Tập hút toàn cục của phương trình khuếch tán không cổ điển với điều kiện tăng trưởng kiểu mũ

NGÀNH TOÁN HỌC Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 63 GLOBAL ATTRACTOR FOR NONCLASSICAL DIFFUSION EQUATION WITH EXPONENTIAL NONLINEARITY TẬP HÚT TOÀN CỤC CỦA PHƯƠNG TRÌNH KHUẾCH TÁN KHÔNG CỔ ĐIỂN VỚI ĐIỀU KIỆN TĂNG TRƯỞNG KIỂU MŨ Nguyen Viet Tuan1, Nguyen Thi Hue1, Nguyen Thi Hong1, Nguyen Xuan Tu2 Email: [email protected] 1Sao Do University 2Hung Vuong University Date received: 30/10/2017 Date received after review: 20/12/2017 Date accept: 28/12/2017 Abstract In this paper, we study the existence and long-time behavior of weak solutions to a nonclasscial diffusion equation with exponential nonlinearity. We prove the existence of a global attractor of the dynamical system associated to the equation. The main novelty of the results obtained is that no restriction on the upper growth of the nonlinearity is imposed. Keywords: Nonclasscial diffusion equation; global attractor; exponential nonlinearity. Tóm tắt Trong bài báo này, chúng tôi nghiên cứu sự tồn tại và dáng điệu tiệm cận của nghiệm yếu phương trình khuếch tán không cổ điển với điều kiện hàm phi tuyến tăng trưởng và tiêu hao kiểu mũ. Chúng tôi chứng minh sự tồn tại của tập hút toàn cục của hệ động lực sinh bởi phương trình. Tính mới lạ của kết quả thu được là hàm phi tuyến không bị giới hạn về tốc độ tăng trưởng. Từ khóa: Phương trình khuếch tán không cổ điển; tập hút toàn cục; tăng trưởng kiểu mũ. 1. INTRODUCTION In this paper, we study the existence and long-time behavior of solutions to the following nonclasscial diffusion equation 0 ( ) ( ), , 0, ( , ) 0, , 0, ( ,0) ( ), , − D −D + = ∈Ω >  = ∈∂Ω >  = ∈Ω t tu u u f u g x x t u x t x t u x u x x (1) where Ω is a bounded domain in N with smooth boundary ∂Ω . This equation arises as a model to describe physical phenomena, such as non- Newtonian flows, soil mechanics and heat conduction theory (see [1]). In the past years, the existence and long-time behavior of solutions to nonclassical diffusion equations has been studied extensively, for both autonomous case [5, 6] and non-autonomous case [2, 3, 4], and even in the case with finite delay. To study problem (1), we assume the following assumptions: (H1) : → f f: is a continuously differentiable function satisfying ( ) ,′ ≥ −f u 2 0( ) , for all ,≥ − − ∈f u u u C ub where 0, C are two positive constants, 10 < <b l with 1l is the first eigenvalue of the operator −D in Ω with the homogeneous Dirichlet boundary condition; (H2) The external force 1( ).−∈ Ωg H Now, we introduce some notations. Unless otherwise specified, it is understood that we consider spaces of functions acting on the domain Ω . Let ,⋅ ⋅ and ⋅‖‖ denote the 2 −L inner product and 2 −L norm, respectively. We will also consider, with standard notation, spaces of functions defined on an interval I with values in Banach space X such as ( , ), ( , )pC I X L I X and , ( , )m pH I X , with the usual norms. The paper is organized as follows. In Section 2, we prove the existence and uniqueness of weak solutions to problem (1) in the space 2 ( )ΩL by utilizing the compactness method and weak 64 NGHIÊN CỨU KHOA HỌC Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 convergence techniques in Orlicz spaces. The existence of a global attractor for the continuous semigroup associated to the problem is studied in the last section. 2. EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS Definition 1. A function ( )u t is called a weak solution of problem (1) on the interval [0, ]T , where 0>T , with initial datum 10 0(0) ( )∈= Ωu u H if 1 1 2 1 0 0([0, ]; ( )), ( ) ( ), (0, ; ( )),∈ Ω ∈ ∈ ΩT tu C T H f u L Q u L T H and 1 1 1 0 , , , , , ( ), , , t t L L H H u u u f u g ϕ ϕ ϕ ϕ ϕ ∞ − 〈 〉 + 〈∇ ∇ 〉 + 〈∇ ∇ 〉 + 〈 〉 = 〈 〉 for all test functions 10 ( ) ( ), ∞∈ = Ω ∩ ΩW H Lϕ and a.e. [0, ]∈t T . We now prove the existence and uniqueness result for problem (1). Theorem 1. Assume that conditions (H1), (H2) hold. Then for any 10 0 ( )Ω∈u H and any 0>T given, there exists a unique weak solution u to problem (1) on the interval [0, ]T . Furthermore, 1 0([0, ]; ( )),∈ Ωu C T H and the mapping 1 1 0 0 0( ) ( ( ), ( )) [0, ],∈ Ω Ω ∀ ∈u u t C H H t T that is, the weak solutions depend continuously on the initial data. Proof. We use the Feado-Galerkin method. We recall that there exists a smooth orthonormal basis 1{ } ∞ =j jw of 2 ( )ΩL which is also orthogonal in 10 ( )ΩH , consisting of normalized eigenfunctions for −D in 10 ( )ΩH . Step 1 (Feado-Galerkin scheme). Given an integer n , denote by the projection on the subspace 11 0span( , , ) ( ) ⊂ Ωn Hw w . We look for a function nu of the form 1 ( ) ( ) = = ∑ n n j j j u t a t w satisfying 1 0 ( ) 0 0 , , ( ), ( , ), 1 , | , t n t n k H n k n k k n t n u u u f u g k n u P u w w w w Ω = 〈∂ − D∂ 〉 = 〈D 〉 − 〈 〉 + ≤ ≤ = for a.e. ≤t T . We get a system of ODEs in the variables ( )ka t of the form (1 ) , ( ),( )+ = − + −k k k k k n kd a a g f udt ν ν w w (2) subject to the initial condition 1 00 ( ) (0) , . Ω =k k Ha u w According to standard existence theory for ODEs, there exists a solution on some interval (0, )nT . The a priori estimates below imply that in fact = +∞nT . Step 2 (Energy estimates). Multiplying the equation (2) by ka , then summing over k and adding the results, we get (3) Using (H1) we deduce that So where 0>ε is small enough so that 1 1 0.− − >b ε l Integrating on (0, ), (0, )∈t t T , leads to the following estimate In particular, we see that { }nu is bounded in 1 0(0, ; ( )) ∞ ΩL T H . Using the boundedness of { }nu in 1 0(0, ; ( )) ∞ ΩL T H , it is easy to check that { }D nu is bounded in 2 1(0, ; ( ))− ΩL T H . Therefore, up to passing to a subsequence, there exists a function u such that 1 0 2 1 weakly-star in (0, ; ( )), weakly in (0, ; ( )). ∞ − Ω D D Ω   n n u u L T H u u L T H Step 3 (Passage to limits). From (3), we get (4) Integrating (4) from 0 to T , we have Hence 0 ( ) . Ω ≤∫ ∫ T n nf u u dxdt C NGÀNH TOÁN HỌC Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 65 We now prove that { ( )}nf u is bounded in 1( )TL Q . Putting ( ) ( ) (0)= − +h s f s f sγ , where > γ . Note that 2 2 2 2 ( ) ( ( ) (0)) ( ) ( ) 0 h s s f s f s s f s s s γ ξ γ γ = − + ′= + ≥ − ≥ for all ∈s , we have 1 2 0 {| | 1} {| | 1} | | 1 ( ) 2 ( ) | | 1 | ( ) | | ( ) | | ( ) | ( ) sup | ( ) | || . | | ( ) | (0) |. || sup | ( ) | . | | . T n T n T TT T T n n n nQ u Q u n n TQ s n n n L QQ n TL Q s h u dxdt h u u dxdt h u dxdt h u u dxdt h s Q f u u dxdt f u u h s Q C γ Ω ∩ > ∩ ≤ ≤ ≤ ≤ + ≤ + ≤ + + + ≤ ∫ ∫ ∫ ∫ ∫ ∫ ＼ ＼ Hence it implies that { ( )}nh u , and therefore { ( )}nf u is bounded in 1( )TL Q . Now, we prove the boundedness of { }∂t nu . In the first equation in (2), replacing kw by ∂t nu , and then using the Cauchy inequality, we get Hence, by choosing ε small enough, we arrive at Integrating from 0 to t , we can deduce that { }∂t nu is bounded in 2 1 0(0, ; ( ))ΩL T H . So, up to a subsequence, 2 1 0 weakly in (0, ; ( ))∂ Ωt n tu u L T H , 2 1 weakly in (0, ; ( )).−D∂ D Ωt n tu u L T H Because 1 2 1 10 ( ) ( ) ( ) ( ) −Ω ⊂⊂ Ω ⊂ Ω + ΩH L H L , by the Aubin-Lions-Simon compactness lemma, we obtain →nu u strongly in 2 2(0, ; ( ))ΩL T L . Hence we may assume, up to a subsequence, that →nu u a.e. in TQ . Since f is continuous, it follows that ( ) ( )→nf u f u a.e. in [0, ]Ω× T . We obtain that 1( ) ( )∈ Th u L Q and for all test functions 1 0 0([0, ]; ( ) ( )) ∞ ∞∈ Ω ∩ ΩC T H Lφ , ( ) ( ) .→∫ ∫ T T nQ Q h u dxdt h u dxdtφ φ Hence 1( ) ( )∈ Tf u L Q and ( ) ( ) , T T nQ Q f u dxdt f u dxdtφ φ→∫ ∫ for all 10 0 ([0, ]; ( ) ( )) ∞ ∞∈ Ω ∩ ΩC T H Lφ . By standard arguments, we can check that u satisfies the initial condition 0(0) =u u and this implies that u is a weak solution of problem (1). Step 4 (Uniqueness and continuous dependence of the solutions). We assume that 1u and 2u are two solutions of (1) with initial data 10u and 20u , respectively. Denote 1 2= −u u u , then it satisfies ( )1 2ˆ ˆ( ( ) ( )) 0, 0, 5t tu u u f u f u u t∂ − D∂ − D + − − = ∀ > where ˆ ( ) ( ) .= + f s f s s Here because ( )u t does not belong to 10: ( ) ( ) ∞= Ω ∩ ΩW H L , we cannot choose ( )u t as a test function. Let if , ( ) if | | , if . > = ≤ − < − k k s k B s s s k k s k Consider the corresponding Nemytskii mapping ˆ : →kB W W defined as follows ˆ ( )( ) ( ( )),=k kB u x B u x for all ∈Ωx . We notice that ˆ ( ) 0− →k WB u u‖ ‖ as →∞k . Now multiplying (5) by ˆ ( )kB u , then integrating over Ω we get Thus Note that ˆ ( ) 0′ ≥f s and ( ) 0≥ksB s for all ∈s , we get ˆ ˆ( ) ( ) 0. Ω ′ ≥∫ kf uB u dxξ Therefore 2 { :| ( , )| } | | 0. ∈Ω ≤ ∇ ≥∫ x u x t k u dx Since the above inequalities, we get Integrating from 0 to t , where (0, )∈t T , then letting →∞k , we obtain 66 NGHIÊN CỨU KHOA HỌC Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 By the Gronwall inequality of integral form, we get for all [0, ].t T∈ Hence 10([0, ] (; ))∈ Ωu C T H , in particular, we get the uniqueness if ( )0 0.=u 3. EXISTENCE OF A GLOBAL ATTRACTOR Theorem 1 allows us to define a continuous semigroup 1 10 0( ) () : )( Ω Ω→S t H H associated to problem (1) by the formula 0( ) : ( ),=S t u u t where (.)u is the unique weak solution of (1) with the initial datum 10 0 ( )Ω∈u H . The aim of this section is to prove the following result. Theorem 2. Assume that (H1), (H2) hold. Then the semigroup 0{ ( )} ≥tS t possesses a compact global attractor in 10 ( )ΩH . To prove this theorem, by the classical abstract results on existence of global, we need to show that the semigroup ( )S t has a bounded absorbing set 0B in 1 0 ( )ΩH and ( )S t is asymptotically compact in 10 ( )ΩH , that is, for any 0>t , it can be decomposed in the form 1 2( ) ( ) ( ),= +S t S t S t where for any bounded subset B in 10 ( )ΩH , we have i) 1( )S t is a continuous mapping from 1 0 ( )ΩH into itself and 1( ) sup ( ) 0 as ; ∈ = → → +∞B y B r t S t y t‖ ‖ ii) The operators 2 ( )S t are uniformly compact for t large, i.e., 0 2 ( ) ≥  t t S t B is relatively compact in 1 0 ( )ΩH for some 0>t . It is clear that we only need to verify conditions i) and ii) above for the absorbing set 0B . Lemma 1. Assume that (H1), (H2) hold. Then there exists a bounded absorbing set in 10 ( )ΩH for the semigroup ( )S t . Proof. Multiplying the equation (1) by ( )u t , we have By the Cauchy inequality, we get Thus, where 1 1 1 min 1 ; 0.   = − − − − >    b γ ε l b l ε l According to Gronwall Lemma, we obtain Now, we can choose 1T and 0ρ such that 2 0( ) ,∇ ≤u t ρ‖ ‖ for all 1≥t T and for all 0 ∈u B . This completes the proof. Recall that in this paper we only assume the external force 1( )−∈ Ωg H . However, we know that for any 1( )−∈ Ωg H and 0>ε given, there is a 2 ( )∈ Ωg Lε , which depends on g and ε, such that 1 ( ) .− Ω− <Hg g ε ε‖ ‖ To make the asymptotic regular estimates, we decompose the solution 0( ) ( )=S t u u t of problem (1) as follows 0 1 0 2 0( ) ( ) ( ) ,= +S t u S t u S t u where 1 0 1( ) ( )=S t u u t and 02 2( ) ( )=S t u u t , that is the decomposition is of the following form = +u v wε ε , where ( )v tε is the unique solution of the following problem ( ) 0 0 ( ) ( ) , , ( , ) | 0, ( , ) | ( , 6 ) t t t v v v f u f w v g g v x t v x t u x ε ε ε ε ε ε ε ε l l ∂Ω =  − D − D + − +  = − >  = =  (6) and ( )w t is the unique solution of the following problem ( ) 0 ( ) ( ) , 7 ( , ) | 0, ( , ) | 0. t t t w w w f w u w g w x t w x t ε ε ε ε ε ε ε ε l l ∂Ω =  − D − D + − − = >  = =  As in the proof of Theorem 1, one can prove the existence and uniqueness of solutions to (6) and (7). Lemma 2. Let hypotheses (H1), (H2) hold. Then the solutions of (6) satisfy the following estimates: there is a constant 0d depending on 1,l , such that for every 0≥t , 0 1 0 2 1 0 0( ) ( ) ( .− Ω ≤ ∇ +d t H S t u Q u e ε‖ ‖ ‖ ‖) Proof. Multiplying the first equation of (6) by v, then integrating over Ω we get NGÀNH TOÁN HỌC Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 67 Note that ( )′ ≥ −f s and 1 1 1 0 2 2 , ( ) 1 1, 2 2− − Ω 〈 − 〉 ≤ ∇ + − H H H g g v v g gε ε ε ε‖ ‖ ‖ ‖ , we get Similarly to the proof of Lemma 1, we obtain 02 1 0 0( ) ( ) . −≤ ∇ +d tS t u Q u e ε‖ ‖ ‖ ‖ The proof is complete. Lemma 3. Let hypotheses (H1), (H2) hold. Then, there exists a positive constant M such that for any 10 0 ( )∈ Ωu H , there exists 0>T large enough, which depends on 12 0( ) , ,− Ω ∇Hg uε‖ ‖ ‖ ‖, such that 2 2 2 0 ( ) ( ) , for all . Ω ≤ ≥ H S t u M t T‖ ‖ Proof. Multiplying the first equation of (7) by −Dw , we get when 0 ( )≥t t B . Notice that, similarly to the proof of Lemma 1, we obtain a 0>T large enough such that 222 0 ( )( ) , .Ω ≤ ∀ ≥HS t u M t T‖ ‖ The proof is complete. Since the embedding 2 1 10 0( ) ( ) ( )Ω ∩ Ω ⊂⊂ ΩH H H is compact, we obtain Lemma 4. Let 2 0{ ( )} ≥tS t be the solution semigroup of (7). Then for large enough, ( )2 S T B is relatively compact in 10 ( )ΩH . By Lemma 1, the semigroup ( )S t has a bounded absorbing set 0B in 1 0 ( )ΩH . Moreover, the semigroup ( )S t is asymptotically compact in 1 0 ( )ΩH due to Lemmas 2 and 4. Therefore, the w - limit set 0( )= Bw is the compact global attractor for ( )S t in 10 ( )ΩH . REFERENCES [1]. E.C. Aifantis (1980). On the problem of diffusion in solids. Acta Mech. 37, 265÷296. [2]. C.T. Anh and T.Q. Bao (2010). Pullback attractors for a class of non-autonomous nonclassical diffusion equations. Nonlinear Anal. 73, 399÷412. [3]. C.T. Anh and T.Q. Bao (2012). Dynamics of non- autonomous nonclassical diffusion equations on  N . Comm. Pure Appl. 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