Pullback and uniform exponential attractors for non-autonomous Oregonator systems

Definition 2.1

A two-parameter family of mappings U ( t , τ ) is said to be a continuous process in X if

1. U ( t , s ) U ( s , τ ) = U ( t , τ ) , ∀ t ≥ s ≥ τ , τ ∈ R ;

2. U ( τ , τ ) = I d X , τ ∈ R ;

3. U ( t , τ ) : X → X is continuous for any t ≥ τ .

Definition 2.2

A family of subsets D = { D ( t ) } t ∈ R of X is said to be pullback absorbing with respect to the process U ( ⋅ , ⋅ ) if for every t ∈ R and all bounded set B ⊂ X , there exists T = T ( t , B ) > 0 such that

for all τ ≥ T .

Definition 2.3

Let D = { D ( t ) } t ∈ R be a family of subsets of X . A process U ( ⋅ , ⋅ ) is said to be pullback D -asymptotically compact in X if for any t ∈ R , any sequence τ n → ∞ and x n ∈ D ( t − τ n ) , the sequence { U ( t , t − τ n ) x n } is relatively compact in X .

Definition 2.4

A family of compact subsets K = { K ( t ) } t ∈ R of X is said to be a pullback attractor if it satisfies the following:

1. K is invariant, i.e., U ( t , τ ) K ( τ ) = K ( t ) for all t ≥ τ ;

2. K is pullback attracting, i.e., for all bounded sets B ⊂ X ,

   lim τ → ∞ dist X ( U ( t , t − τ ) B , K ( t ) ) = 0 .

Definition 2.5

A family of subsets { A ( t ) } t ∈ R of X is called a pullback exponential attractor for the continuous process U ( t , τ ) if

1. A ( t ) ( t ∈ R ) is a compact subset of X and its fractal dimension is finite and uniformly bounded on t , i.e., sup t ∈ R dim f A ( t ) < ∞ ;

2. it is positively invariant, i.e., U ( t , τ ) A ( τ ) ⊂ A ( t ) for all t ≥ τ ;

3. there exist an exponent α > 0 and two functions Q , T : R + → R + such that for any bounded set B ⊂ X ,

   dist X ( U ( t , τ ) B , A ( t ) ) ≤ Q ( ‖ B ‖ X ) e − α ( t − τ ) , τ ∈ R , τ + T ( ‖ B ‖ X ) ≤ t ,

   where ‖ B ‖ X = sup ω ∈ B ‖ ω ‖ X .

Proposition 2.1

Let U ( t , τ ) be a continuous process in X. Assume that there exists a family of bounded subsets { Y ( t ) } t ∈ R of X satisfying:

1. the diameter ‖ Y ( t ) ‖ X of Y ( t ) is uniformly bounded, i.e., sup t ∈ R ‖ Y ( t ) ‖ X ≤ R < ∞ ;

2. it is positively invariant, i.e., U ( t , τ ) Y ( τ ) ⊂ Y ( t ) for all t ≥ τ ;

3. it is absorbing in the sense that for any bounded set B ⊂ X and τ ∈ R , there exists T B , τ ≥ 0 such that

   U ( t , τ ) B ⊂ Y ( t ) , τ + T B , τ ≤ t < ∞ ;

4. there exist t * > 0 and L = L ( t * ) > 0 such that for any τ ∈ R , t ∈ [ 0 , t * ] and φ τ , ψ τ ∈ Y ( τ ) ,

   ‖ U ( t + τ , τ ) φ τ − U ( τ + t , τ ) ψ τ ‖ ≤ L ‖ φ τ − ψ τ ‖ ;

5. there exist a constant γ ∈ ( 0 , 1 2 ) and a N-dimensional subspace X N ( N ∈ N ) of X such that the bounded projection P N : X → X N satisfies that for every τ ∈ R and φ τ , ψ τ ∈ Y ( τ ) ,

   ‖ ( I − P N ) ( U ( τ + t * , τ ) φ τ − U ( τ + t * , τ ) ψ τ ) ‖ ≤ γ ‖ φ τ − ψ τ ‖ ,

   where γ , N depend on t * but not on τ ;

6. for any given τ ∈ R ,

   lim t → τ sup ψ τ ∈ Y ( τ ) ‖ U ( t , τ ) ψ τ − ψ τ ‖ = 0 .

1. A ( t ) ⊂ Y ( t ) ¯ is compact;

2. dim f ( A ( t ) ) ≤ − ln N θ ln a θ , where a θ = 2 ( γ + θ L ) with 0 < θ < 1 − 2 γ 2 L and N θ is the minimal number of closed balls of X with radius θ covering the closed unit ball of X N , B N ( 0 , 1 ) = { ψ ∈ X N : ‖ ψ ‖ ≤ 1 } ;

3. U ( t , τ ) A ( τ ) ⊂ A ( t ) for all t ≥ τ ;

4. for any bounded set B ⊂ X ,

   dist X ( U ( t , τ ) B , A ( t ) ) ≤ L Re − T B , τ ln a θ t * a θ 2 e ln a θ t * ( t − τ ) , − ∞ < τ + T B , τ ≤ t < ∞ ;

5. lim s → t dist X ( A ( s ) , A ( t ) ) = 0 .

Definition 2.6

A set ℳ ⊂ ℬ ⊂ X is called a uniform exponential attractor for the family of processes { U σ ( t , τ ) } σ ∈ T m , t ≥ τ on X if

1. ℳ is a compact set;

2. ℳ has a finite fractal dimension, i.e., dim f ℳ < ∞ ;

3. there exist two positive constants a 1 and a 2 such that

   sup σ ∈ T m dist X ( U σ ( t , τ ) ℬ , ℳ ) ≤ a 1 e − a 2 ( t − τ ) .

Proposition 2.2

Let { U σ ( t , τ ) } σ ∈ T m , t ≥ τ ∈ R be a family of continuous processes on X. Assume that

1. { U σ ( t , τ ) } σ ∈ T m , t ≥ τ ∈ R and { T ( s ) } s ∈ R satisfy

   (2.1) U σ ( t + s , τ + s ) = U T ( s ) σ ( t , τ ) , s ∈ R , t ≥ τ , τ ∈ R ;

2. there exists a bounded set ℬ 0 ⊂ X such that for any σ ∈ T m , τ ∈ R ,

   U σ ( t , τ ) ℬ 0 ⊂ ℬ 0 , t ≥ τ ;

3. there exist t * > 0 , L = L ( t * ) > 0 and a N-dimensional subspace X N of X such that for any ψ 1 , ψ 2 ∈ ℬ 0 , σ 1 , σ 2 ∈ T m ,

   ‖ U σ 1 ( t , 0 ) ψ 1 − U σ 2 ( t , 0 ) ψ 2 ‖ ≤ L ( ‖ ψ 1 − ψ 2 ‖ 2 + ‖ σ 1 − σ 2 ‖ T m 2 ) 1 2 , ∀ t ∈ [ 0 , t * ] , ‖ ( I − P N ) U σ 1 ( t * , 0 ) ψ 1 − U σ 2 ( t * , 0 ) ψ 2 ‖ ≤ 1 8 2 ( ‖ ψ 1 − ψ 2 ‖ 2 + ‖ σ 1 − σ 2 ‖ T m 2 ) 1 2 ,

Lemma 3.1

Let p i ∈ L loc 2 ( R ; L 2 ( Ω ) ) , i = 1 , 2 , 3 . Then, for any ( u τ , v τ , w τ ) ∈ H and T ≥ τ , there exists a unique weak solution of (3.1) which satisfies

and for any θ i ∈ H 0 1 ( Ω ) , i = 1 , 2 , 3 and a.s. t ≥ τ ,

Moreover, the weak solution continuously depends on the initial data in H.

Theorem 3.1

There exists a uniformly bounded closed absorbing set B 0 = B ( 0 , r 0 ) = { ϕ ∈ H : ‖ ϕ ‖ ≤ r 0 } ( r 0 is independent of t and τ ) such that for any τ ∈ R and any bounded set B ⊂ X , there is a function T : R + → R + such that U ( t + τ , τ ) B ⊂ B 0 for all t ≥ T ( ‖ B ‖ ) , where ‖ B ‖ = sup ϕ ∈ B ‖ ϕ ‖ .

Proof

Let w ˜ = c 2 b 2 w . Clearly, system (1.1) becomes

Taking the inner-products ( ( 3.2 ) , u ) , ( ( 3.3 ) , v ) , ( 3.4 ) , c 2 w ˜ c 3 and summing them up, by Young’s inequality, we obtain

Let M 1 = a 1 + b 1 2 b 2 + a 3 2 c 2 2 c 3 2 b 2 + 1 2 and M 2 = max { 1 2 , 1 b 2 , c 2 2 c 3 2 b 2 } . We obtain

Therefore, by (1.6) and letting

one has

Integrating (3.5) from τ to t + τ gives