ) =

t



p−1 (s)ds, 0

then P and P˜ are complementary N-functions (see [11–13]). In order to construct an Orlicz–Sobolev space for problem (P), we impose the following condition on p(t ): tp(t ) tp(t ) (p) 1 < p− := inft >0 P (t ) ≤ p+ := supt >0 P (t ) < +∞.

Under the condition (p), the function P (t ) satisfies ∆2 -condition, i.e. P (2t ) ≤ kP (t ),

t > 0,

for some constant k > 0. The Orlicz space LP coincides with the set (equivalence classes) of measurable functions u : Ω → R such that

 Ω

P (|u|)dx < +∞

(2.1)

and is equipped with the (Luxemburg) norm, i.e.

     |u| dx < 1 . |u|P := inf k > 0, P Ω

k

We will denote by W 1,P (Ω ) the corresponding Orlicz–Sobolev space with the norm

  ∥u∥W 1,P (Ω ) := |u|P + ∇ uP 1 ,P

and define W0 (Ω ) as the closure of C0∞ (Ω ) in W 1,P (Ω ). The Orlicz–Sobolev conjugate function P∗ of P is given by P∗−1 (t ) :=

t

 0

P −1 (τ ) N +1 N

τ

dτ .

(2.2)

Let p− ∗ := inf

tP∗′ (t )

t >0

P∗ (t )

,

p+ ∗ := sup t >0

tP∗′ (t ) P∗ (t )

.

(2.3)

Throughout this paper, we assume that p+ and p− ∗ satisfy the following condition: (p∗ ) p+ < p− . ∗ 1,P

In this paper, using the Poincaré inequalities in Orlicz–Sobolev spaces we can use the following equivalent norm on W0 (Ω ):



∥u∥W 1,P (Ω ) := inf k > 0 :



 P

0

|∇ u| k

Ω





dx < 1 .

We now state some useful lemmas. 1,P

Lemma 2.1 (See [11–13]). Under the condition (p), the spaces LP (Ω ), W0 (Ω ) and W 1,P (Ω ) are separable and reflexive Banach spaces. Lemma 2.2 (See [11–13]). Let P and Q be N-functions. (1) If

 +∞ 1

P −1 (t ) N +1 t N

= ∞ and Q grow essentially more slowly than P∗ , then the embedding W 1,P (Ω ) ↩→ LQ (Ω ) is compact and

the embedding W 1,P (Ω ) ↩→ LP∗ (Ω ) is continuous;  +∞ P −1 (t ) (2) If 1 < ∞, then the embedding W 1,P (Ω ) ↩→ LQ (Ω ) is compact and the embedding W 1,P (Ω ) ↩→ L∞ (Ω ) is N +1 t N

continuous.

Lemma 2.3 (See [4]). Let ρ(u) = p+ P p− P

(1) If |u|P < 1, then |u|

(2) If |u|P > 1, then |u|



Ω

P (u)dx, we have −

≤ ρ(u) ≤ |u|pP ; + ≤ ρ(u) ≤ |u|pP ;

F. Fang, C. Ji / Nonlinear Analysis: Real World Applications 22 (2015) 307–318 +

309

−

(3) If 0 < t < 1, then t p P (u) ≤ P (tu) ≤ t p P (u); − + (4) If t > 1, then t p P (u) ≤ P (tu) ≤ t p P (u). Lemma 2.4 (See [4]). Let ρ( ˜ u) = p− p− −1

(1) If |u| P < 1, then |u| ˜

P

p+

p+ −1

(2) If |u| P > 1, then |u| ˜

P

(3) If 0 < t < 1, then t

p− p− −1

p+



Ω

P˜ (u)dx, we have p+ p+ −1

≤ ρ( ˜ u) ≤ |u|P˜

;

p−

− −1

≤ ρ( ˜ u) ≤ |u|P˜p

; p+

P˜ (u) ≤ P˜ (tu) ≤ t p+ −1 P˜ (u); p−

(4) If t > 1, then t p+ −1 P˜ (u) ≤ P˜ (tu) ≤ t p− −1 P˜ (u). Lemma 2.5 (See [11–13]). Assume that A(t ) and A˜ (t ) are complementary N-functions. We have (1) (2) (3) (4)

Young inequalities: uv ≤ A(u) + A˜ (v); Hölder inequalities: | Ω u(x)v(x)dx| ≤ 2|u|A |v|A˜ ; A(u) A˜ ( u ) ≤ A(u); A (u) A˜ ∗ ( ∗u ) ≤ A∗ (u).

Remark 2.1. Since problem (P) possesses the inhomogeneous operator, we utilize Lemmas 2.3–2.5 to overcome the nonhomogeneous difficulty. 3. The existence results of solutions Throughout this paper, we assume that N ≥ 1, p+ − p− < 1 and p− > 2. For simplicity, we denote M := L2 (Ω ), X := (Ω ) and −∆P u := −div(a(|∇ u|)∇ u). In this section we will prove some properties and estimates of the operator −∆P and use these properties and estimates to obtain three existence results of solutions for problem (P).

1,P W0

Definition 3.1. (1) (See [14]) Let A be an operator acting in a Hilbert space H and f ∈ L1 (0, T ; H ). A function u ∈ C ([0, T ]; H ) is a strong solution of the inclusion du dt

+ Au ∋ f

(3.1)

(t ) + Au(t ) ∋ f (t ) for if u is absolutely continuous in any compact subset of (0, T ), u(t ) ∈ D (A) for a.e. t ∈ (0, T ) and du dt a.e. t ∈ (0, T ). (2) (See [14]) We say that u ∈ C ([0, T ]; H ) is a weak solution of the inclusion (3.1) if there exist sequences fn ∈ L1 (0, T ; H ) and un ∈ C ([O, T ]; H ) such that un is a strong solution of the inclusion dun dt

+ Aun ∋ fn

(3.2)

fn → f in L1 (0, T ; H ) and un → u uniformly in [0, T ]. (3) A function u ∈ C ([O, T ]; H ) is a strong solution to

  du 

(t ) + A(u(t )) + B(u(t )) = 0, dt u(0) = u0 ∈ H

t >0

(3.3)

where A is a maximal monotone operator and B is a globally Lipschitz map on a Hilbert space H, if u is absolutely continuous in any compact subinterval of (0, T ), u(t ) ∈ D (A) for a.e. t ∈ (0, T ), and du

(t ) + A(u(t )) + B(u(t )) = 0 (3.4) dt for a.e. t ∈ (0, T ). A function u ∈ C ([0, T ]; H ) is called a weak solution of (3.3) if there is a sequence {un } of strong solutions convergent to u in C ([0, T ]; H ). Now we have the following theorem: Theorem 3.1. When T = +∞ and g : M → M is a globally Lipschitz map with Lipschitz constant L ≥ 0, problem (P) has a global weak solution. Consider the following operator A: Au(v) = for all u, v ∈ X .

 Ω

a(|∇ u|)∇ u∇v dx,

(3.5)

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F. Fang, C. Ji / Nonlinear Analysis: Real World Applications 22 (2015) 307–318

In order to prove Theorem 3.1, we need to prove the following lemmas: Lemma 3.1. (1) The operator A : X → X ∗ is monotone; (2) The operator A : X → X ∗ is coercive; (3) The operator A : X → X ∗ is hemicontinuous. Proof. (1) For α, β ∈ RN , α ̸= 0, we have the following inequality for a constant k0 > 0 (see [4,3])

(a(|α|)α − a(|β|)β)(α − β) ≥ k0

P (|α − β|)(l+1)/l

(P (|α|) + P (|β|))1/l

.

(3.6)

Let u, v ∈ X , then using (3.6) for each fixed x ∈ Ω we obtain

⟨Au − Av, u − v⟩X ∗, ,X = ⟨Au, u − v⟩X ∗, ,X − ⟨Av, u − v⟩X ∗, ,X   = a(|∇ u|)∇ u(∇ u − ∇v)dx − a(|∇v|)∇v(∇ u − ∇v)dx Ω Ω = (a(|∇ u|)∇ u − a(|∇v|)∇v) (∇ u − ∇v)dx Ω

P (|∇ u − ∇v|)(l+1)/l

 ≥ k0

Ω

(P (|∇ u|) + P (|∇v|))1/l

dx ≥ 0

(3.7)

and the conclusion (1) is proved. (2) Using Lemma 2.3, for u ∈ X and |∇ u|P > 1 we obtain

⟨Au, u⟩X ∗ ,X =

 Ω

≥ p−

a(|∇ u|)∇ u∇ udx

 Ω

P (|∇ u|)dx −

−

≥ p− |∇ u|pP = p− ∥u∥pX .

(3.8)

Therefore, if {uj } ⊂ X is sequence such that limn-sss lim ∥ui ∥X = lim |∇ ui |P = +∞,

i→∞

(3.9)

i→∞

there exists i0 ∈ N such that ∥ui ∥X > 1, for all i ≥ i0 . So for all i > i0 we have − ⟨Aui , ui ⟩X ∗ ,X ≥ p− ∥ui ∥pX −1 ∥ ui ∥ X

(3.10)

⟨Aui ,ui ⟩ ∗

and limi→∞ ∥u ∥X ,X = ∞, and the conclusion (2) is proved. i X (3) We will show that w − limt →0 A(u + t v) = A(u) for u, v ∈ X . Since X is reflexive, we need to prove that lim ⟨A(u + t v), φ⟩X ∗ ,X = ⟨A(u), φ⟩X ∗ ,X

t →0

∀φ ∈ X .

(3.11)

Let u, v, φ ∈ X and t ∈ (−1, +1). For convenience we denote gt (x) = a(|∇ u + t ∇v|)(∇ u + t ∇v)∇φ

(3.12)

g (x) = a(|∇ u|)∇ u∇φ.

(3.13)

and

Hence,

|⟨A(u + t v), φ⟩X ∗ ,X

      − ⟨A(u), φ⟩X ∗ ,X | =  gt (x)dx − g (x)(dx) Ω Ω      =  [gt (x) − g (x)]dx . Ω

Using the conditions on p(t ), for all x ∈ Ω , we have

|gt (x)| = |a(|∇ u + t ∇v|)(∇ u + t ∇v)∇φ| ≤ |a(|∇ u + t ∇v|)| |(∇ u + t ∇v)| |∇φ|

(3.14)

F. Fang, C. Ji / Nonlinear Analysis: Real World Applications 22 (2015) 307–318

= |p(|∇ u + t ∇v|)| |∇φ| ≤ |p(|∇ u| + |t ∇v|)| |∇φ| ≤ |p(|∇ u| + |∇v|)| |∇φ|.

311

(3.15)

Obviously, the function on the right term of (3.15) is in L (Ω ) and limt →0 gt (x) = g (x) for all x ∈ Ω . Then by the Dominated Convergence Theorem, we get 1

lim ⟨A(u + t v), φ⟩X ∗ ,X = ⟨A(u), φ⟩X ∗ ,X

∀φ ∈ X .

t →0

This completes our proof.

(3.16)



Remark 3.1. (1) According to Theorem 1.3, p. 40 in [15], one can show that A : X → X ∗ , with domain X , is maximal monotone and R(A) := A(X ) = X ∗ . (2) In virtue of Example 2.3.7, p. 26 in [14], we conclude that the operator AM , which is the realization of A at M = L2 (Ω ) given by



D(AM ) := {u ∈ X : A(u) ∈ M } AM (u) = A(u), if u ∈ D(AM ),

(3.17)

is maximal monotone in M . In what follows we will write −∆P u := −div(a(|∇ u|)∇ u) to mean AM (u) as we have just defined. Lemma 3.2. If we assume that ∥v∥X ≤ 1, then we have p+

(1) ⟨Av, v⟩X ∗ ,X ≥ ∥v∥X ; p−

p−

(2) ∥Av∥X ∗ ≤ (p+ ) p− −1 ∥v∥X + 1; p− p+ −p− + p− p+ −p+

(3) ∥Av∥X ∗ ≤ 2(p )

− −1

|∇v|pP

.

Proof. (i) Let v ∈ X with ∥v∥X ≤ 1. Lemma 2.3 implies that +

+

⟨Av, v⟩X ∗, ,X = ρ(∇v) ≥ |∇v|pP = ∥v∥pX and (1) is proved. (2) According to Lemmas 2.3–2.5, we have

∥Av∥X ∗ = sup |⟨Av, w⟩X ∗ ,X | ∥w∥X ≤1

    = sup  a(|∇v|)∇v∇wdx ∥w∥X ≤1 Ω      ≤ sup  p(|∇v|)|∇w|dx ∥w∥X ≤1

≤ ≤ ≤ ≤

Ω

    P (|∇v|) sup  p+ |∇w|dx |∇v| ∥w∥X ≤1 Ω      + P (|∇v|) ˜ sup P p dx + P (|∇w|)dx |∇v| ∥w∥X ≤1 Ω Ω      p− P (|∇v|) sup (p+ ) p− −1 P˜ dx + P (|∇w|)dx |∇v| ∥w∥X ≤1 Ω Ω    − p sup (p+ ) p− −1 P (|∇v|)dx + P (|∇w|)dx ∥w∥X ≤1

Ω

Ω

p−

≤ (p+ ) p− −1 ρ(∇v) + 1 p−

−

p−

−

= (p+ ) p− −1 |∇v|pP + 1 = (p+ ) p− −1 ∥v∥pX + 1 and (2) is proved.

(3.18)

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F. Fang, C. Ji / Nonlinear Analysis: Real World Applications 22 (2015) 307–318

(3) According to Lemmas 2.3–2.5, we have

∥Av∥X ∗ = sup |⟨Aw, w⟩X ∗ ,X | ∥w∥X ≤1     = sup  a(|∇v|)∇v∇wdx ∥w∥X ≤1 Ω     = sup  p(|∇v|)|∇w|dx ∥w∥ ≤1 Ω

X

= sup 2|p(|∇v|)|P˜ |∇w|P ∥w∥X ≤1

= 2|p(|∇v|)|P˜   p−−−1  p    P˜ (p(|∇v|))dx  Ω ≤2  p++−1    p    P˜ (p(|∇v|))dx Ω

if |p(|∇v|)|P˜ < 1 if |p(|∇v|)|P˜ > 1

   p−−−1   p  + P (|∇v|)  ˜  dx if |p(|∇v|)|P˜ < 1 P p  |∇v| Ω ≤2   p++−1     p   ˜P p+ P (|∇v|) dx  if |p(|∇v|)|P˜ > 1 |∇v| Ω     p−−−1   p P (|∇v|)  +  ˜ p dx if |p(|∇v|)|P˜ < 1 P  |∇v| Ω ≤2     p++−1   p− p+ −p− p  P (|∇v|)  + − + + p p − p ˜ (p ) P if |p(|∇v|)|P˜ > 1 dx |∇v| Ω    p−−−1  p  +   if |p(|∇v|)|P˜ < 1 P (|∇v|)dx p Ω ≤2   p++−1   p− p+ −p− p   + − + + (p ) p p −p if |p(|∇v|)|P˜ > 1 P (|∇v|)dx Ω  − p+ |∇v|pP −1 ≤2 p− p+ −p−  + pp−− pp++ −−pp+− + (p ) |∇v|P p p− p+ −p−

− −1

≤ 2(p+ ) p− p+ −p+ |∇v|pP and (3) is proved.

(3.19)



Remark 3.2. Similar to Lemma 3.2, if ∥v∥X ≥ 1 we have the following inequalities: p−

(1) ⟨Av, v⟩X ∗ ,X ≥ ∥v∥X ; p−

p+

(2) ∥Av∥X ∗ ≤ (p+ ) p− −1 ∥v∥X + 1; p− p+ −p−

p+ −1

(3) ∥Av∥X ∗ ≤ 2(p+ ) p− p+ −p+ |∇v|P

.

Lemma 3.3. D(AM ) is dense in M . If we denote the closure of D(AM ) by chM (D (AM )), then chM (D (AM )) = M . Proof. Let u be an arbitrary element of M and take ε ∈ (0, 1). Let uε := (1 + ε AM )−1 (u) ∈ D(AM ),

(3.20)

where 1 denotes the identity operator. Since uε + ε AM (uε ) = u

(3.21)

F. Fang, C. Ji / Nonlinear Analysis: Real World Applications 22 (2015) 307–318

313

then

∥uε ∥2M + ε⟨AM (uε ), uε ⟩ = ⟨u, uε ⟩.

(3.22)

This yields

∥uε ∥2M + ε⟨AM (uε ), uε ⟩ = ∥u∥M ∥uε ∥M ≤

1 2

1

∥u∥2M + ∥uε ∥2M . 2

(3.23)

So we have 1 2

∥uε ∥2M + ε⟨AM (uε ), uε ⟩ ≤

1 2

∥u∥2M .

(3.24)

By Lemma 3.2(1) and Remark 3.2(1), we easily know that ε⟨Av, v⟩ ≥ 0 for all v ∈ X . In particular, ε⟨AM (uε ), uε ⟩ ≥ 0 for all ε ∈ (0, 1). Thus by (3.24)

∥uε ∥M ≤ ∥u∥M ,

∀ε ∈ (0, 1).

(3.25)

If ε ∈ (0, 1) is such that ∥uε ∥X ≤ 1, then using (3.24) and Lemma 3.2(1), we obtain 1 2

+

∥uε ∥2M + ε∥uε ∥pX ≤

1 2

∥uε ∥2M + ε⟨AM (uε ), uε ⟩ ≤

1 2

∥u∥2M := C1

(3.26)

and then +

ε∥uε ∥pX ≤ C1 ,

(3.27)

where C1 is a constant independent of ε . From (3.21), Lemma 3.2(ii) and (3.27) we get further

∥uε − u∥X ∗ = ε∥AM (uε )∥X ∗   p− − ≤ ε (p+ ) p− −1 ∥uε ∥pX + 1     p−+ p− p C 1 ≤ ε (p+ ) p− −1 + 1 ε = (p ) +

p− p− −1

p− p+

C1 ε

p− 1− + p

+ ε.

(3.28)

If ε ∈ (0, 1) is such that ∥uε ∥X ≥ 1, then using (3.24) and Remark 3.2(1), we obtain 1 2

−

∥uε ∥2M + ε∥uε ∥pX ≤

1 2

∥uε ∥2M + ε⟨AM (uε ), uε ⟩ ≤

1 2

∥u∥2M := C1

(3.29)

and then −

ε∥uε ∥pX ≤ C1 .

(3.30)

From (3.21), Remark 3.2(3) and (3.30) we obtain

∥uε − u∥X ∗ = ε∥AM (uε )∥X ∗   p− p+ −p− p+ −1 + p− p+ −p+ ≤ ε 2(p ) ∥ uε ∥ P     p+−−1 p− p+ −p− p C 1  ≤ ε 2(p+ ) p− p+ −p+ ε p− p+ −p−

p+ −1

= 2(p+ ) p− p+ −p+ C1 p

−

ε

1−

p+ −1 p−

.

(3.31)

Obviously, p−

p− p+

lim (p+ ) p− −1 C1 ε

p− 1− + p

ε→0

p− p+ −p−

p+ −1 p−

lim 2(p+ ) p− p+ −p+ C1

ε→0

+ ε = 0, ε

1−

p+ −1 p−

= 0.

(3.32)

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F. Fang, C. Ji / Nonlinear Analysis: Real World Applications 22 (2015) 307–318

Hence, (3.28) and (3.31) imply that lim ∥uε − u∥X ∗ = 0.

(3.33)

ε→0

This implies that uε → u in X ∗ . Since M is a Hilbert space and {un } is bounded in M , any sequence {uεn } has a subsequence weakly convergent in M . As a consequence uε converges to u weakly in M when ε → 0. By (3.25), we have 0 ≤ ∥uε ∥M ≤ ∥u∥M for all ε ∈ (0, 1), moreover lim sup ∥uε − u∥M ≤ ∥u∥M ,

(3.34)

ε→0

and uε converges to u strongly in M (according to [16, Proposition 1.4, p. 14]). So the theorem is proved.



Proof of Theorem 3.1. By Lemma 3.1 and Proposition 1, p. 695 in [17], it follows that problem (P) determines a continuous semigroup of nonlinear operators

{T (t ) : chM (D (AM )) → chM (D(AM )), t > 0},

(3.35)

where for each u0 ∈ chM (D (AM )), t → T (t )u0 is a weak global solution to problem (P) beginning at u0 . This semigroup is such that R+ × chM (D (AM )) ∋ (t , u0 ) → T (t )u0 ∈ chM (D (AM )) is a continuous map and, if u0 ∈ D (AM ), then u(·) := T (·)u0 is a Lipschitz continuous strong solution of problem (P).

(3.36) 

Remark 3.3. Result similar to Theorem 3.1 has been obtained p(x)-Laplacian parabolic equations by J. Simsen and M.S. Simsen in [18]. Compared with p(x)-Laplacian operator, the operator −∆P has more strong nonlinearity. So some new techniques were used to overcome the difficulties, for example, some inequalities were managed to make hard estimate of the formula (3.18). 4. The existence results of the minimal global B-attractor In this section, we will use nonlinear semigroups theory to study the asymptotic behavior of problem (P). The reader is referred to [19–21] for more details on nonlinear semigroups theory. In past two decades, nonlinear semigroups theory was widely applied in nonlinear differential equations (see [16,22–25] and references therein). For the readers’ convenience we enumerate four definitions from Ladyzhenskaya [20] for nonlinear semigroups theory. Definition 4.1. Let O and M be subsets of X . We say that A attracts M or M is attracted to O by semigroup {T (T )}t ≥0 if for every ε > 0 there exists a t1 (ε, M ) ∈ R+ such that T (t )M ⊂ Uε (O) := {x ∈ X : d(x, O) ≤ ε} for all t ≥ t1 (ε, M ). The set O ⊂ X attracts the point x ∈ X if O attracts the one-point set {x}. Definition 4.2. If O attracts each point x of X then O is called a global attractor (for the semigroup). O is called a global B-attractor if O attracts each bounded set in X . Definition 4.3. A semigroup is called bounded dissipative or B-dissipative (respectively pointwise dissipative) if it has a bounded global B-attractor (respectively a bounded global attractor). Definition 4.4. A semigroup {T (t )}t ≥0 belongs to the class K if for each t > 0 the operator T (t ) is compact, i.e., for any bounded set B ∈ X its image T (t )B is precompact. Then we will prove Theorem 4.1. When T = +∞, and g : M → M is a globally Lipschitz map with Lipschitz constant L ≥ 0, then the semigroup {T (t )} with respect to problem (P) has a minimal closed global B-attractor N , which is compact and invariant. We will use the following proposition to prove Theorem 4.1. Proposition 4.1. Let {T (t ) : X → X , t ≥ 0} be a semigroup of class K . If it is B-dissipative, then {T (t ) : X → X , t ≥ 0} has a minimal closed global B-attractor M , which is compact and invariant. Next we will prove the existence of the minimal closed global B-attractor, which is compact and invariant for problem (P). To do this we need to prove that the semigroup determined by problem (P) is of class K and is bounded dissipative in H. So we have to verify the following lemmas. Lemma 4.1. Let u0 ∈ D (−∆P ) and u(·) = T (·)u0 be the global solution of problem (P). For all T > 0 we have:

T

p−

(1) 0 ∥u∥X ds ≤ c1 (|u0 |2 , T ); (2) ∥u∥L∞ (0,T ;M) ≤ c2 (|u0 |2 , T ); where c1 and c2 are locally bounded functions.

F. Fang, C. Ji / Nonlinear Analysis: Real World Applications 22 (2015) 307–318

315

Proof. Assume that u0 ∈ D (−∆P ) and T > 0. We have that u(·) := T (·)u0 is a Lipschitz continuous strong solution of problem (P). So, u(t ) ∈ C ([0, T ]; M ) and u(t ) ∈ D (−∆P ) for a.e. t ∈ I := (0, T ). Let us consider the measurable sets I1 := {t ∈ I ; ∥u(t )∥X < 1} and I2 := {t ∈ I ; ∥u(t )∥X ≥ 1}. So I = I1 ∪ I2 . Multiplying the two sides of (1.2) by u, applying Lemma 3.2 and Remark 3.2, one has 1 d 2 dt

|u(t )|22 = −⟨−∆P (u(t )), u(t )⟩M + ⟨g (u(t )), u(t )⟩M = −⟨−∆P (u(t )), u(t )⟩X ∗ ,X + ⟨g (u(t )), u(t )⟩M  + −∥u∥pX + |g (u(t ))|2 |u(t )|2 if t ∈ I1 ≤ − −∥u∥pX + |g (u(t ))|2 |u(t )|2 if t ∈ I2  + −∥u∥pX + d1 ∥(u(t ))∥2X + d2 ∥u(t )∥X if t ∈ I1 ≤ − −∥u∥pX + d1 ∥(u(t ))∥2X + d2 ∥u(t )∥X if t ∈ I2 ,

(4.1)

where d1 = d1 (L, σ ) > 0 and d2 = d2 (σ ) ≥ 0 are constants (d2 = 0 if only if, B(0) = 0). We now consider ε > 0 arbitrarily small, θ :=

ab ≤

ε ap

+

p

ε

−q p

q

bq

,

p+ 2

a, b, ε > 0, p, q > 1

and α := 1

and

+

p

p− . 2

1

So, using following Young’s inequalities,

= 1,

q

(4.2)

we get

  + 1 θ 1 p+   − 1 + ε + ε ∥u(t )∥pX + d3 + d4  + 1 d θ p  |u(t )|22 ≤  +  1 1 p− 2 dt   −1 + ε α + ε ∥u(t )∥pX + d5 + d6 − α p

if t ∈ I1 (4.3) if t ∈ I2 ,

where d3 = d3 (ε, p+ ) > 0, d4 = d4 (ε, p+ ) > 0, d5 = d5 (ε, p− ) > 0 and d6 = d6 (ε, p− ) > 0 are constants. Now, since ε > 0 is arbitrary, in the case B(0) ̸= 0 there always exists sufficiently small ε0 such that θ1 ε0θ + p−

and α1 ε0α + p1− ε0 < 12 . When B(0) = 0, one can choose ε0 > 0 sufficiently small such that θ1 ε0θ < in both cases, we get 1 d 2 dt

|u(t )|22 ≤

 + 1  − ∥u(t )∥pX + d7 2 1

 − ∥u(t )∥ 2

p− X

+ d8

1 2

1 p+

and α1 ε0α <

+

ε0p <

1 . 2

1 2

Hence,

if t ∈ I1 (4.4) if t ∈ I2 ,

where d7 = d7 (ε, p ) > 0 and d8 = d8 (ε, p− ) > 0 are constants. Therefore, +

T

 0

1 d 2 dt

|u(t )|22 dt =



1 d 2 dt

I1

≤−

1



2

|u(t )|22 dt +



1 d I2

∥u(t )∥

p+ X dt

I1

2 dt

|u(t )|22 dt

+ d7 T −

1 2



−

∥u(t )∥pX dt + d8 T .

(4.5)

I2

This implies

|u(T )| + 2 2



p+ X dt

∥u(t )∥



−

∥u(t )∥pX dt ≤ |u0 |22 + d9 ,

+

I1

(4.6)

I2

where d9 = d9 (p+ , p− , ε0 , T ) > 0 is a constant. Furthermore,

|u(T )|22 +



−

∥u(t )∥pX dt ≤ |u0 |22 + d9 ,

(4.7)

I2 −

−

and u ∈ Lp (I2 ; X ). Similarly, we have u ∈ Lp (I1 ; X ). Hence,

|u(T )|22 +



−

∥u(t )∥pX dt + I1



−

∥u(t )∥pX dt ≤ |u0 |22 + d9 + I2



−

∥u(t )∥pX dt . I1

(4.8)

316

F. Fang, C. Ji / Nonlinear Analysis: Real World Applications 22 (2015) 307–318

Then

|u(T )|22 +

T



+

∥u(t )∥pX dt ≤ |u0 |22 + d9 + m(I1 ) 0

≤ |u0 |22 + d9 + T = |u0 |22 + d10 ,

(4.9)

where d10 = d10 (p , p , ε0 , T ) > 0 is a constant and m(I1 ) = |I1 | is the Lebesgue measure of I1 . So (i) is satisfied. Now, we take the same arguments as above, and obtain +

−

|u(t )| ≤ |u(t )| + 2 2

2 2

t



−

∥u(s)∥pX ds 0

≤ |u0 |22 + d˜ 9 + t ≤ |u0 |22 + d˜ 9 + T = |u0 |22 + d10 ∀t ∈ I ,

(4.10)

where d9 = d9 (p , p , ε0 , T ) > 0 is a constant and d˜ 9 < d9 . This shows that (ii) is satisfied. +

−



Lemma 4.2. Let {T (t )} be the semigroup with respect to problem (P) on M . Then T (t ) is bounded dissipative in M . Proof. It is sufficient to consider initial value u0 ∈ D (−∆P ). Now consider the embedding constant σ > 0 from X ↩→ M



2

−



−

and the numbers t1 = 1 and r0 = (2σ p d8 ) p− + σ −p



p− 2

−1



−2 p− −2

 12

. Let t > t1 and u(·) = T (·)u0 .

If ∥u∥X ≤ 1, then |u|2 ≤ σ ∥u∥X ≤ σ . If ∥u∥X ≥ 1, then from (4.4) 1 d 2 dt

−

1

+

|u(t )|22 ≤ − ∥u(t )∥pX + d8 ≤ −

σ −p

2

2

−

|u(t )|p2 + d8 ,

(4.11)

where d8 = d8 (ε, p− ) > 0 is a constant. So the function y(t ) := |u(t )|22 satisfies the following differential inequality −

y′ (t ) ≤ −σ −p y(t )

p− 2

+ 2d8 .

(4.12)

Therefore, applying Lemma 5.1, p. 163 in [21], we get y(t ) = |u(t )| ≤ (2σ 2 2

p−

d8 )

2 p−



+ σ

−p−



p− 2

  −1 t

−2 p− −2

.

(4.13)

Hence, let r1 > max{r0 , σ }, one has

|u(t )|2 ≤ r1 ,

∀t ≥ t 1 .

(4.14)

Note that r1 does not depend on the initial value. Therefore, the set {w0 ∈ M ; |w0 |2 ≤ r } attracts bounded subsets of M in the M -norm. This completes the proof of Lemma 4.2.  Lemma 4.3. Let {T (t )} be the semigroup with respect to problem (P) on M . Then T (t ) : M → M is of class K . Proof. Let B ∈ M be a bounded set and t > 0. Since D (−∆P ) is dense in M for p+ − p− < 1 and p− > 2, then by Lemma 2, p. 697, in [17], it is sufficient to check compactness of the semigroup considering initial data u0 from D (−∆P ), that is, it is sufficient to prove that T (t )(B ∩ D (−∆P )) is precompact in M . Take any sequence {u0n } ⊂ B ∩ D (−∆P ) and consider the sequence {T (t )u0n }. Obviously, ∥u0n ∥ ≤ r , ∀n ∈ N. Consider T > t, δ1 > 0 and δ2 > 0 such that 0 < δ1 < δ2 < t < T . We define un (·) = T (·)u0n . Using (i) of Lemma 4.1 we have for all E ⊂ (0, T ) with m(E ) ≥ δ1 that 1 m(E )



1

−

∥un (s)∥pX ds ≤ E

m(E ) 1

c1 (|u0n |2 , T )

c1 (|u0n |22 + C10 ) m(E ) 1 ≤ c1 (|u0n |22 + C10 )

=

δ1

≤

1

δ1

c1 (|r |2 + C10 ) := a0 ,

where a0 > 0 is a constant. Now, we choose a constant a1 such that a1 > a0 .

(4.15)

F. Fang, C. Ji / Nonlinear Analysis: Real World Applications 22 (2015) 307–318

Statement. For each n ∈ N, there exists sn ∈ δ2 such that ∥un (sn )∥ In fact, if ∥un (s)∥ 1

δ2



δ2

p− X

p− X

317

≤ a1 .

> a1 , ∀s ∈ (0, δ2 ), then −

∥un (s)∥pX ds ≥

0

1

δ2

a1 δ2 = a1 .

But, since m((0, δ2 )) = δ2 > δ1 , we have by (4.15) that 1

δ2

δ2



−

∥un (s)∥pX ds ≤ a0 .

0 p−

This is a contradiction. So, there exists sn ∈ (0, δ2 ) such that ∥un (sn )∥X ≤ a1 and the proof of the statement is completed. Now, since X ⊂ M compactly, there is a subsequence {unj (snj )} of {un (sn )} such that T (snj )u0nj = unj (snj ) → v0 in M as j → +∞. As snj ∈ (0, δ2 ) ⊂ [0, δ2 ], there is a subsequence, which we do not relabel, such that snj → s0 ∈ [0, δ2 ]. Since the semigroup is such that R+ × chM (D (−∆P )) ∋ (t , u0 ) → T (t )u0 ∈ chM (D (−∆P )) = M

(4.16)

is a continuous map and (t − snj , T (snj )u0nj ) → (t − s0 , v0 ) as j → +∞, we have that T (t )u0nj = T (t − snj )T (snj )u0nj → T (t − s0 )v0 in M as j → +∞. The proof is completed.  Remark 4.1. In this paper we utilize Lemma 4.3 to obtain the existence of a global B-attractor, but this lemma is very important by itself. If we consider B ≡ 0, we have especially that the operator −∆P generates a compact semigroup for p+ − p− < 1 and p− > 2. This kind of result can be used in compactness research; for example, Theorems 2.2.2 and 2.3.3 in [26]. Hence, Proposition 4.1, Lemmas 4.2 and 4.3 imply Theorem 4.1. Acknowledgments The first author was supported by China Postdoctoral Science Foundation (No. 2014M550538). The second author was supported by the China Scholarship Council and NSFC (No. 11301181) and the Fundamental Research Funds for the Central Universities. References [1] P. Clément, M. García-Huidobro, R. Manásevich, K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations 11 (2000) 33–62. [2] F. Fang, Z. Tan, Existence and multiplicity of solutions for a class of quasilinear elliptic equations: an Orlicz–Sobolev space setting, J. Math. Anal. Appl. 389 (1) (2012) 420–428. [3] N. Fukagai, K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. Mat. Pura Appl. (4) 186 (2007) 539–564. [4] N. Fukagai, M. Ito, K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz–Sobolev nonlinearity on RN , Funkcial. Ekvac. 49 (2006) 235–267. [5] M. García-Huidobro, V.K. Le, R. Manásevich, K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz–Sobolev space setting, NoDEA Nonlinear Differential Equations Appl. 6 (1999) 207–225. [6] M. Degiovanni, S. Lancelotti, Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 907–919. [7] O. Gil, J.L. Vázquez, Focusing solutions for the p-Laplacian evolution equation, Adv. Differential Equations 2 (1997) 183–202. [8] R.G. Iagar, J.L. Vázquez, Asymptotic analysis of the p-Laplacian flow in an exterior domain, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 497–520. [9] E.H. Papageorgiou, N.S. Papageorgiou, A multiplicity theorem for problems with the p-Laplacian, J. Funct. Anal. 244 (2007) 63–77. [10] K. Perera, Nontrivial critical groups in p-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal. 21 (2003) 301–309. [11] R.A. Adams, J.J.F. Fournier, Sobolev Spaces, second ed., in: Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. [12] M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces, in: Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker Inc., New York, 1991. [13] M.M. Rao, Z.D. Ren, Applications of Orlicz Spaces, in: Monographs and Textbooks in Pure and Applied Mathematics, vol. 250, Marcel Dekker Inc., New York, 2002. [14] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, in: North-Holland Mathematics Studies, vol. 5, North-Holland Publishing Co., Amsterdam, 1973, Notas de Matemática (50). [15] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest, 1976, Translated from the Romanian. [16] M. Brandau, Stochastic differential equations with nonlinear semigroups, ZAMM Z. Angew. Math. Mech. 82 (2002) 737–743. 4th GAMM-Workshop ‘‘Stochastic Models and Control Theory’’ (Lutherstadt Wittenberg, 2001). [17] A.N. Carvalho, J.W. Cholewa, T. Dlotko, Global attractors for problems with monotone operators, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 2 (1999) 693–706. [18] J. Simsen, M.S. Simsen, On p(x)-Laplacian parabolic problems, Nonlinear Stud. 18 (2011) 393–403. [19] J.K. Hale, Asymptotic Behavior of Dissipative Systems, in: Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. [20] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, in: Lezioni Lincee (Lincei Lectures), Cambridge University Press, Cambridge, 1991. [21] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, in: Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988.

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