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.
NGÀNH TOÁN HỌC GLOBAL ATTRACTOR FOR NONCLASSICAL DIFFUSION EQUATION WITH EXPONENTIAL NONLINEARITY TẬP HÚT TỒN 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: nguyentuandhsd@gmail.com Sao Do University Hung 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 báo này, chúng tơi nghiên cứu tồn dáng điệu tiệm cận nghiệm yếu phương trình khuếch tán khơng cổ điển với điều kiện hàm phi tuyến tăng trưởng tiêu hao kiểu mũ Chúng chứng minh tồn tập hút toàn cục hệ động lực sinh phương trình Tính lạ kết thu hàm phi tuyến không bị giới hạn tốc độ tăng trưởng Từ khóa: Phương trình khuếch tán khơng cổ điển; tập hút toàn cục; tăng trưởng kiểu mũ INTRODUCTION In this paper, we study the existence and long-time behavior of solutions to the following nonclasscial diffusion equation + f (u ) g ( x), x ∈ Ω, t > 0, ut − Dut − Du= = u ( x , t ) 0, x ∈ ∂Ω, t > 0, = x ∈ Ω, u ( x, 0) u0 ( 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 nonNewtonian 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 ) ≥ −, f (u )u ≥ − b u − C0 , for all u ∈ , where , C0 are two positive constants, < b < l1 with l1 is the first eigenvalue of the operator −D in Ω with the homogeneous Dirichlet boundary condition; (H2) The external force g ∈ H −1 (Ω) 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 L2 − inner product and 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 C ( I , X ), Lp ( I , X ) and H m , p ( 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 L2 (Ω) by utilizing the compactness method and weak Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 63 NGHIÊN CỨU KHOA HỌC 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 EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS Definition A function u (t ) is called a weak solution of problem (1) on the interval [0, T ] , where T > , with initial datum = u (0) u0 ∈ H 01 (Ω) if u ∈ C ([0, T ]; H 01 (Ω)), f (u ) ∈ L1 (QT ), ut ∈ L2 (0, T ; H 01 (Ω)), there exists a solution on some interval (0, Tn ) The a priori estimates below imply that in fact Tn = +∞ Step (Energy estimates) Multiplying the equation (2) by ak , then summing over k and adding the results, we get (3) Using (H1) we deduce that and 〈ut , ϕ 〉 + 〈∇ut , ∇ϕ 〉 + 〈∇u , ∇ϕ 〉 + 〈 f (u ), ϕ 〉 L1 , L∞ So = 〈 g , ϕ 〉 H −1 , H , for all test functions ϕ ∈ W= H 01 (Ω) ∩ L∞ (Ω), and a.e t ∈ [0, T ] We now prove the existence and uniqueness result for problem (1) Theorem Assume that conditions (H1), (H2) hold Then for any u0 ∈ H 01 (Ω) and any T > given, there exists a unique weak solution u to problem (1) on the interval [0, T ] Furthermore, u ∈ C ([0, T ]; H 01 (Ω)), and the mapping u0 u (t ) ∈ C ( H 01 (Ω), H 01 (Ω)) ∀t ∈ [0, T ], that is, the weak solutions depend continuously on the initial data where ε > is small enough so that 1− b − ε > l1 Integrating on (0, t ), t ∈ (0, T ) , leads to the following estimate Proof We use the Feado-Galerkin method We recall that there exists a smooth orthonormal basis {w j }∞j =1 of L2 (Ω) which is also orthogonal in H 01 (Ω) , consisting of normalized eigenfunctions for −D in H 01 (Ω) In particular, we see that {un } is bounded in L∞ (0, T ; H 01 (Ω)) Step (Feado-Galerkin scheme) Therefore, up to passing to a subsequence, there exists a function u such that Given an integer n , denote by the projection on the subspace span(w1 , …, wn ) ⊂ H 01 (Ω) We look n for a function un of the form un (t ) = ∑ a j (t )w j j =1 satisfying 〈∂ t un − D∂ t un , wk 〉 H ( Ω ) Using the boundedness of {un } in L∞ (0, T ; H 01 (Ω)) , it is easy to check that {Dun } is bounded in L2 (0, T ; H −1 (Ω)) weakly-star in un u weakly in Dun Du L∞ (0, T ; H 01 (Ω)), L2 (0, T ; H −1 (Ω)) Step (Passage to limits) From (3), we get = 〈Dun , wk 〉 − 〈 f (un ), wk 〉 + ( g , wk ), ≤ k ≤ n, un |t = = Pn u0 , (4) for a.e t ≤ T We get a system of ODEs in the variables ak (t ) of the form d ((1 +ν k )ak ) = −ν k ak + g , wk − f (un ), wk dt Integrating (4) from to T , we have (2) subject to the initial condition ak (0) = u0 , wk H 01 ( Ω ) According to standard existence theory for ODEs, Hence T ∫ ∫ Ω f (un )un dxdt ≤ C 64 Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 NGÀNH TOÁN HỌC We now prove that { f (un )} is bounded in L1 (QT ) Putting h( s ) = f ( s ) − f (0) + γ s , where γ > satisfies the initial condition u (0) = u0 and this implies that u is a weak solution of problem (1) Note that h( s ) s =( f ( s ) − f (0)) s + γ s Step (Uniqueness and continuous dependence of the solutions) = f ′(ξ ) s + γ s ≥ (γ − ) s ≥ We assume that u1 and u2 are two solutions of (1) with initial data u10 and u20 , respectively Denote u= u1 − u2 , then it satisfies for all s ∈ , we have T ∫ ∫ Ω |h(un ) | dxdt ≤∫ QT ∩{|un | >1} |h(un )un | dxdt + ∫ QT ∩{|un | ≤1} ∂ t u − D∂ t u − Du + ( fˆ (u1 ) − fˆ (u2 )) − u= 0, ∀t > 0, ( ) |h(un ) | dxdt where fˆ= ( s ) f ( s ) + s Here because u (t ) does not belong to W=: H 01 (Ω) ∩ L∞ (Ω) , we cannot choose u (t ) as a test function ≤ ∫ h(un )un dxdt + sup | h( s ) | | QT | QT ≤∫ QT | s | ≤1 f (un )un dxdt + | f (0) | || un ||L1 (Q T ) Let + γ ＼un ＼2L2 (Q ) + sup | h( s ) | | QT | if s > k , k = Bk ( s ) s if | s |≤ k , −k if s < −k Hence it implies that {h(un )} , and therefore { f (un )} is bounded in L1 (QT ) Consider the corresponding Nemytskii mapping Bˆ k : W → W defined as follows Bˆ k (u )( x) = Bk (u ( x)), Now, we prove the boundedness of {∂ t un } for all x ∈ Ω We notice that ‖Bˆ k (u ) − u‖W → In the first equation in (2), replacing wk by ∂ t un , as k → ∞ Now multiplying (5) by Bˆ k (u ) , then and then using the Cauchy inequality, we get integrating over Ω we get T | s | ≤1 ≤ C Hence, by choosing ε small enough, we arrive at Integrating from to t , we can deduce that {∂ t un } is bounded in L2 (0, T ; H 01 (Ω)) Thus So, up to a subsequence, ∂ t un ut D∂ t un Dut weakly in L2 (0, T ; H 01 (Ω)) weakly in , L2 (0, T ; H −1 (Ω)) Because H 01 (Ω) ⊂⊂ L2 (Ω) ⊂ H −1 (Ω) + L1 (Ω) , by the Aubin-Lions-Simon compactness lemma, we obtain un → u strongly in L2 (0, T ; L2 (Ω)) Hence we may assume, up to a subsequence, that un → u a.e in QT Since f is continuous, it follows that f (un ) → f (u ) a.e in Ω × [0, T ] We obtain that h(u ) ∈ L1 (QT ) and for all test functions , φ ∈ C0∞ ([0, T ]; H 01 (Ω) ∩ L∞ (Ω)) ∫ QT QT Ω Therefore fˆ ′( s ) ≥ and sBk ( s ) ≥ for all s ∈ , ˆf ′(ξ )uBˆ (u )dx ≥ k ∫ { x∈Ω:|u ( x , t )| ≤ k } |∇u |2 dx ≥ Since the above inequalities, we get h(un )φ dxdt → ∫ h(u )φ dxdt QT Hence f (u ) ∈ L1 (QT ) and ∫ Note that we get ∫ f (un )φ dxdt → ∫ QT Integrating from to t , where t ∈ (0, T ) , then letting k → ∞ , we obtain f (u )φ dxdt , for all φ ∈ C0∞ ([0, T ]; H 01 (Ω) ∩ L∞ (Ω)) By standard arguments, we can check that u Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 65 NGHIÊN CỨU KHOA HỌC By the Gronwall inequality of integral form, we get for all t ∈ [0, T ] Hence u ∈ C ([0, T ]; H 01 (Ω)) , in particular, we get the uniqueness if u ( ) = b γ 1 − − ε ; l1 − b − l1ε > where= l1 According to Gronwall Lemma, we obtain EXISTENCE OF A GLOBAL ATTRACTOR Theorem allows us to define a continuous semigroup S (t ) : H 01 (Ω) → H 01 (Ω) associated to problem (1) by the formula S (t )u0 := u (t ), where u (.) is the unique weak solution of (1) with the initial datum u0 ∈ H 01 (Ω) The aim of this section is to prove the following result Theorem Assume that (H1), (H2) hold Then the semigroup {S (t )}t ≥ possesses a compact global attractor in H 01 (Ω) 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 B0 in H 01 (Ω) and S (t ) is asymptotically compact in H 01 (Ω) , that is, for any t > , it can be decomposed in the form S= (t ) S1 (t ) + S (t ), where for any bounded subset B in H 01 (Ω) , we have i) S1 (t ) is a continuous mapping from H 01 (Ω) into itself= and rB (t ) sup‖S1 (t ) y‖→ as t → +∞; Now, we can choose T1 and ρ0 such that ‖∇u (t )‖2 ≤ ρ0 , for all t ≥ T1 and for all u0 ∈ B This completes the proof Recall that in this paper we only assume the external force g ∈ H −1 (Ω) However, we know that for any g ∈ H −1 (Ω) and ε > given, there is a g ε ∈ L2 (Ω), which depends on g and ε , such that ‖g − g ε ‖H −1 ( Ω ) < ε To make the asymptotic regular estimates, we decompose the solution S (t )u0 = u (t ) of problem (1) as follows S= (t )u0 S1 (t )u0 + S (t )u0 , where S1 (t )u0 = u1 (t ) and S (t )u0 = u2 (t ) , that is the decomposition is of the following form u= vε + wε , where vε (t ) is the unique solution of the following problem H (Ω) for some t > vtε − Dvtε − Dvε + f (u ) − f ( wε ) + l vε g − g ε , l > , = ε = vε ( x, t ) |t = u0 ( x), v ( x, t ) |∂Ω 0,= It is clear that we only need to verify conditions i) and ii) above for the absorbing set B0 and w(t ) is the unique solution of the following problem y∈B ii) The operators S (t ) are uniformly compact for t large, i.e., S (t ) B is relatively compact in t ≥ t0 Lemma Assume that (H1), (H2) hold Then there exists a bounded absorbing set in H 01 (Ω) for the semigroup S (t ) Proof Multiplying the equation (1) by u (t ) , we have By the Cauchy inequality, we get (6) (6) ε ε ε ε wε ) g ε , l > wt − Dwt − Dw + f ( w ) − l (u −= (7) ε = wε ( x, t ) |t = 0 w ( x, t ) |∂Ω 0,= As in the proof of Theorem 1, one can prove the existence and uniqueness of solutions to (6) and (7) Lemma Let hypotheses (H1), (H2) hold Then the solutions of (6) satisfy the following estimates: there is a constant d depending on l1 , , such that for every t ≥ , ‖S1 (t )u0‖H ( Ω ) ≤ Q‖ ( ∇u0‖)e − d0t + ε Thus, Proof Multiplying the first equation of (6) by v, then integrating over Ω we get 66 Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 NGÀNH TOÁN HỌC Lemma Let {S (t )}t ≥ be the solution semigroup of (7) Then for large enough, S (T ) B is relatively compact in H 01 (Ω) Note that f ′( s ) ≥ −and 1 〈 g − g ε , vε 〉 H −1 , H ≤ ‖∇vε ‖2 + ‖g − g ε ‖2H −1 ( Ω ) , 2 we get By Lemma 1, the semigroup S (t ) has a bounded absorbing set B0 in H 01 (Ω) Moreover, the semigroup S (t ) is asymptotically compact in H 01 (Ω) due to Lemmas and Therefore, the w limit set = w ( B0 ) is the compact global attractor for S (t ) in H 01 (Ω) REFERENCES Similarly to the proof of Lemma 1, we obtain solids Acta Mech 37, 265÷296 ‖S1 (t )u0‖2 ≤ Q‖ ( ∇u0‖)e − d0t + ε [2] C.T Anh and T.Q Bao (2010) Pullback attractors The proof is complete Lemma Let hypotheses (H1), (H2) hold Then, there exists a positive constant M such that for any u0 ∈ H 01 (Ω) , there exists T > large enough, which depends on ‖g‖2H −1 ( Ω ) , ε ,‖∇u0‖, such that ‖S (t )u0‖2H ( Ω ) ≤ M , for all t ≥ T Proof Multiplying the first equation −Dw , we get [1] E.C Aifantis (1980) On the problem of diffusion in of (7) by 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 nonautonomous nonclassical diffusion equations on N Comm Pure Appl Anal 11, 1231÷1252 [4] C.T Anh and N.D Toan (2014) Existence and upper semicontinuity of uniform attractors in H ( N ) for non-autonomous nonclassical diffusion equations Ann Polon Math 113, 271÷295 [5] Y Xie, Q Li and K Zhu (2016) Attractors for when t ≥ t0 ( B) Notice that, similarly to the proof of Lemma 1, we obtain a T > large enough such that ‖S (t )u0‖2H ( Ω ) ≤ M , ∀ t ≥ T The proof is complete Since the embedding H (Ω) ∩ H 01 (Ω) ⊂⊂ H 01 (Ω) is compact, we obtain nonclassical diffusion equations with arbitrary polynomial growth nonlinearity Nonlinear Anal Real World Appl 31, 23÷37 [6] F Zhang, L Wang and J Gao (2016) Attractors and asymptotic regularity for nonclassical diffusion equations in locally uniform spaces with critical exponent Asymptot Anal 99, 241÷262 Tạp chí Nghiên cứu khoa học - Đại học Sao Đỏ, ISSN 1859-4190 Số 4(59).2017 67 ... M , for all t ≥ T Proof Multiplying the first equation −Dw , we get [1] E.C Aifantis (1980) On the problem of diffusion in of (7) by for a class of non-autonomous nonclassical diffusion equations... nonautonomous nonclassical diffusion equations on N Comm Pure Appl Anal 11, 1231÷1252 [4] C.T Anh and N.D Toan (2014) Existence and upper semicontinuity of uniform attractors in H ( N ) for non-autonomous... compact, we obtain nonclassical diffusion equations with arbitrary polynomial growth nonlinearity Nonlinear Anal Real World Appl 31, 23÷37 [6] F Zhang, L Wang and J Gao (2016) Attractors and asymptotic