Existence of the global attractor for the plate equation with nonlocal nonlinearity in R^{n}
aa r X i v : . [ m a t h . A P ] M a r EXISTENCE OF THE GLOBAL ATTRACTOR FOR THE PLATE EQUATIONWITH NONLOCAL NONLINEARITY IN R n AZER KHANMAMEDOV AND SEMA SIMSEK
Abstract.
We consider Cauchy problem for the semilinear plate equation with nonlocal nonlin-earity. Under mild conditions on the damping coefficient, we prove that the semigroup generatedby this problem possesses a global attractor. Introduction
In this paper, we study the long-time behavior of the solutions for the following plate equationwith localized damping and nonlocal nonlinearity in terms of global attractors: u tt + ∆ u + α ( x ) u t + λu + f ( k u ( t ) k L p ( R n ) ) | u | p − u = h ( x ) , ( t, x ) ∈ (0 , ∞ ) × R n . (1.1)Plate equations have been investigated for many years due to their importance in some physicalareas such as vibration and elasticity theories of solid mechanics. For instance, in the case when f ( · ) is identically constant, equation (1.1) becomes an equation with local polynomial nonlinearitywhich arises in aeroelasticity modeling (see, for example, [7], [8]), whereas in the case when p = 2,the nonlinearity f ( k u k L p ( R n ) ) | u | p − u appears in the models of Kerr-like medium (see [15], [21]).The study of the long-time dynamics of evolution equations has become an outstanding areaduring the recent decades. As it is well known, the attractors can be used as a tool to describethe long-time dynamics of these equations. In particular, there have been many works on theinvestigation of the attractors for the plate equations over the last few years. For the attractorsof the plate equations with local and nonlocal nonlinearities in bounded domains, we refer to [3],[5], [14], [16-19] and [22]. In the case of unbounded domains, owing to the lack of Sobolev compactembedding theorems, there are difficulties in applying the methods given for bounded domains. Inorder to overcome these difficulties, the authors of [9-10], [13] and [23] established the uniform tailestimates for the plate equations with local nonlinearities and then used the weak continuity of thenonlinear source operators.In the case when the domain is unbounded and the equation includes nonlocal nonlinearity, anadditional obstacle occurs. For equation (1.1), this obstacle is caused by the operator defined by F ( u ) := f ( k u k L p ( R n ) ) | u | p − u , because the operator F , besides being not compact, is not alsoweakly continuous from H ( R n ) to L ( R n ). This situation does not allow us to apply the standardsplitting method and the energy method devised in [2]. Recently in [1], the obstacle mentionedabove is handled for the nonlinearity f ( k∇ u k L ( R n ) )∆ u by using compensated compactness methodintroduced in [11]. In that paper, the strictly positivity condition on the damping coefficient α ( · )is critically used. In the present paper, we replace this condition with the weaker conditions (see(2.3), (2.4)), and by using effectiveness of the dissipation for large enough x , we prove the existenceof the global attractor which equals the unstable manifold of the set of stationary points.The paper is organized as follows: In Section 2, we give the statement of the problem and themain result. In Section 3, we firstly prove two auxiliary lemmas and then establish the asymptoticcompactness of the solution, which together with the presence of the strict Lyapunov function leadsto the existence of the global attractor. Mathematics Subject Classification.
Key words and phrases. plate equation, global attractor. Statement of the problem and the main result
We consider the following initial value problem u tt + ∆ u + α ( x ) u t + λu + f ( k u ( t ) k L p ( R n ) ) | u | p − u = h ( x ) , ( t, x ) ∈ (0 , ∞ ) × R n , (2.1) u (0 , x ) = u ( x ), u t (0 , x ) = u ( x ), x ∈ R n , (2.2)where λ > , h ∈ L ( R n ) , p ≥ , p ( n − ≤ n − α ( · ), f ( · ) satisfy the followingconditions: α ∈ L ∞ ( R n ), α ( · ) > R n , (2.3) α ( · ) ≥ α > { x ∈ R n : | x | ≥ r } , for some r , (2.4) f ∈ C ( R + ), f ( · ) ≥
0. (2.5)Applying the semigroup theory (see [4, p.56-58]) and repeating the arguments done in the intro-duction of [1], one can prove the following well-posedness result.
Theorem 2.1.
Assume that the conditions (2.3)-(2.5) hold. Then for every
T > and ( u , u ) ∈ H ( R n ) × L ( R n ) , problem (2.1)-(2.2) has a unique weak solution u ∈ C (cid:0) [0 , T ]; H ( R n ) (cid:1) ∩ C (cid:0) [0 , T ]; L ( R n ) (cid:1) which satisfies the energy equality E R n ( u ( t )) + 1 p F (cid:16) k u ( t ) k pL p ( R n ) (cid:17) − Z R n h ( x ) u ( t, x ) dx + t Z s Z R n α ( x ) | u t ( τ, x ) | dxdτ = E R n ( u ( s )) + 1 p F (cid:16) k u ( s ) k pL p ( R n ) (cid:17) − Z R n h ( x ) u ( s, x ) dx , ∀ t ≥ s ≥ , (2.6) where F ( z ) = z R f ( p √ s ) ds , for all z ∈ R + and E Ω ( u ( t )) = R Ω ( | u t ( t, x ) | + | ∆ u ( t, x ) | + λ | u ( t, x ) | ) dx ,for subset Ω ⊂ R n . Moreover, if ( u , u ) ∈ H ( R n ) × H ( R n ) , then u ∈ C (cid:0) [0 , T ]; H ( R n ) (cid:1) ∩ C (cid:0) [0 , T ]; H ( R n ) (cid:1) .In addition, if v, w ∈ C (cid:0) [0 , T ]; H ( R n ) (cid:1) ∩ C (cid:0) [0 , T ]; L ( R n ) (cid:1) are the weak solutions to (2.1)-(2.2) with initial data ( v , v ) ∈ H ( R n ) × L ( R n ) and ( w , w ) ∈ H ( R n ) × L ( R n ) , then k v ( t ) − w ( t ) k H ( R n ) + k v t ( t ) − w t ( t ) k L ( R n ) ≤ c ( T, e r ) (cid:16) k v − w k H ( R n ) + k v − w k L ( R n ) (cid:17) , ∀ t ∈ [0 , T ] ,where c : R + × R + → R + is a nondecreasing function with respect to each variable and e r =max n k ( v , v ) k H ( R n ) × L ( R n ) , k ( w , w ) k H ( R n ) × L ( R n ) o . Thus, according to Theorem 2.1, by the formula ( u ( t ) , u t ( t )) = S ( t ) ( u , u ), problem (2.1)-(2.2)generates a strongly continuous semigroup { S ( t ) } t ≥ in H ( R n ) × L ( R n ), where u ( t, · ) is the weaksolution of (2.1)-(2.2), determined by Theorem 2.1, with initial data ( u , u ).Now, we are in a position to state our main result. Theorem 2.2.
Under the conditions (2.3)-(2.5), the semigroup { S ( t ) } t ≥ generated by the problem(2.1)-(2.2) possesses a global attractor A in H ( R n ) × L ( R n ) and A = M u ( N ) . Here M u ( N ) isunstable manifold emanating from the set of stationary points N (for definition, see [6, p.359]). Remark 2.1.
We note that by using the method of this paper, one can prove the existence of theglobal attractors for the initial boundary value problems u itt + ( − ∆) i u i + α ( x ) u it + λu i + f ( k u i ( t ) k L p ( R n ) ) | u i | p i − u i = h ( x ) , ( t, x ) ∈ (0 , ∞ ) × Ω , u i ( t, x ) = (cid:0) ∂∂ν (cid:1) i − u i ( t, x ) = 0 , ( t, x ) ∈ (0 , ∞ ) × ∂ Ω , u i (0 , x ) = u i ( x ) , u it (0 , x ) = u i ( x ) , x ∈ Ω , XISTENCE OF THE GLOBAL ATTRACTOR 3 where Ω ⊂ R n is an unbounded domain with smooth boundary, ν is outer unit normal vector, λ > , h ∈ L (Ω) , p i ≥ , p i ( n − i ) ≤ n − i , i = 1 , , the function f ( · ) satisfies the condition (2.5) andthe damping coefficient α ( · ) satisfies the following conditions α ∈ L ∞ (Ω) , α ( · ) > , a.e. in Ω , α ( · ) ≥ α > , a.e. in ω , for some ω ⊂ Ω , such that ω is the union of a neighbourhood of the boundary ∂ Ω and { x ∈ Ω : | x | ≥ r } , for some r . Proof of Theorem 2.2
We start with the following lemmas.
Lemma 3.1.
Assume that the condition (2.5) holds. Also, assume that the sequence { v m } ∞ m =1 isweakly star convergent in L ∞ (cid:0) , ∞ ; H ( R n ) (cid:1) , the sequence { v mt } ∞ m =1 is bounded in L ∞ (cid:0) , ∞ ; L ( R n ) (cid:1) and the sequence n k v m ( t ) k L p ( R n ) o ∞ m =1 is convergent, for all t ≥ . Then, for all r > m →∞ lim sup l →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t Z Z B (0 ,r ) τ (cid:16) f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) | v m ( τ, x ) | p − v m ( τ, x ) − f (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) | v l ( τ, x ) | p − v l ( τ, x ) (cid:17) ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ (cid:12)(cid:12)(cid:12) = 0 , ∀ t ≥ , where B (0 , r ) = { x ∈ R n : | x | < r } .Proof. Denote f ε ( u ) = (cid:26) f ( u ) , u ≥ ε,f ( ε ) , ≤ u < ε, for ε >
0. Then, we have (cid:12)(cid:12)(cid:12) f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) − f ε (cid:16) k v m ( τ ) k L p ( R n ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ max ≤ s ,s ≤ ε | f ( s ) − f ( s ) | , and consequently (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t Z Z B (0 ,r ) τ (cid:16) f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) | v m ( τ, x ) | p − v m ( τ, x ) − f (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) | v l ( τ, x ) | p − v l ( τ, x ) (cid:17) × ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ |≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t Z Z B (0 ,r ) τ (cid:16) f ε (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) | v m ( τ, x ) | p − v m ( τ, x ) − f ε (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) | v l ( τ, x ) | p − v l ( τ, x ) (cid:17) × ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ | + c t max ≤ s ,s ≤ ε | f ( s ) − f ( s ) | , ∀ t ≥
0. (3.1)Let us estimate the first term on the right hand side of (3.1). t Z Z B (0 ,r ) τ (cid:16) f ε (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) | v m ( τ, x ) | p − v m ( τ, x ) − f ε (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) | v l ( τ, x ) | p − v l ( τ, x ) (cid:17) × ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ = t Z τ f ε (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) Z B (0 ,r ) | v m ( τ, x ) | p − v m ( τ, x ) v mt ( τ, x ) dxdτ + t Z τ f ε (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) Z B (0 ,r ) | v l ( τ, x ) | p − v l ( τ, x ) v lt ( τ, x ) dxdτ AZER KHANMAMEDOV AND SEMA SIMSEK − t Z τ f ε (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) Z B (0 ,r ) | v m ( τ, x ) | p − v m ( τ, x ) v lt ( τ, x ) dxdτ − t Z τ f ε (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) Z B (0 ,r ) | v l ( τ, x ) | p − v l ( τ, x ) v mt ( τ, x ) dxdτ. (3.2)For the first two terms on the right hand side of (3.2), we have t Z τ f ε (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) Z B (0 ,r ) | v m ( τ, x ) | p − v m ( τ, x ) v mt ( τ, x ) dxdτ + t Z τ f ε (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) Z B (0 ,r ) | v l ( τ, x ) | p − v l ( τ, x ) v lt ( τ, x ) dxdτ = 1 p tf ε (cid:16) k v m ( t ) k L p ( R n ) (cid:17) k v m ( t ) k pL p ( B (0 ,r )) + 1 p tf ε (cid:16) k v l ( t ) k L p ( R n ) (cid:17) k v l ( t ) k pL p ( B (0 ,r )) − p t Z f ε (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) k v m ( τ ) k pL p ( B (0 ,r )) dτ − p t Z f ε (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) k v l ( τ ) k pL p ( B (0 ,r )) dτ − p t Z τ ddt (cid:16) f ε (cid:16) k v m ( τ ) k L p ( R n ) (cid:17)(cid:17) k v m ( τ ) k pL p ( B (0 ,r )) dτ − p t Z τ ddt (cid:16) f ε (cid:16) k v l ( τ ) k L p ( R n ) (cid:17)(cid:17) k v l ( τ ) k pL p ( B (0 ,r )) dτ. Since the sequence n k v m ( t ) k L p ( R n ) o ∞ m =1 is convergent, by continuity of f ε , it follows that the se-quence n f ε (cid:16) k v m ( t ) k L p ( R n ) (cid:17)o ∞ m =1 also converges for all t ∈ [0 , ∞ ). Moreover, by the conditionsof the lemma and the definition of f ε , we obtain that the sequence n f ε (cid:16) k v m ( . ) k L p ( R n ) (cid:17)o ∞ m =1 isbounded in W , ∞ (0 , ∞ ). So, the sequence n f ε (cid:16) k v m ( . ) k L p ( R n ) (cid:17)o ∞ m =1 converges weakly star in W , ∞ (0 , ∞ ) and we have f ε (cid:16) k v m ( . ) k L p ( R n ) (cid:17) → Q weakly star in W , ∞ (0 , ∞ ) ,v m → v weakly star in L ∞ (cid:0) , ∞ ; H ( R n ) (cid:1) ,v mt → v t weakly star in L ∞ (cid:0) , ∞ ; L ( R n ) (cid:1) , (3.3)for some Q ∈ W , ∞ (0 , ∞ ) and v ∈ L ∞ (cid:0) , ∞ ; H ( R n ) (cid:1) ∩ W , ∞ (cid:0) , ∞ ; L ( R n ) (cid:1) . Applying Aubin–Lions–Simon lemma (see [20]), by (3.3) -(3.3) , we find v m → v strongly in C ([0 , T ] ; L q ( B (0 , r ))) , ∀ T ≥ , (3.4)where q < n ( n − + . Then, considering (3.3) and (3.4), we getlim m →∞ t Z τ f ε (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) Z B (0 ,r ) | v m ( τ, x ) | p − v m ( τ, x ) v mt ( τ, x ) dxdτ + lim l →∞ t Z τ f ε (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) Z B (0 ,r ) | v l ( τ, x ) | p − v l ( τ, x ) v lt ( τ, x ) dxdτ XISTENCE OF THE GLOBAL ATTRACTOR 5 = 2 p tQ ( t ) k v ( t ) k pL p ( B (0 ,r )) − p t Z Q ( τ ) k v ( τ ) k pL p ( B (0 ,r )) dτ − p t Z τ ddt ( Q ( τ )) k v ( τ ) k pL p ( B (0 ,r )) dτ . (3.5)For the last two terms on the right hand side of (3.2), by using (3.3), we havelim m →∞ lim l →∞ t Z τ f ε (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) Z B (0 ,r ) | v m ( τ, x ) | p − v m ( τ, x ) v lt ( τ, x ) dxdτ + lim m →∞ lim l →∞ t Z τ f ε (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) Z B (0 ,r ) | v l ( τ, x ) | p − v l ( τ, x ) v mt ( τ, x ) dxdτ = 2 t Z τ Q ( τ ) Z B (0 ,r ) | v ( τ, x ) | p − v ( τ, x ) v t ( τ, x ) dxdτ = 2 p tQ ( t ) k v ( t ) k pL p ( B (0 ,r )) − p t Z Q ( τ ) k v ( τ ) k pL p ( B (0 ,r )) dτ − p t Z τ ddt ( Q ( τ )) k v ( τ ) k pL p ( B (0 ,r )) dτ . (3.6)Hence, taking into account (3.5)-(3.6) and passing to limit in (3.2), we obtainlim m →∞ lim l →∞ t Z Z B (0 ,r ) τ f ε (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) | v m ( τ, x ) | p − v m ( τ, x ) − f ε (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) | v l ( τ, x ) | p − v l ( τ, x ) ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ = 0. (3.7)Then, by (3.1) and (3.7), for all r >
0, we havelim sup m →∞ lim sup l →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t Z Z B (0 ,r ) τ f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) | v m ( τ, x ) | p − v m ( τ, x ) − f (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) | v l ( τ, x ) | p − v l ( τ, x ) ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ (cid:12)(cid:12)(cid:12) ≤ c t max ≤ s ,s ≤ ε | f ( s ) − f ( s ) | , ∀ t ≥ ε →
0, we obtain the claim of the lemma. (cid:3)
Lemma 3.2.
Assume that in addition to the conditions of Lemma 3.1, conditions (2.3) and (2.4)also hold. Then, for every γ > there exists c γ > such that t Z Z R n τ (cid:16) f (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) | v l ( τ, x ) | p − v l ( τ, x ) − f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) | v m ( τ, x ) | p − v m ( τ, x ) (cid:17) × ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ ≤ γ t Z τ E R n \ ( B (0 ,r )) ( v m ( τ ) − v l ( τ )) dτ + c γ t Z E R n \ ( B (0 ,r )) ( v m ( τ ) − v l ( τ )) dτ AZER KHANMAMEDOV AND SEMA SIMSEK + c γ t Z τ (cid:16)(cid:13)(cid:13) √ av mt ( τ ) (cid:13)(cid:13) L ( R n ) + (cid:13)(cid:13) √ av lt ( τ ) (cid:13)(cid:13) L ( R n ) (cid:17) E R n \ B (0 ,r ) ( v m ( τ ) − v l ( τ )) dτ + K m,lr ( t ) , ∀ t ≥ , ∀ r ≥ r , where K m,lr ∈ C [0 , t ] , sup m,l (cid:13)(cid:13) K m,lr (cid:13)(cid:13) C [0 ,t ] < ∞ and lim m →∞ lim sup l →∞ (cid:12)(cid:12) K m,lr ( t ) (cid:12)(cid:12) = 0 , for all t ≥ .Proof. Firstly, we have t Z Z R n τ (cid:16) f (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) | v l ( τ, x ) | p − v l ( τ, x ) − f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) | v m ( τ, x ) | p − v m ( τ, x ) (cid:17) × ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ = t Z Z R n \ B (0 ,r ) τ (cid:16) f (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) | v l ( τ, x ) | p − v l ( τ, x ) − f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) | v m ( τ, x ) | p − v m ( τ, x ) (cid:17) × ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ + K m,l ,r ( t ), ∀ r > , (3.8)where K m,l ,r ( t ) := t Z Z B (0 ,r ) τ (cid:16) f (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) | v l ( τ, x ) | p − v l ( τ, x ) − f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) | v m ( τ, x ) | p − v m ( τ, x ) (cid:17) ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ ,and by Lemma 3.1, it follows thatsup m,l (cid:13)(cid:13)(cid:13) K m,l ,r (cid:13)(cid:13)(cid:13) C [0 ,t ] < ∞ and lim m →∞ lim sup l →∞ (cid:12)(cid:12)(cid:12) K m,l ,r ( t ) (cid:12)(cid:12)(cid:12) = 0 , ∀ t ≥ . On the other hand, for the first term on the right hand side of (3.8), we get t Z Z R n \ B (0 ,r ) τ (cid:16) f (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) | v l ( τ, x ) | p − v l ( τ, x ) − f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) | v m ( τ, x ) | p − v m ( τ, x ) (cid:17) × ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ = − ( p − t Z τ f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) Z R n \ B (0 ,r ) Z | v m ( τ, x ) + σ ( v l ( τ, x ) − v m ( τ, x )) | p − dσ × ddτ | v m ( τ, x ) − v l ( τ, x ) | dxdτ + K m,l ,r ( t ) , (3.9)where K m,l ,r ( t ) := t Z τ (cid:16) f (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) − f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17)(cid:17) Z R n \ B (0 ,r ) | v l ( τ, x ) | p − v l ( τ, x ) × ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ .By the conditions of the lemma, we findsup m,l (cid:13)(cid:13)(cid:13) K m,l ,r (cid:13)(cid:13)(cid:13) C [0 ,t ] < ∞ and lim m →∞ lim sup l →∞ (cid:12)(cid:12)(cid:12) K m,l ,r ( t ) (cid:12)(cid:12)(cid:12) = 0, ∀ t ≥ K m,lr ( t ) := K m,l ,r ( t ) + K m,l ,r ( t ), by (3.8) and (3.9), we have t Z Z R n τ (cid:16) f (cid:16) k v l ( τ ) k L p ( R n ) (cid:17) | v l ( τ, x ) | p − v l ( τ, x ) − f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) | v m ( τ, x ) | p − v m ( τ, x ) (cid:17) XISTENCE OF THE GLOBAL ATTRACTOR 7 × ( v mt ( τ, x ) − v lt ( τ, x )) dxdτ = − ( p − t Z τ f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) Z R n \ B (0 ,r ) Z | v m ( τ, x ) + σ ( v l ( τ, x ) − v m ( τ, x )) | p − dσ × ddτ | v m ( τ, x ) − v l ( τ, x ) | dxdτ + K m,lr ( t ) . (3.10)Now, let us estimate the first term on the right hand side of (3.10). Denote ϕ M ( u ) = (cid:26) u, | u | ≤ M,M, | u | > M, and Ψ ε ( u ) = (cid:26) ε p − , | u | ≤ ε, | u | p − , | u | > ε . Then, we get (cid:12)(cid:12)(cid:12) k v m ( τ ) k L p ( R n ) − k ϕ M ( v m ( τ )) k L p ( R n ) (cid:12)(cid:12)(cid:12) ≤ Z { x ∈ R n : | v m ( τ,x ) | >M } | v m ( τ, x ) | p dx p ≤ M β Z { x ∈ R n : | v m ( τ,x ) | >M } | v m ( τ, x ) | p + β dx p ≤ M β k v ( τ ) k p + βp H ( R n ) , (3.11)where β ∈ (cid:16) , n ( n − + − p (cid:17) . Also, it is clear that (cid:12)(cid:12)(cid:12) | w | p − − Ψ ε ( w ) (cid:12)(cid:12)(cid:12) ≤ ω ( ε ) , (3.12)where ω ( ε ) = (cid:26) ε p − , p > , p = 2 .By (3.11) and (3.12), it is easy to see that t Z τ f (cid:16) k v m ( τ ) k L p ( R n ) (cid:17) Z R n \ B (0 ,r ) Z | v m ( τ, x ) + σ ( v l ( τ, x ) − v m ( τ, x )) | p − dσ × ddτ | v m ( τ, x ) − v l ( τ, x ) | dxdτ ≥ t Z τ f ε (cid:16) k ϕ M ( v m ( τ )) k L p ( R n ) (cid:17) Z R n \ B (0 ,r ) Z Ψ ε ( v m ( τ, x ) + σ ( v l ( τ, x ) − v m ( τ, x ))) dσ × ddτ | v m ( τ, x ) − v l ( τ, x ) | dxdτ − c (cid:18) max
0. (3.17)Since, by Lebesgue dominated convergence theorem, lim λ → + R B (0 ,r ) (cid:16) λa ( x )+ λ (cid:17) dx = 0, we can choosepositive parameters ε, M, µ and λ such that c (cid:18) µε p − + µε max { , − p } + max ≤ s ,s ≤ ε | f ( s ) − f ( s ) | + M p − ε p − µ + 1 M β + ω ( ε ) (cid:19) + c c M p − ε p − µ Z B (0 ,r ) (cid:18) λa ( x ) + λ (cid:19) dx ≤ γ .Thus, by (3.10), (3.16) and (3.17), the proof of the lemma is complete. (cid:3) Now, we prove the following theorem on the asymptotic compactness of { S ( t ) } t ≥ in H ( R n ) × L ( R n ), which plays a key role in the existence of the global attractor. Theorem 3.1.
Assume that the conditions (2.3)-(2.5) hold and B is a bounded subset of H ( R n ) × L ( R n ) . Then for every sequence of the form { S ( t k ) ϕ k } ∞ k =1 , where { ϕ k } ∞ k =1 ⊂ B , t k → ∞ , has aconvergent subsequence in H ( R n ) × L ( R n ) .Proof. To get the claim of the theorem, it is sufficient to prove the following sequential limit estimatelim inf k →∞ lim inf m →∞ k S ( t k ) ϕ k − S ( t m ) ϕ m k H ( R n ) × L ( R n ) = 0, (3.18)for every { ϕ k } ∞ k =1 ⊂ B and t k → ∞ . Indeed, establishing (3.18) and using the argument at the endof the proof of [12, Lemma 3.4], we obtain the desired result.Now, by (2.3), (2.5) and (2.6), we havesup t ≥ sup ϕ ∈ B k S ( t ) ϕ k H ( R n ) × L ( R n ) < ∞ . (3.19)Since { ϕ k } ∞ k =1 is bounded in H ( R n ) × L ( R n ), by (3.19), the sequence { S ( . ) ϕ k } ∞ k =1 is boundedin C b (cid:0) , ∞ ; H ( R n ) × L ( R n ) (cid:1) , where C b (cid:0) , ∞ ; H ( R n ) × L ( R n ) (cid:1) is the space of continuously XISTENCE OF THE GLOBAL ATTRACTOR 11 bounded functions from [0 , ∞ ) to H ( R n ) × L ( R n ). Then for any T ≥ { k m } ∞ m =1 such that t k m ≥ T , and v m → v weakly star in L ∞ (cid:0) , ∞ ; H ( R n ) (cid:1) , v mt → v t weakly star in L ∞ (cid:0) , ∞ ; L ( R n ) (cid:1) , k v m ( t ) k pL p ( R n ) → q ( t ) weakly star in W , ∞ (0 , ∞ ) , v m ( t ) → v ( t ) weakly in H ( R n ) , ∀ t ≥
0, (3.20)for some q ∈ W , ∞ (0 , ∞ ) and v ∈ L ∞ (cid:0) , ∞ ; H ( R n ) (cid:1) ∩ W , ∞ (cid:0) , ∞ ; L ( R n ) (cid:1) , where ( v m ( t ) , v mt ( t )) = S ( t + t k m − T ) ϕ k m . By (2.1) , we also have v mtt ( t, x ) − v ltt ( t, x ) + ∆ ( v m ( t, x ) − v l ( t, x )) + α ( x ) ( v mt ( t, x ) − v lt ( t, x )) + λ ( v m ( t, x ) − v l ( t, x ))= f ( k v l ( t ) k L p ( R n ) )) | v l ( t, x ) | p − v l ( t, x ) − f ( k v m ( t ) k L p ( R n ) ) | v m ( t, x ) | p − v m ( t, x ) . (3.21)We obtain (3.18) by means of the sequential limit estimate of the energy of v m − v l which isproved in the following three steps. In the first step, we get the tail estimates, by using the effectof the damping term. In the second step, we obtain the interior estimates. Finally, in the last step,we get the sequential limit estimate of the energy in R n , by considering the results obtained in theprevious steps. Note that we establish these estimates for the smooth solutions of (2.1)-(2.2) withthe initial data in H ( R n ) × H ( R n ) , for which the estimates in the following text are justified.These estimates can be extended to the weak solutions with the initial data in H ( R n ) × L ( R n )by the standard density arguments. Step 1 (Tail estimates):
Taking into account (2.3), (2.4), (2.5) and (2.6) we get T Z k v mt ( t ) k L ( R n \ B (0 ,r )) dt ≤ c , ∀ T ≥
0. (3.22)Now, putting v m instead of v in (2.1), we have v mtt ( t, x ) + ∆ v m ( t, x ) + α ( x ) v mt ( t, x ) + λv m ( t, x )+ f ( k v m ( t ) k L p ( R n ) ) | v m ( t, x ) | p − v m ( t, x ) = h ( x ) .Let η ∈ C ∞ ( R n ), 0 ≤ η ( x ) ≤ η ( x ) = (cid:26) , | x | ≤ , | x | ≥ η r ( x ) = η (cid:0) xr (cid:1) . Multiplying aboveequation by η r v m and integrating over (0 , T ) × R n , we get T Z (cid:16) k η r ∆ v m ( t ) k L ( R n ) + λ k η r v m ( t ) k L ( R n ) (cid:17) dt = T Z k η r v mt ( t ) k L ( R n ) dt − Z R n η r ( x ) v mt ( t, x ) v m ( t, x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T − r n X i =1 T Z η r ( x ) η x i (cid:16) xr (cid:17) ∆ v m ( t, x ) v mx i ( t, x ) dxdt − T Z Z R n ∆ (cid:0) η r ( x ) (cid:1) ∆ v m ( t, x ) v m ( t, x ) dxdt − Z R n η r ( x ) α ( x ) ( v m ( t, x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T − T Z f ( k v m ( t ) k L p ( R n ) ) Z R n | v m ( t, x ) | p η r ( x ) dxdt + T Z Z R n h ( x ) η r ( x ) v m ( t, x ) dxdt. Taking into account (2.3), (2.5), (3.19) and (3.22) we obtain T Z (cid:16) k ∆ ( v m ( t )) k L ( R n \ B (0 , r )) + λ k v m ( t ) k L ( R n \ B (0 , r )) (cid:17) dt ≤ c (cid:18) Tr + T k h k L ( R n \ B (0 ,r )) (cid:19) , ∀ T ≥ ∀ r ≥ r . (3.23) Step 2 (Interior estimates):
Multiplying (3.21) by P ni =1 x i (1 − η r ) ( v m − v l ) x i + ( n −
1) (1 − η r ) ( v m − v l ), and integrating over (0 , T ) × R n , we find32 T Z k ∆ ( v m ( t ) − v l ( t )) k L ( B (0 , r )) dt + 12 T Z k v mt ( t ) − v lt ( t ) k L ( B (0 , r )) dt ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ni =1 Z B (0 , r ) (1 − η r ( x )) x i ( v m ( T, x ) − v l ( T, x )) x i ( v mt ( T, x ) − v lt ( T, x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ni =1 Z B (0 , r ) (1 − η r ( x )) x i ( v m (0 , x ) − v l (0 , x )) x i ( v mt (0 , x ) − v lt (0 , x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 12 ( n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B (0 , r ) (1 − η r ( x )) ( v mt ( T, x ) − v lt ( T, x )) ( v m ( T, x ) − v l ( T, x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 12 ( n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B (0 , r ) (1 − η r ( x )) ( v mt (0 , x ) − v lt (0 , x )) ( v m (0 , x ) − v l (0 , x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 14 r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ni =1 T Z Z B (0 , r ) \ B (0 , r ) η x i (cid:16) x r (cid:17) x i ( v mt ( t, x ) − v lt ( t, x )) dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 14 r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ni =1 T Z Z B (0 , r ) \ B (0 , r ) η x i (cid:16) x r (cid:17) x i (∆ v m ( t, x ) − ∆ v l ( t, x )) dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ni =1 T Z Z B (0 , r ) ∆ ((1 − η r ( x )) x i ) ( v m ( t, x ) − v l ( t, x )) x i ∆ ( v m ( t, x ) − v l ( t, x )) dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 1 r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ni,j =1 T Z Z B (0 , r ) \ B (0 , r ) η x j (cid:16) x r (cid:17) x i ( v m ( t, x ) − v l ( t, x )) x i x j ∆ ( v m ( t, x ) − v l ( t, x )) dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 12 ( n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z Z B (0 , r ) \ B (0 , r ) ∆ ((1 − η r ( x ))) ( v m ( t, x ) − v l ( t, x )) ∆ ( v m ( t, x ) − v l ( t, x )) dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 12 r ( n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ni =1 T Z Z B (0 , r ) \ B (0 , r ) η x i (cid:16) x r (cid:17) ( v m ( t, x ) − v l ( t, x )) x i ∆ ( v m ( t, x ) − v l ( t, x )) dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) XISTENCE OF THE GLOBAL ATTRACTOR 13 + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ni =1 T Z Z B (0 , r ) (1 − η r ( x )) x i ( v m ( t, x ) − v l ( t, x )) x i a ( x ) ( v mt ( t, x ) − v lt ( t, x )) dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 12 ( n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z Z B (0 , r ) (1 − η r ( x )) ( v m ( t, x ) − v l ( t, x )) a ( x ) ( v mt ( t, x ) − v lt ( t, x )) dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ni =1 T Z Z B (0 , r ) (1 − η r ( x )) x i ( v m ( t, x ) − v l ( t, x )) x i ( v m ( t, x ) − v l ( t, x )) dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ni =1 T Z Z B (0 , r ) (1 − η r ( x )) x i ( v m ( t, x ) − v l ( t, x )) x i × (cid:16) f ( k v m ( t ) k L p ( R n ) ) | v m ( t, x ) | p − v m ( t, x ) − f ( k v l ( t ) k L p ( R n ) ) | v l ( t, x ) | p − v l ( t, x ) (cid:17) dxdt (cid:12)(cid:12)(cid:12) + 12 ( n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z Z B (0 , r ) (1 − η r ( x )) ( v m ( t, x ) − v l ( t, x )) × (cid:16) f ( k v m ( t ) k L p ( R n ) ) | v m ( t, x ) | p − v m ( t, x ) − f ( k v l ( t ) k L p ( R n ) ) | v l ( t, x ) | p − v l ( t, x ) (cid:17) dxdt (cid:12)(cid:12)(cid:12) ≤ c r (cid:16) k∇ v m ( T ) − ∇ v l ( T ) k L ( B (0 , r )) + k∇ v m (0) − ∇ v l (0) k L ( B (0 , r )) (cid:17) c k v mt − v lt k L (0 ,T ; L ( B (0 , r ) \ B (0 , r ))) + c k v m − v l k L (0 ,T ; H ( B (0 , r ) \ B (0 , r ))) + c r √ T k∇ v m − ∇ v l k L ((0 ,T ) × B (0 , r )) , (3.24)since, by (2.5) and (3.19), (cid:13)(cid:13)(cid:13) f ( k v m ( t ) k L p ( R n ) ) | v m ( t ) | p − v m ( t ) − f ( k v l ( t ) k L p ( R n ) ) | v l ( t ) | p − v l ( t ) (cid:13)(cid:13)(cid:13) L ( B (0 , r )) ≤ e c .Since the sequence { v m } ∞ m =1 is bounded in C (cid:0) [0 , T ] ; H ( R n ) (cid:1) and the sequence { v mt } ∞ m =1 is boundedin C (cid:0) [0 , T ] ; L ( R n ) (cid:1) , by the generalized Arzela-Ascoli theorem, the sequence { v m } ∞ m =1 is relativelycompact in C (cid:0) [0 , T ] ; H ( B (0 , r )) (cid:1) for every r >
0. So, according to (3.20) -(3.20) , the sequence { v m } ∞ m =1 strongly converges to v in C (cid:0) [0 , T ] ; H ( B (0 , r )) (cid:1) . Then, by using (3.22) and (3.23) in(3.24) , we getlim sup m →∞ lim sup l →∞ T Z h k ∆ ( v m ( t ) − v l ( t )) k L ( B (0 , r )) + k v mt ( t ) − v lt ( t ) k L ( B (0 , r )) i dt ≤ c (cid:18) Tr + T k h k L ( R n \ B (0 ,r )) (cid:19) , ∀ T ≥ ∀ r ≥ r . Step 3 (Estimates in R n ): By using (3.22), (3.23) and the last estimate of the previous step, weobtain lim sup m →∞ lim sup l →∞ T Z h k v m ( t ) − v l ( t ) k H ( R n ) + k v mt ( t ) − v lt ( t ) k L ( R n ) i dt ≤ c (cid:18) Tr + T k h k L ( R n \ B (0 ,r )) (cid:19) , ∀ T ≥ ∀ r ≥ r . Passing to limit as r → ∞ in the last inequality, we getlim sup m →∞ lim sup l →∞ T Z h k v m ( t ) − v l ( t ) k H ( R n ) + k v mt ( t ) − v lt ( t ) k L ( R n ) i dt ≤ c , ∀ T ≥ . (3.25)Multiplying (3.21) by 2 t ( v mt − v lt ), integrating over (0 , T ) × R n , using integration by parts andconsidering (2.4), we find T k ∆ ( v m ( T ) − v l ( T )) k L ( R n ) + T k v mt ( T ) − v lt ( T ) k L ( R n ) + T λ k v m ( T ) − v l ( T ) k L ( R n ) +2 α T Z Z R n \ B (0 ,r ) t ( v mt ( t ) − v lt ( t )) dxdt ≤ T Z k v mt ( t ) − v lt ( t ) k L ( R n ) dt + T Z k ∆ ( v m ( t ) − v l ( t )) k L ( R n ) dt + λ T Z k v m ( t ) − v l ( t ) k L ( R n ) dt +2 T Z Z R n t (cid:16) f ( k v l ( t ) k L p ( R n ) ) | v l ( t, x ) | p − v l ( t, x ) − f ( k v m ( t ) k L p ( R n ) ) | v m ( t, x ) | p − v m ( t, x ) (cid:17) × ( v mt ( t, x ) − v lt ( t, x )) dxdt . (3.26)Multiplying (3.21) by tη r ( v m − v l ), integrating over (0 , T ) × R n and using integration by parts, weget T Z R n ( v mt ( T, x ) − v lt ( T, x )) η r ( x ) ( v m ( T, x ) − v l ( T, x )) dx − T Z Z R n tη r ( x ) ( v mt ( t, x ) − v lt ( t, x )) dxdt − T Z Z R n η r ( x ) ( v mt ( t, x ) − v lt ( t, x )) ( v m ( t, x ) − v l ( t, x )) dxdt + T Z Z R n t (∆ ( v m ( t, x ) − v l ( t, x ))) η r ( x ) dxdt +2 X ni =1 T Z Z R n (∆ ( v m ( t, x ) − v l ( t, x ))) t ( η r ( x )) x i ( v m ( t, x ) − v l ( t, x )) x i dxdt + T Z Z R n (∆ ( v m ( t, x ) − v l ( t, x ))) t ∆ ( η r ( x )) ( v m ( t, x ) − v l ( t, x )) dxdt + T Z R n α ( x ) ( v m ( T, x ) − v l ( T, x )) η r ( x ) dx − T Z Z R n α ( x ) η r ( x ) ( v m ( t, x ) − v l ( t, x )) dxdt + λ T Z Z R n t ( v m ( t, x ) − v l ( t, x )) η r ( x ) dxdt + T Z Z R n t (cid:16) f ( k v m ( t ) k L p ( R n ) ) | v m ( t, x ) | p − v m ( t, x ) − f ( k v l ( t ) k L p ( R n ) ) | v l ( t, x ) | p − v l ( t, x ) (cid:17) × η r ( x ) ( v m ( t, x ) − v l ( t, x )) dxdt = 0 . XISTENCE OF THE GLOBAL ATTRACTOR 15
Then, considering (2.3), we obtain T Z Z R n t (∆ ( v m ( t, x ) − v l ( t, x ))) η r ( x ) dxdt + λ T Z Z R n t ( v m ( t, x ) − v l ( t, x )) η r ( x ) dxdt ≤ − T Z R n ( v mt ( T, x ) − v lt ( T, x )) η r ( x ) ( v m ( T, x ) − v l ( T, x )) dx + T Z Z R n tη r ( x ) ( v mt ( t, x ) − v lt ( t, x )) dxdt + T Z Z R n η r ( x ) ( v mt ( t, x ) − v lt ( t, x )) ( v m ( t, x ) − v l ( t, x )) dxdt − X ni =1 T Z Z R n (∆ ( v m ( t, x ) − v l ( t, x ))) t ( η r ( x )) x i ( v m ( t, x ) − v l ( t, x )) x i dxdt − T Z Z R n (∆ ( v m ( t, x ) − v l ( t, x ))) t ∆ ( η r ( x )) ( v m ( t, x ) − v l ( t, x )) dxdt + 12 T Z Z R n α ( x ) η r ( x ) ( v m ( t, x ) − v l ( t, x )) dxdt − T Z Z R n t (cid:16) f ( k v m ( t ) k L p ( R n ) ) | v m ( t, x ) | p − v m ( t, x ) − f ( k v l ( t ) k L p ( R n ) ) | v l ( t, x ) | p − v l ( t, x ) (cid:17) × η r ( x ) ( v m ( t, x ) − v l ( t, x )) dxdt, ∀ T ≥ ∀ r ≥ r .Taking into account (2.5) and (3.19) in the above inequality, we find T Z Z R n t (∆ ( v m ( t, x ) − v l ( t, x ))) η r ( x ) dxdt + λ T Z Z R n t ( v m ( t, x ) − v l ( t, x )) η r ( x ) dxdt ≤ T (cid:16) k v mt ( T, x ) − v lt ( T, x ) k L ( R n ) + k v m ( T, x ) − v l ( T, x ) k L ( R n ) (cid:17) + T Z Z R n tη r ( x ) ( v mt ( t, x ) − v lt ( t, x )) dxdt + T Z Z R n η r ( x ) ( v mt ( t, x ) − v lt ( t, x )) dxdt + T Z Z R n η r ( x ) ( v m ( t, x ) − v l ( t, x )) dxdt + 12 T Z Z R n α ( x ) η r ( x ) ( v m ( t, x ) − v l ( t, x )) dxdt + c Tr + f K rm,l ( T ) , ∀ T ≥ ∀ r ≥ r , (3.27)where f K rm,l ( T ) := T Z t (cid:16) f ( k v l ( t ) k L p ( R n ) − f ( k v m ( t ) k L p ( R n ) (cid:17) Z R n | v l ( t, x ) | p − v l ( t, x ) η r ( x ) × ( v m ( t, x ) − v l ( t, x )) dxdt , and considering (3.19) -(3.20) , it is easy to see thatsup m,l (cid:13)(cid:13)(cid:13) f K rm,l (cid:13)(cid:13)(cid:13) C [0 ,T ] < ∞ and lim m →∞ lim sup l →∞ (cid:12)(cid:12)(cid:12) f K rm,l ( T ) (cid:12)(cid:12)(cid:12) = 0 , ∀ T ≥ . Now, multiplying (3.27) by δ > T k ∆ ( v m ( T ) − v l ( T )) k L ( R n ) + T k v mt ( T ) − v lt ( T ) k L ( R n ) + T λ k v m ( T ) − v l ( T ) k L ( R n ) +2 α T Z Z R n \ B (0 ,r ) t ( v mt ( t ) − v lt ( t )) dxdt + δ T Z Z R n t (∆ ( v m ( t, x ) − v l ( t, x ))) η r ( x ) dxdt + δλ T Z Z R n t ( v m ( t, x ) − v l ( t, x )) η r ( x ) dxdt ≤ T Z k v mt ( T ) − v lt ( T ) k L ( R n ) dt + T Z k ∆ ( v m ( t ) − v l ( t )) k L ( R n ) dt + λ T Z k v m ( t ) − v l ( t ) k L ( R n ) dt +2 T Z Z R n t (cid:16) f ( k v l ( t ) k L p ( R n ) ) | v l ( t, x ) | p − v l ( t, x ) − f ( k v m ( t ) k L p ( R n ) ) | v m ( t, x ) | p − v m ( t, x ) (cid:17) × ( v mt ( t, x ) − v lt ( t, x )) dxdt + δT (cid:16) k v mt ( T, x ) − v lt ( T, x ) k L ( R n ) + k v m ( T, x ) − v l ( T, x ) k L ( R n ) (cid:17) + δ T Z Z R n tη r ( x ) ( v mt ( t, x ) − v lt ( t, x )) dxdt + δ T Z Z R n η r ( x ) ( v mt ( t, x ) − v lt ( t, x )) dxdt + δ T Z Z R n η r ( x ) ( v m ( t, x ) − v l ( t, x )) dxdt + δ T Z Z R n α ( x ) η r ( x ) ( v m ( t, x ) − v l ( t, x )) dxdt + c δ Tr + δ f K rm,l ( T ) , ∀ T ≥ ∀ r ≥ r . (3.28)Considering Lemma 3.2 in (3.28), for every γ >
0, we get T k ∆ ( v m ( T ) − v l ( T )) k L ( R n ) + T k v mt ( T ) − v lt ( T ) k L ( R n ) + T λ k v m ( T ) − v l ( T ) k L ( R n ) +2 α T Z Z R n \ B (0 ,r ) t ( v mt ( t ) − v lt ( t )) dxdt + δ T Z Z R n t (∆ ( v m ( t, x ) − v l ( t, x ))) η r ( x ) dxdt + δλ T Z Z R n t ( v m ( t, x ) − v l ( t, x )) η r ( x ) dxdt ≤ T Z k v mt ( t ) − v lt ( t ) k L ( R n ) dt + T Z k ∆ ( v m ( t ) − v l ( t )) k L ( R n ) dt + λ T Z k v m ( t ) − v l ( t ) k L ( R n ) dt + γ T Z τ E R n \ ( B (0 , r )) ( v m ( t ) − v l ( t )) dt + c γ T Z E R n \ ( B (0 , r )) ( v m ( t ) − v l ( t )) dt + c γ T Z t (cid:16)(cid:13)(cid:13) √ av mt ( t ) (cid:13)(cid:13) L ( R n ) + (cid:13)(cid:13) √ av lt ( t ) (cid:13)(cid:13) L ( R n ) (cid:17) E R n \ B (0 , r ) ( v m ( t ) − v l ( t )) dt + (cid:12)(cid:12) K m,lr ( T ) (cid:12)(cid:12) XISTENCE OF THE GLOBAL ATTRACTOR 17 + δT (cid:16) k v mt ( T, x ) − v lt ( T, x ) k L ( R n ) + k v m ( T, x ) − v l ( T, x ) k L ( R n ) (cid:17) + δ T Z Z R n tη r ( x ) ( v mt ( t, x ) − v lt ( t, x )) dxdt + δ T Z Z R n η r ( x ) ( v mt ( t, x ) − v lt ( t, x )) dxdt + δ T Z Z R n η r ( x ) ( v m ( t, x ) − v l ( t, x )) dxdt + δ T Z Z R n α ( x ) η r ( x ) ( v m ( t, x ) − v l ( t, x )) dxdt + c δ Tr + δ f K rm,l ( T ) , ∀ T ≥ ∀ r ≥ r . Then, for sufficiently small γ and δ, we obtain T E R n ( v m ( T ) − v l ( T )) ≤ c T Z E R n ( v m ( t ) − v l ( t )) dt + c γ T Z t (cid:16)(cid:13)(cid:13) √ av mt ( t ) (cid:13)(cid:13) L ( R n ) + (cid:13)(cid:13) √ av lt ( t ) (cid:13)(cid:13) L ( R n ) (cid:17) E R n ( v m ( t ) − v l ( t )) dt + c (cid:18) Tr + (cid:12)(cid:12) K m,lr ( T ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f K rm,l ( T ) (cid:12)(cid:12)(cid:12)(cid:19) , ∀ T ≥ ∀ r ≥ r .Now, denoting y m,l ( t ) := tE R n ( v m ( t ) − v l ( t )) , from the previous inequality, we have y m,l ( T ) ≤ c γ T Z (cid:16)(cid:13)(cid:13) √ av mt ( t ) (cid:13)(cid:13) L ( R n ) + (cid:13)(cid:13) √ av lt ( t ) (cid:13)(cid:13) L ( R n ) (cid:17) y m,l ( t ) dt + c T Z E R n ( v m ( t ) − v l ( t )) dt + c (cid:18) Tr + (cid:12)(cid:12) K m,lr ( T ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f K rm,l ( T ) (cid:12)(cid:12)(cid:12)(cid:19) , ∀ T ≥ ∀ r ≥ r .Applying Gronwall inequality and considering (2.6) and (3.19) in the above inequality, we get T E R n ( v m ( T ) − v l ( T )) ≤ c T Z E R n ( v m ( t ) − v l ( t )) dt + c (cid:18) Tr + (cid:12)(cid:12) K m,lr ( T ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f K rm,l ( T ) (cid:12)(cid:12)(cid:12)(cid:19) + c T Z t Z E R n ( v m ( s ) − v l ( s )) ds + tr + (cid:12)(cid:12) K m,lr ( t ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f K rm,l ( t ) (cid:12)(cid:12)(cid:12) × (cid:16)(cid:13)(cid:13) √ av mt ( t ) (cid:13)(cid:13) L ( R n ) + (cid:13)(cid:13) √ av lt ( t ) (cid:13)(cid:13) L ( R n ) (cid:17) e c γ R Tt (cid:16) k √ av mt ( τ ) k L R n ) + k √ av lt ( τ ) k L R n ) (cid:17) dτ ds ≤ c T Z E R n ( v m ( t ) − v l ( t )) dt + c (cid:18) Tr + (cid:12)(cid:12) K m,lr ( T ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f K rm,l ( T ) (cid:12)(cid:12)(cid:12)(cid:19) + c T Z E R n ( v m ( t ) − v l ( t )) dt T Z (cid:16)(cid:13)(cid:13) √ av mt ( t ) (cid:13)(cid:13) L ( R n ) + (cid:13)(cid:13) √ av lt ( t ) (cid:13)(cid:13) L ( R n ) (cid:17) + c Tr T Z (cid:16)(cid:13)(cid:13) √ av mt ( t ) (cid:13)(cid:13) L ( R n ) + (cid:13)(cid:13) √ av lt ( t ) (cid:13)(cid:13) L ( R n ) (cid:17) dt + c T Z (cid:16)(cid:12)(cid:12) K m,lr ( t ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f K rm,l ( t ) (cid:12)(cid:12)(cid:12)(cid:17) (cid:16)(cid:13)(cid:13) √ av mt ( t ) (cid:13)(cid:13) L ( R n ) + (cid:13)(cid:13) √ av lt ( t ) (cid:13)(cid:13) L ( R n ) (cid:17) dt ≤ c (cid:16)(cid:12)(cid:12) K m,lr ( T ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f K rm,l ( T ) (cid:12)(cid:12)(cid:12)(cid:17) + c T Z E R n ( v m ( t ) − v l ( t )) dt + c Tr + c T Z (cid:16)(cid:12)(cid:12) K m,lr ( t ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f K rm,l ( t ) (cid:12)(cid:12)(cid:12)(cid:17) dt , ∀ T ≥ ∀ r ≥ r .By using Lebesgue dominated convergence theorem and considering (3.25) in the last inequality, weobtain lim sup m →∞ lim sup l →∞ T E R n ( v m ( T ) − v l ( T )) ≤ c (1 + Tr ), ∀ T ≥ ∀ r ≥ r .By passing to limit as r → ∞ in the above inequality, we findlim sup m →∞ lim sup l →∞ T E R n ( v m ( T ) − v l ( T )) ≤ c , ∀ T ≥ , which giveslim sup m →∞ lim sup l →∞ k S ( T + t k m − T ) ϕ k m − S ( t + t k l − T ) ϕ k l k H ( R n ) × L ( R n ) ≤ c √ T , ∀ T > . Choosing T = T in the previous inequality, we havelim sup m →∞ lim sup l →∞ k S ( t k m ) ϕ k m − S ( t k l ) ϕ k l k H ( R n ) × L ( R n ) ≤ c √ T , ∀ T > . As a consequence, from the above sequential limit inequality, we get (3.18) which completes theproof. (cid:3)
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