Stability of fractionally dissipative 2D quasi-geostrophic equation with infinite delay

Journal of Dynamics and Differential Equations manuscript No. (will be inserted by the editor) Stability of fractionally dissipative 2D quasi-geostrophic equation with infinite delay Tongtong Liang1·Yejuan Wang1· Tom´as Caraballo2 Received: date / Accepted: date Abstract In this paper, fractionally dissipative 2D quasi-geostrophic equations with an external force containing infinite delay is considered in the space Hswith s≥2−2αand α∈(1 2,1). First, we investigate the existence and regularity of solutions by Galerkin approximation and the energy method. The continuity of solutions with respect to initial data and the uniqueness of solutions are also established. Then we prove the existence and uniqueness of a stationary solution by the Lax-Milgram theorem and the Schauder fixed point theorem. Using the classical Lyapunov method, the construction method of Lyapunov functionals and the Razumikhin-Lyapunov technique, we analyze the local stability of stationary solutions. Finally, the polynomial stability of stationary solutions is verified in a particular case of unbounded variable delay. Keywords Quasi-geostrophic equation ·Infinite delay ·Stationary solution · Stability ·Asymptotic stability ·Polynomial stability This work was supported by NSF of China (Grant No. 41875084), and by FEDER and Ministerio de Ciencia, Innovaci´on y Universidades of Spain (Grant PGC2018-096540-B-I00). Tongtong Liang [email protected] Yejuan Wang (Corresponding author) w[email protected] Tom´as Caraballo [email protected] 1School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China 2Depto. de Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, U- niversidad de Sevilla, c/ Tarfia s/n, 41012 Seville, Spain 2 Liang, Wang and Caraballo 1 Introduction We consider the following 2D quasi-geostrophic equation with infinite delay in the periodic domain T2=R2/(2πZ)2: {θt+κ(−∆)αθ+u·∇θ=f+g(t, θt),in (0,∞)×T2, θ(t, x) = φ(t, x), t ∈(−∞,0], x ∈T2,(1.1) where α∈(1 2,1), κ > 0, θ=θ(t, x) represents the potential temperature, fis a nondelayed external force independent of time, gis the external force containing some hereditary characteristics, and u= (u1, u2) is the velocity field determined by θvia the following relation: u= (u1, u2) = (−∂ψ ∂x2 ,∂ψ ∂x1 ),and (−∆)1 2ψ=−θ. (1.2) The fractionally dissipative 2D quasi-geostrophic equation describes a kind of dynamics of large-scale phenomena in the atmosphere and ocean, see [31] for more details. Due to potential applications in meteorology, 2D quasigeostrophic equations have been receiving much attention over the last decades, see, e.g. [13,16–18,30,32,34,36] and the references therein. The existence, uniqueness and regularity of the quasi-geostrophic equation have been considered in [12,14,15,19,20]. The asymptotic analysis of the systems, as a key method to explore the evolution of the systems in the future, has also been investigated in [11,34], where the large time behavior of solutions has been discussed by several decay estimates. For the existence and regularity of attractors of the quasi-geostrophic equation, we refer the reader to [10,18,21]. It is well known that stability theory is an important issue in the study of the asymptotic behaviour of the systems. The asymptotic stability for the weak solution of the quasi-geostrophic equation with respect to large perturbation has been investigated in [17,32]. The existence and uniqueness of steady-state solutions to the quasi-geostrophic equation with finite energy (L2norm), and the nonlinear stability of such a solution have been proved in [13]. Here we are interested in a quasi-geostrophic model containing some hereditary features in the forcing term. These situations may appear, for instance, when the evolution of the systems depends not only on the present state of the systems, but also on the history of the solutions, see [28,30,36], for instance. There has, however, been little mention of the existence and stability of stationary solutions to the quasi-geostrophic equation with infinite delay in the space Hseven in the case of the non-delayed quasi-geostrophic equation. The quasi-geostrophic equation exhibits strikingly similar features (singularities) as 3D Navier-Stokes equations. 2D Navier-Stokes models with hereditary characteristics were proposed in [4], and developed in [2,3,5,24], where the existence, uniqueness and stability of stationary solutions have been established. For the 3D case, the stationary problem has been analyzed for a globally modified version of Navier-Stokes equations with delay terms within a locally Lipschitz operator in [26,27]. Stability of quasi-geostrophic equation 3 In this paper, we aim to study the stability of the quasi-geostrophic equation with unbounded variable delay in the phase space C(Hs) with s≥2−2α and α∈(1 2,1), see Section 2 for more details. The first purpose of this paper is to investigate the existence and uniqueness of the solution to Eq. (1.1), which are proved by the classic Galerkin approximation and the energy method combined with auxiliary Lpestimates, p > 2 2α−1. Compared to some recent works for the Navier-Stokes equation, the essential difficulty is the lack of cancellation property for the quadratic type of nonlinear term in Hs. On the other hand, the problems become harder thanks to the dissipative term (−∆)α,1 2< α < 1, and the nonlinear term u·∇θ. In order to overcome these difficulties, we give some complicated estimates on solutions by using appropriate commutator estimates. Then we introduce some new suitable assumptions to guarantee the existence, uniqueness and regularity of stationary solutions to Eq. (1.1) by the Lax-Milgram theorem and the Schauder fixed point theorem. Lyapunov’s method is a fundamental tool to study the stability of systems. Along with the birth of functional differential equations, Razumikhin, and Kolmanovskii together with Shaikhet have proposed the method of Razumikhin type and the construction method of Lyapunov functionals, respectively, which have been successfully used to investigate the stability of delay evolution equations, for instance, [6,22,29]. Finally, we use different methods to deduce different sufficient conditions ensuring the stability of stationary solutions to Eq. (1.1), and discuss the polynomial stability of the quasi-geostrophic equation with proportional delay. The article is organized as follows. In Section 2, we introduce some notations, and briefly recall some useful estimates and relevant mathematical preliminaries from functional analysis. The existence and uniqueness of solutions to Eq. (1.1) are considered in Section 3. In Section 4, the existence and uniqueness of stationary solutions to Eq. (1.1) are established, and several methods are used to show the stability of stationary solutions. 2 Preliminaries Throughout this manuscript and notation, we consider mean-zero (zero average) solutions of Eq. (1.1), that is ∫T2 θ(t, x)dx = 0,for any t∈R. Denote Λ≡(−∆)1 2. The fractional Laplacian Λs, with s∈Rcan be defined by d Λsf(k) = |k|sb f(k), where b fdenotes the Fourier transform of f, i.e., b f(k) = 1 (2π)2∫T2 f(x)e−ikxdx. 4 Liang, Wang and Caraballo For 1 ≤p≤ ∞,Lpdenotes the Banach space of pth-power integrable functions if p < ∞and the essentially bounded functions when p=∞. The following standard notations are used: ∥f∥p Lp=∫T2|f|pdx, ∥f∥L∞= ess sup x∈T2|f(x)|. For any tempered distribution fon T2and s∈R, we define ∥f∥2 Hs=∥Λsf∥2 L2=∑ k∈Z2|k|2s|b f(k)|2,(2.1) and Hsdenotes the Sobolev space of all ffor which ∥f∥Hsis finite. In particular, we denote by H:= H0=L2. For 1 ≤p≤ ∞ and s∈R, the space Hs,p is a subspace of Lp, consisting of all fwhich can be written in the form f=Λ−sg,g∈Lp, and the Hs,p norm of fis defined by ∥f∥Hs,p =∥Λsf∥Lp. We denote by ⟨·,·⟩ the inner product of L2. Given a Banach space Xand its dual X′, we also denote the dual pairing between Xand X′by ⟨·,·⟩, unless noted otherwise. The equality relating uto θin (1.2) can be rewritten in terms of periodic Riesz transforms as: u= (∂x2Λ−1θ, −∂x1Λ−1θ)=(−R2θ, R1θ)≡ R⊥θ, where Rj,j= 1,2 denote the Riesz transforms defined by d Rjf(k) = −ikj |k|b f(k), k ∈Z2\{0}. The following result can be obtained by the fact that the Riesz transforms commute with (−∆)land the boundedness of the Riesz transforms in Lp, see [35, Chapter III] for more details. Lemma 1 Let 1< p < ∞and l≥0. Then there exists a constant C(l, p) such that ∥(−∆)lu∥Lp≤C(l, p)∥(−∆)lθ∥Lp.(2.2) If p= 2, the inequality (2.2) can be strengthened to ∥(−∆)lu∥L2=∥(−∆)lθ∥L2.(2.3) We recall some important estimates which will be used frequently in the section below. Stability of quasi-geostrophic equation 5 Lemma 2 (Commutator Estimate) Suppose that s > 0and p∈(1,∞). If f, g ∈ S, the Schwartz class, then ∥Λs(fg)−fΛsg∥Lp≤C1(∥∇f∥Lp1∥g∥Hs−1,p2+∥f∥Hs,p3∥g∥Lp4) (2.4) and ∥Λs(fg)∥Lp≤C2(∥f∥Lp1∥g∥Hs,p2+∥f∥Hs,p3∥g∥Lp4),(2.5) with p2, p3∈(1,∞)such that 1 p=1 p1 +1 p2 =1 p3 +1 p4 . Lemma 3 (Fractional LpPoincar´e) For α∈[0,1],p≥2,ϕ∈H2αwith |ϕ|p−1∈Hα, the following estimate holds: ∫T2 (−∆)αϕsgnϕ|ϕ|p−1dx ≥4(p−1) p2∫T2[(−∆)α 2(|ϕ|p 2)]2dx. (2.6) Lemma 3 is a version of the famous Kato-Beurling-Deny inequality, which has been proven in [8]. When pis even, we can also refer to [9,12] for more details. Next, we recall some spaces which allow us to deal with infinite delay. Given real numbers a < b, we denote by C([a, b]; X) the Banach space of all continuous functions from [a, b] into Xequipped with sup norm. For any given number T > 0, if a function x∈C((−∞, T]; X), for each t∈[0, T], we denote by xt∈C((−∞,0]; X) the function defined by xt(r) = x(t+r), ∀r∈(−∞,0]. In this paper, we aim to establish well-posedness and stability results for 2D quasi-geostrophic equation with infinite delay in the phase space C(X) = {ψ∈C((−∞,0]; X) : lim t→−∞ψ(t) exists in X}, which is endowed with the norm ∥ψ∥C(X)= sup t∈(−∞,0] ∥ψ(t)∥X. To describe the conditions imposed on the delay term g, we introduce some notations. Let Xand Ybe two Banach spaces. We denote by Lip0(X,Y) the collection of all mappings g: [0, T]×X→Ysatisfying the following conditions: –for any ξ∈X, the mapping t∈[0, T]→g(t, ξ)∈Yis measurable; –for each t∈[0, T], g(t, 0) = 0; –there exists a constant Lg>0 such that ∥g(t, ξ)−g(t, η)∥Y≤Lg∥ξ−η∥X,∀t∈[0, T] and ∀x, y ∈X. Remark 1 It is clear that if g∈Lip0(X,Y), then for any ξ∈X, ∥g(t, ξ)∥Y=∥g(t, ξ)−g(t, 0)∥Y≤Lg∥ξ∥X,∀t∈[0, T], and therefore |g(·, ξ)| ∈ L∞(0, T). Throughout the paper, we denote by Ca real positive constant which can vary from a line to another and even in the same line. If the constant Cdepends on some variable x, we denote it by Cx. 6 Liang, Wang and Caraballo 3 Existence, regularity and uniqueness of solutions In this section, we first study the existence and regularity of weak solutions to Eq. (1.1). Next the continuity of solutions with respect to initial datum is established and the uniqueness of solutions follows directly. Definition 1 For any given number T > 0 and initial datum φ∈C(Hs), a weak solution θto Eq. (1.1) in the interval (−∞, T) is a function θ∈ C((−∞, T]; Hs)∩L2(0, T;Hs+α) with θ0=φsuch that, for all v∈Hs+α, the equality d dt⟨θ(t), v⟩+κ⟨(−∆)αθ(t), v⟩+⟨u(t)·∇θ(t), v⟩=⟨f, v⟩+⟨g(t, θt), v⟩, holds true in the sense of scalar distribution D′(0, T). The existence and regularity of weak solutions to Eq. (1.1) can be proved by the classic Galerkin method. Theorem 1 Fix α∈(1 2,1),s≥2−2αand p > 2 2α−1. Assume that f∈ Hs−α∩Lpand g∈Lip0(C(Hs), Hs)∩Lip0(C(Lp), Lp). Then for any given T > 0and φ∈C(Hs)∩C(Lp), there exists a weak solution θof Eq. (1.1) in the sense of Definition 1. Furthermore, if f∈Hs∩Lpand φ∈C(Hs)∩C(Lp) with φ(0) ∈Hs+α, then θ∈C([0, T]; Hs+α)∩L2(0, T;Hs+2α). Proof We will split the proof into several steps. Step 1. The Galerkin approximation. By the classical spectral theory of elliptic operators, it follows that −∆possesses a sequence of eigenvalues {λi}∞ i=1 and a corresponding family of eigenfunctions {ei}∞ i=1, which forms an orthonormal basis of H. Denote by Pnthe projection from Honto the linear span PnH of {e1, . . . , en}, i.e., Pnθ= n ∑ i=1 θieifor θ=∞ ∑ i=1 θiei. We define the n-th Galerkin approximation of Eq. (1.1) as the following ODE system with infinite delay: {d dt θn(t) + κ(−∆)αθn(t) + Pn(un(t)·∇θn(t)) = Pnf+Png(t, θn t), t ≥0, θn(t) = Pnφ(t), t ∈(−∞,0], (3.1) with un=R⊥θnsatisfying ∇·un= 0. Step 2. A priori estimates. Multiplying Eq. (3.1) by sgnθn(t)|θn(t)|p−1, with p > 2 2α−1, and taking the inner product in L2, we have 1 p d dt ∫T2|θn(t)|pdx +κ∫T2 (−∆)αθn(t)sgnθn(t)|θn(t)|p−1dx =∫T2 Pnfsgnθn(t)|θn(t)|p−1dx +∫T2 Png(t, θn t)sgnθn(t)|θn(t)|p−1dx, (3.2) Stability of quasi-geostrophic equation 7 since the term un·∇θnvanishes in the calculation due to the fact: ∫T2 2 ∑ i=1 un i(t)∂θn(t) ∂xi sgnθn(t)|θn(t)|p−1dx =1 p∫T2 2 ∑ i=1 un i(t)∂|θn(t)|p ∂xi dx =−1 p∫T2|θn(t)|p 2 ∑ i=1 ∂un i(t) ∂xi dx = 0.(3.3) It follows from (2.6) that κ∫T2 (−∆)αθn(t)sgnθn(t)|θn(t)|p−1dx ≥0. Then using the condition g∈Lip0(C(Lp), Lp) and the Young inequality, (3.2) can be transformed as follows: d dt∥θn(t)∥p Lp≤ ∥θn(t)∥p Lp+Cp∥Pnf∥p Lp+Cp∥Png(t, θn t)∥p Lp ≤ ∥θn(t)∥p Lp+Cp∥f∥p Lp+Cp∥g(t, θn t)∥p Lp ≤ ∥θn(t)∥p Lp+Cp∥f∥p Lp+CpLg∥θn t∥p C(Lp) ≤ ∥θn(t)∥p Lp+Cp∥f∥p Lp+CpLg(∥Pnφ∥p C(Lp)+ sup 0≤τ≤t∥θn(τ)∥p Lp). Integrating the above inequality between 0 and t∈[0, T ], then replacing tby τ∈[0, t] and taking the supremum over [0, t], we have sup 0≤τ≤t∥θn(τ)∥p Lp≤ ∥θn(0)∥p Lp+∫t 0∥θn(σ)∥p Lpdσ +CpT∥f∥p Lp +CpLg∫t 0(∥Pnφ∥p C(Lp)+ sup 0≤τ≤σ∥θn(τ)∥p Lp)dσ ≤(1 + CpLgT)∥φ∥p C(Lp)+CpT∥f∥p Lp+ (1 + CpLg)∫t 0 sup 0≤τ≤σ∥θn(τ)∥p Lpdσ. Applying the Gronwall lemma results in sup 0≤τ≤t∥θn(τ)∥p Lp≤((1 + CpLgT)∥φ∥p C(Lp)+CpT∥f∥p Lp)e(1+CpLg)t. Thus for any given T > 0, there exists a constant Cp,T such that sup 0≤τ≤t∥θn(τ)∥Lp≤Cp,T ,∀t∈[0, T],∀n≥1.(3.4) Applying Λsto Eq. (3.1) and taking the inner product in L2with Λsθn, we have 1 2 d dt∥θn(t)∥2 Hs+κ∥θn(t)∥2 Hs+α=−⟨ΛsPn(un(t)·∇θn(t)), Λsθn(t)⟩ +⟨ΛsPnf, Λsθn(t)⟩+⟨ΛsPng(t, θn t), Λsθn(t)⟩ := I1+I2+I3.(3.5) 8 Liang, Wang and Caraballo Note that Pncommutes with Λson Hs, and Pnis self-adjoint in L2, then |I1|=|⟨PnΛs(un(t)·∇θn(t)), Λsθn(t)⟩| =|⟨Λs(un(t)·∇θn(t)), PnΛsθn(t)⟩| =|⟨Λs(un(t)·∇θn(t)), Λsθn(t)⟩|.(3.6) Taking into account that ∇·un= 0, by the Schwarz inequality and Lemmas 1 and 2, we obtain |I1| ≤ ∥Λs+1−α(un(t)θn(t))∥L2∥Λs+αθn(t)∥L2 ≤C∥un(t)∥Lp∥θn(t)∥Hs+1−α,q ∥θn(t)∥Hs+α +C∥un(t)∥Hs+1−α,q ∥θn(t)∥Lp∥θn(t)∥Hs+α ≤C∥θn(t)∥Lp∥θn(t)∥Hs+1−α,q ∥θn(t)∥Hs+α ≤C∥θn(t)∥1+η Hs+α∥θn(t)∥1−η L2∥θn(t)∥Lp,(3.7) where 1 p+1 q=1 2, and we have used the Nirenberg-Gagliardo inequality (cf. [7]): ∥θn(t)∥Hs+1−α,q ≤C∥θn(t)∥η Hs+α∥θn(t)∥1−η L2, where η=s+2−α−2 q s+α. Since 1 <1 + η < 2 and Lp⊂L2, in view of the Young inequality, (3.7) can be further estimated by |I1| ≤ C∥θn(t)∥1+η Hs+α∥θn(t)∥2−η Lp ≤κ 4∥θn(t)∥2 Hs+α+C∥θn(t)∥ 4−2η 1−η Lp.(3.8) For the term I2, since Pncommutes with Λson Hsand Pnis self-adjoint in L2, by the Schwarz inequality and the Young inequality we have |I2|=|⟨PnΛsf, Λsθn(t)⟩| =|⟨Λsf, Λsθn(t)⟩| ≤ ∥f∥Hs−α∥θn(t)∥Hs+α≤κ 4∥θn(t)∥2 Hs+α+C∥f∥2 Hs−α,(3.9) where Hs−αdenotes the dual space of Hα−sfor the case s−α < 0. For the last term, arguing as in (3.9), in view of g∈Lip0(C(Hs), Hs), we find that |I3|=|⟨PnΛsg(t, θn t), Λsθn(t)⟩| =|⟨Λsg(t, θn t), Λsθn(t)⟩| ≤ ∥g(t, θn t)∥Hs∥θn(t)∥Hs≤Lg∥θn t∥2 C(Hs).(3.10) Inserting (3.8)-(3.10) into (3.5) gives 1 2 d dt∥θn(t)∥2 Hs+κ 2∥θn(t)∥2 Hs+α ≤C∥θn(t)∥ 4−2η 1−η Lp+C∥f∥2 Hs−α+Lg∥θn t∥2 C(Hs). Stability of quasi-geostrophic equation 9 Integrating the above inequality and using (3.4), we infer that ∥θn(t)∥2 Hs+κ∫t 0∥θn(σ)∥2 Hs+αdσ ≤ ∥θn(0)∥2 Hs +CT(1 + ∥f∥2 Hs−α)+2Lg∫t 0∥θn σ∥2 C(Hs)dσ. (3.11) Hence, ∥θn t∥2 C(Hs)≤max{sup −∞<τ≤0∥θn(τ)∥2 Hs,sup 0≤τ≤t∥θn(τ)∥2 Hs} ≤ ∥φ∥2 C(Hs)+CT(1 + ∥f∥2 Hs−α)+2Lg∫t 0∥θn σ∥2 C(Hs)dσ, and applying the Gronwall lemma, ∥θn t∥2 C(Hs)≤(∥φ∥2 C(Hs)+CT(1 + ∥f∥2 Hs−α))e2Lgt. Then we obtain the following estimates. Firstly, there exists a constant CT such that ∥θn t∥2 C(Hs)≤CT,∀t∈[0, T],∀n≥1,(3.12) which implies that {θn}is bounded in L∞(0, T;Hs).(3.13) Secondly, it follows from (3.11) and (3.12) that κ∫T 0∥θn(σ)∥2 Hs+αdσ ≤ ∥θn(0)∥2 Hs+CT(1 + ∥f∥2 Hs−α) + 2Lg∫T 0∥θn σ∥2 C(Hs)dσ ≤ ∥θn(0)∥2 Hs+CT(1 + ∥f∥2 Hs−α) + 2LgCT, and consequently there exists another constant CTsuch that ∥θn∥2 L2(0,T ;Hs+α)≤CT,∀t∈[0, T],∀n≥1.(3.14) Finally, we prove the boundedness of {(θn)′}. From Lemmas 1 and 2, we find that ∥Λs−α(un·∇θn)∥L2=∥Λs−α∇(unθn)∥L2≤ ∥Λs+1−α(unθn)∥L2 ≤C(∥un∥Lp1∥θn∥Hs+1−α,p2+∥un∥Hs+1−α,p2∥θn∥Lp1) ≤C∥θn∥Hs∥θn∥Hs+α, where p1=2 2α−1,p2=1 1−α, and we have used the fact ∇ · un= 0 and the Sobolev embeddings Hs⊂Lp1,Hs+α⊂Hs+1−α,p2. Then, by using the generalized Poincar´e inequality: λβ−γ 1∫T2 [(−∆)γ 2ϕ]2dx ≤∫T2 [(−∆)β 2ϕ]2dx, ∀β > γ, (3.15) 16 Liang, Wang and Caraballo By the H¨older inequality and Lemmas 1-2, in view of the fact that ∇·u(1) = ∇·u(2) = 0, we estimate the second term in the right-hand side of (3.31) as follows: |⟨(u(1)(t)−u(2)(t)) ·∇(Λsθ(2)(t)), Λsw(t)⟩| =|⟨∇((u(1)(t)−u(2)(t))Λsθ(2)(t)), Λsw(t)⟩| ≤ ∥Λ1−α((u(1)(t)−u(2)(t))Λsθ(2)(t))∥L2∥Λs+αw(t)∥L2 ≤C∥u(1)(t)−u(2)(t)∥Lp1∥θ(2)(t)∥Hs+1−α,p2∥w(t)∥Hs+α +∥u(1)(t)−u(2)(t)∥H1−α,p3∥θ(2)(t)∥Hs,p4∥w(t)∥Hs+α ≤C∥w(t)∥Hs∥θ(2)(t)∥Hs+α∥w(t)∥Hs+α,(3.32) where p1=2 2α−1,p2=1 1−α,p3=2 α,p4=2 1−α, and we have used the Sobolev embeddings Hs⊂Lp1,Hs+α⊂Hs+1−α,p2,Hs⊂H1−α,p3,Hs+α⊂Hs,p4. Substituting (3.32) into (3.31) gives |J2| ≤ C∥θ(2)(t)∥Hs+α∥w(t)∥Hs+α∥w(t)∥Hs.(3.33) For J3, using the H¨older inequality and the assumption on g, we deduce that |J3|≤∥g(t, θ(1) t)−g(t, θ(2) t)∥Hs∥w(t)∥Hs ≤Lg∥wt∥C(Hs)∥w(t)∥Hs≤Lg∥wt∥2 C(Hs).(3.34) Combining estimates (3.30), (3.33) and (3.34), by the Young inequality, it follows from (3.29) that 1 2 d dt∥w(t)∥2 Hs+1 2κ∥w(t)∥2 Hs+α ≤C(∥θ(1)(t)∥2 Hs+α+∥θ(2)(t)∥2 Hs+α)∥w(t)∥2 Hs+Lg∥wt∥2 C(Hs).(3.35) After integrating (3.35) between 0 and t, if we substitute tby σ∈[0, t] and take the supremum, we obtain sup 0≤σ≤t∥w(σ)∥2 Hs≤ ∥w(0)∥2 Hs +C∫t 0 (∥θ(1)(σ)∥2 Hs+α+∥θ(2)(σ)∥2 Hs+α)∥w(σ)∥2 Hsdσ + 2Lg∫t 0∥wσ∥2 C(Hs)dσ. (3.36) Note that ∥wt∥C(Hs)= sup −∞<σ≤0∥w(t+σ)∥Hs = max {sup −∞<σ≤0∥φ(1)(σ)−φ(2)(σ)∥Hs,sup 0≤σ≤t∥w(σ)∥Hs} = max {∥φ(1) −φ(2)∥C(Hs),sup 0≤σ≤t∥w(σ)∥Hs}.(3.37) Stability of quasi-geostrophic equation 17 Then we conclude from (3.36) that ∥wt∥2 C(Hs)≤ ∥φ(1) −φ(2)∥2 C(Hs) +∫t 0 (C∥θ(1)(σ)∥2 Hs+α+C∥θ(2)(σ)∥2 Hs+α+ 2Lg)∥wσ∥2 C(Hs)dσ. Thus the estimate (3.28) follows from the Gronwall lemma.  The uniqueness of weak solutions follows immediately from Lemma 4. Theorem 2 Under the assumptions of Theorem 1, the weak solution obtained in Theorem 1 is unique. Proof Let θ(1) and θ(2) be two solutions of Eq. (1.1). Since they have the same initial datum φ, it follows from Lemma 4 that ∥θ(1) t−θ(2) t∥2 C(Hs)= 0, and thus the proof is complete.  4 Asymptotic behavior of solutions In this section we consider the long time behavior of solutions in a neighborhood of a stationary solution to Eq. (1.1) when the delay term has a special form. First of all, we establish the existence, uniqueness and regularity of stationary solutions under some restrictions. Then we present various methods to obtain the stability properties: the Lyapunov function method, the construction method of appropriate Lyapunov functionals and the Razumikhin- Lyapunov technique. Finally, the polynomial stability of stationary solutions is addressed in the case of proportional variable delays. 4.1 Existence, uniqueness and regularity of stationary solutions In order to investigate the properties of stationary solutions to Eq. (1.1), we assume that gfulfills (H) Let i:Hs→C(Hs) be the trivial immersion given by i(θ) = e θwith e θ(t) = θfor all t≤0. Then g(t1, ξ) = g(t2, ξ),for any t1, t2∈R+and all ξ∈i(Hs). For example, given measurable functions ρ:R+→R+and G:Hs→Hs, we can define g:R+×C(Hs)→Hsby g(t, ξ) := G(ξ(−ρ(t))). It is clear that gsatisfies the above assumption (H). If (H) holds, for g∈Lip0(C(Hs), Hs), we define eg:Hs→Hsas eg(θ) = g(0, i(θ)). Then eg=g|R+×i(Hs)is of course autonomous, and it also belongs to Lip0(C(Hs), Hs) with the same Lipschitz constant as g. 18 Liang, Wang and Caraballo Definition 2 A stationary solution to Eq. (1.1) is a function θ∗∈Hs+αsuch that κ(−∆)αθ∗+R⊥θ∗·∇θ∗=f+eg(θ∗).(4.1) We now establish the existence, uniqueness and regularity of stationary solutions to Eq. (1.1) as stated in the next result. Theorem 3 Fix α∈(1 2,1) and s≥2−2α. Let g∈Lip0(C(Hs), Hs)and the condition (H)be satisfied. Suppose that κ > λ−α 1Lgand ∆:= (κ−λ−α 1Lg)2−4Υ∥f∥Hs−α>0,(4.2) where Υ= 4C2M1M2(C(0, p5) + C(s+1−α 2, p6)) and the related constants will be given later on. Then (i)for all f∈Hs−α, there exists at least one solution to Eq. (4.1); (ii)for all f∈Hs−α, the solution to Eq. (4.1) is unique; (iii)if f∈Hs, the solution to Eq. (4.1) belongs to Hs+2α. Proof (i) Let a positive number Rbe fixed later on, and set B={z∈Hs+α:∥z∥Hs+α≤R}. It is clear that Bis a closed and convex subset of Hs+α. Then for each z∈B, by the Lax-Milgram theorem there exists a unique θ∈Hs+αsuch that κ⟨Λs+αθ, Λs+αv⟩+⟨Λs(R⊥z·∇θ), Λsv⟩=⟨Λsf, Λsv⟩+⟨Λseg(z), Λsv⟩,(4.3) with v∈Hs+α. Hence we define a mapping L:z∈B→θ∈Hs+α. The rest of the proof can be divided into three steps. Step 1. The mapping Lmaps Binto itself. Taking v=θin (4.3), it follows that κ∥θ∥Hs+α≤ ∥Λs−α(R⊥z·∇θ)∥L2+∥f∥Hs−α+∥eg(z)∥Hs−α.(4.4) By Lemmas 1 and 2, in view of ∇·(R⊥z) = 0, we deduce that ∥Λs−α(R⊥z·∇θ)∥L2≤ ∥Λs+1−α(R⊥zθ)∥L2 ≤C2(∥R⊥z∥Lp5∥θ∥Hs+1−α,p6+∥R⊥z∥Hs+1−α,p6∥θ∥Lp5) ≤C2(C(0, p5)∥z∥Lp5∥θ∥Hs+1−α,p6+C(s+ 1 −α 2, p6)∥z∥Hs+1−α,p6∥θ∥Lp5) ≤C2M1M2(C(0, p5) + C(s+ 1 −α 2, p6))∥z∥Hs+α∥θ∥Hs+α := Υ∥z∥Hs+α∥θ∥Hs+α,(4.5) where p5=2 2α−−1,p6=1 1−α−with α−∈(1 2, α), C(0, p5) and C(s+1−α 2, p6) are the constants appearing in Lemma 1, C2is the constant appearing in Stability of quasi-geostrophic equation 19 Lemma 2, Mi,i= 1,2, are the constants related with the following Sobolev embedding inequalities: ∥z∥Lp5≤M1∥z∥Hs+α,∥z∥Hs+1−α,p6≤M2∥z∥Hs+α. Thanks to the assumption on g, we have ∥eg(z)∥Hs−α≤λ−α 2 1∥eg(z)∥Hs≤λ−α 2 1Lg∥z∥Hs≤λ−α 1Lg∥z∥Hs+α.(4.6) Inserting (4.5)-(4.6) into (4.4) results in κ∥θ∥Hs+α≤Υ∥z∥Hs+α∥θ∥Hs+α+∥f∥Hs−α+λ−α 1Lg∥z∥Hs+α.(4.7) Note that ∆= (κ−λ−α 1Lg)2−4Υ∥f∥Hs−α>0 and (κ−λ−α 1Lg)−√∆ 2Υ<κ Υ. We can pick 0 < R < κ Υsuch that (κ−λ−α 1Lg)R−ΥR2≥ ∥f∥Hs−α. By (4.7), the mapping Ldefined by (4.3), maps Binto B. Step 2. The mapping Lis continuous in B. Let zi∈Band θi∈Bbe such that κ⟨Λs+αθi, Λs+αv⟩+⟨Λs(R⊥zi·∇θi), Λsv⟩=⟨Λsf, Λsv⟩+⟨Λseg(zi), Λsv⟩, with v∈Hs+α, where i= 1,2. Taking v=θ1−θ2and using the bilinearity of R⊥zi·∇θi, we find that κ∥θ1−θ2∥2 Hs+α≤ |⟨Λs(R⊥z1·∇(θ1−θ2)), Λs(θ1−θ2)⟩| +|⟨Λs((R⊥z1−R⊥z2)·∇θ2), Λs(θ1−θ2)⟩| +|⟨Λs(eg(z1)−eg(z2)), Λs(θ1−θ2)⟩| := I1+I2+I3.(4.8) Modifying slightly the arguments in (4.5), in view of the H¨older inequality, we deduce that I1≤ ∥Λs−α(R⊥z1·∇(θ1−θ2))∥L2∥Λs+α(θ1−θ2)∥L2 ≤Υ∥z1∥Hs+α∥θ1−θ2∥2 Hs+α,(4.9) and I2≤ ∥Λs−α((R⊥z1−R⊥z2)·∇θ2)∥L2∥Λs+α(θ1−θ2)∥L2 ≤C2M2C(0, p5)∥z1−z2∥Lp5∥θ2∥Hs+α∥θ1−θ2∥Hs+α +C2M1C(s+ 1 −α 2, p6)∥z1−z2∥Hs+1−α,p6∥θ2∥Hs+α∥θ1−θ2∥Hs+α. (4.10) By the H¨older inequality and the assumption on g, we have I3≤ ∥eg(z1)−eg(z2)∥Hs−α∥θ1−θ2∥Hs+α ≤λ−α 2 1∥eg(z1)−eg(z2)∥Hs∥θ1−θ2∥Hs+α ≤λ−α 2 1Lg∥z1−z2∥Hs∥θ1−θ2∥Hs+α.(4.11) 20 Liang, Wang and Caraballo Substituting (4.9)-(4.11) into (4.8) gives κ∥θ1−θ2∥2 Hs+α≤Υ∥z1∥Hs+α∥θ1−θ2∥2 Hs+α +C2M2C(0, p5)∥z1−z2∥Lp5∥θ2∥Hs+α∥θ1−θ2∥Hs+α +C2M1C(s+ 1 −α 2, p6)∥z1−z2∥Hs+1−α,p6∥θ2∥Hs+α∥θ1−θ2∥Hs+α +λ−α 2 1Lg∥z1−z2∥Hs∥θ1−θ2∥Hs+α.(4.12) Note that ∥zi∥Hs+α≤R < κ Υ. Hence we can take 0 < ε < κ −ΥR such that (κ−ΥR −ε)∥θ1−θ2∥2 Hs+α≤Cε∥z1−z2∥2 Lp5∥θ2∥2 Hs+α +Cε∥z1−z2∥2 Hs+1−α,p6∥θ2∥2 Hs+α+Cε∥z1−z2∥2 Hs,(4.13) thanks to the Young inequality. Since Hs+α⊂Lp5,Hs+α⊂Hs+1−α,p6and Hs+α⊂Hsare continuous embeddings, the continuity of the mapping z7→ θ in Bfollows from (4.13). Step 3. L(B) is sequentially compact. To ensure the sequential compactness of the set L(B), it suffices to prove that any sequence in L(B) has a convergent subsequence. Let {θn} ∈ L(B) be given arbitrarily. Then there exists a sequence {zn} ∈ Bsuch that θn=L(zn). Since the embeddings Hs+α⊂Lp5,Hs+α⊂Hs+1−α,p6and Hs+α⊂Hsare compact (see, e.g. [7, Chapter 1]), we can find a subsequence zniconverging in Lp5,Hs+1−α,p6and Hs. By (4.13), we conclude that (κ−ΥR −ε)∥θni−θnj∥2 Hs+α≤Cε∥zni−znj∥2 Lp5∥θnj∥2 Hs+α +Cε∥zni−znj∥2 Hs+1−α,p6∥θnj∥2 Hs+α+Cε∥zni−znj∥2 Hs→0 (4.14) when iand jtend to infinity. It follows that the subsequence {θni}is a Cauchy sequence in Hs+α, and therefore it is convergent as desired. By the Schauder fixed point theorem, Step 1-Step 3 ensure the existence of a fixed point in B, which clearly is a stationary solution to Eq. (4.1). (ii) Arguing as in (4.4)-(4.7), the solution θto Eq. (4.1) satisfies κ∥θ∥Hs+α≤Υ∥θ∥2 Hs+α+∥f∥Hs−α+λ−α 1Lg∥θ∥Hs+α. Note that ∆= (κ−λ−α 1Lg)2−4Υ∥f∥Hs−α>0, then we deduce that ∥θ∥Hs+α≤ β1or ∥θ∥Hs+α≥β2, where β1=(κ−λ−α 1Lg)−√∆ 2Υand β2=(κ−λ−α 1Lg)+√∆ 2Υ. According to the proof of (i), we know that the solution θto Eq. (4.1) lies in Band consequently, ∥θ∥Hs+α≤β1.(4.15) Stability of quasi-geostrophic equation 21 Consider two solutions θ1and θ2to Eq. (4.1), similar to the arguments of (4.8), (4.9) and (4.11), we have κ∥θ1−θ2∥2 Hs+α≤ |⟨Λs(R⊥θ1·∇(θ1−θ2)), Λs(θ1−θ2)⟩| +|⟨Λs((R⊥θ1−R⊥θ2)·∇θ2), Λs(θ1−θ2)⟩| +|⟨Λs(eg(θ1)−eg(θ2)), Λs(θ1−θ2)⟩| ≤Υ(∥θ1∥Hs+α+∥θ2∥Hs+α)∥θ1−θ2∥2 Hs+α +λ−α 2 1Lg∥θ1−θ2∥Hs∥θ1−θ2∥Hs+α ≤Υ(∥θ1∥Hs+α+∥θ2∥Hs+α)∥θ1−θ2∥2 Hs+α+λ−α 1Lg∥θ1−θ2∥2 Hs+α ≤2Υβ1∥θ1−θ2∥2 Hs+α+λ−α 1Lg∥θ1−θ2∥2 Hs+α. Therefore, √∆∥θ1−θ2∥2 Hs+α≤0, and the uniqueness of solutions to Eq. (4.1) follows. (iii) If f∈Hs∩Lp, then every solution θ∗to Eq. (4.1) is also a solution to Eq. (1.1), but with initial datum φ(t) = θ∗for t∈(−∞,0]. Similar to Step 5 in Theorem 1, the strong regularity of θ∗follows immediately.  As commented in the introduction of this section, our goal for the rest of our current work is to analyze the stability properties of stationary solutions to Eq. (1.1). For reader’s convenience, we restate the following notions taken from [6]. Definition 3 (i) A stationary solution θ∗to Eq. (1.1) is stable if for any ε > 0 there exists δ > 0 such that if φ∈C(Hs) satisfies ∥φ−i(θ∗)∥C(Hs)< δ, then the solution θ(·;φ) to Eq. (1.1) exists for all t≥0 and satisfies ∥θ(t;φ)−θ∗∥Hs< ε for any t≥0. (ii) A stationary solution θ∗to Eq. (1.1) is attractive if there exists e δ > 0 such that if φ∈C(Hs) satisfies ∥φ−i(θ∗)∥C(Hs)<e δ, then the solution θ(·;φ) to Eq. (1.1) exists for all t≥0 and satisfies limt→∞ ∥θ(t;φ)−θ∗∥Hs= 0. (iii) A stationary solution θ∗to Eq. (1.1) is asymptotically stable if it is stable and attractive. 4.2 Local stability: Lyapunov function method In this subsection we use a direct approach to prove that if the stationary solution to Eq. (1.1) exists, then it is stable in Hswith the topology induced by Lp,p > 2 2α−1, under some additional assumptions. Theorem 4 Fix α∈(1 2,1),s≥2−2αand p > 2 2α−1. Consider f∈Hs−α∩Lp and G∈Lip0(Hs, Hs)∩Lip0(Lp, Lp). Let the delay forcing term gbe given by 22 Liang, Wang and Caraballo g(t, θt) = G(θ(t−ρ(t))), with ρ∈C1([0,∞),R+)such that ρ∗= supt≥0ρ′(t)< 1. Suppose (4.2) and 4κ(p−1) p≥pKβ1+Lg(p−(p−1)ρ∗) λα 1(1 −ρ∗)(4.16) hold true, where β1=(κ−λ−α 1Lg)−√∆ 2Υis as shown in Theorem 3 and Kis given later on. Then for any solution θ∗∈Hs+αto Eq. (4.1) (whose existence is guaranteed by Theorem 3), and any φ∈C(Hs)∩C(Lp), the solution to Eq. (1.1) satisfies that for all t≥0, ∥θ(t)−θ∗∥p Lp≤ ∥φ(0) −θ∗∥p Lpfor ρ(0) = 0, and ∥θ(t)−θ∗∥p Lp≤max {1,Lg 1−ρ∗}(∥φ(0) −θ∗∥p Lp+∥φ−i(θ∗)∥p Lp(−ρ(0),0;Lp)) for ρ(0) >0. Proof Let θbe the solution to Eq. (1.1) and θ∗be the solution to Eq. (4.1). Set w(t) = θ(t)−θ∗, we observe that d dtw(t) + κ(−∆)αw(t) + R⊥θ(t)·∇θ(t)−R⊥θ∗·∇θ∗ =G(θ(t−ρ(t))) −G(θ∗). Multiplying the above equation by |w(t)|p−1sgnw(t) with p > 2 2α−1, and taking the inner product in L2, we obtain 1 p d dt∥w(t)∥p Lp=−κ∫T2 (−∆)αw(t)|w(t)|p−1sgnw(t)dx −∫T2 (R⊥θ(t)·∇θ(t)−R⊥θ∗·∇θ∗)|w(t)|p−1sgnw(t)dx +∫T2 (G(θ(t−ρ(t))) −G(θ∗))|w(t)|p−1sgnw(t)dx := J1+J2+J3.(4.17) Using Lemma 3, we deduce that J1≤ −4κ(p−1) p2∫T2[(−∆)α 2(|w(t)|p 2)]2dx =−4κ(p−1) p2∥|w(t)|p 2∥2 Hα. (4.18) Stability of quasi-geostrophic equation 23 Since the quadratic nonlinear term is bilinear, by similar arguments to (3.3) we have J2=−∫T2R⊥θ(t)·∇w(t)|w(t)|p−1sgnw(t)dx −∫T2R⊥w(t)·∇θ∗|w(t)|p−1sgnw(t)dx =−∫T2R⊥w(t)·∇θ∗|w(t)|p−1sgnw(t)dx. Thanks to the H¨older inequality and Lemma 1, the term J2is bounded by J2≤ ∥R⊥w(t)∥L2p∥∇θ∗∥L2∥|w(t)∥p−1 L2p≤C(0,2p)∥w(t)∥p L2p∥∇θ∗∥L2 =C(0,2p)∥|w(t)|p 2∥2 L4∥θ∗∥H1≤K∥|w(t)|p 2∥2 Hα∥θ∗∥Hs+α,(4.19) with K=C(0,2p)M3M4, where M3and M4are the constants associated with the following Sobolev embedding inequalities: ∥|w(t)|p 2∥2 L4≤M3∥|w(t)|p 2∥2 Hαand ∥θ∗∥H1≤M4∥θ∗∥Hs+α. By the H¨older inequality, the Young inequality and the assumption on g, we have J3≤ ∥G(θ(t−ρ(t))) −G(θ∗)∥Lp∥w(t)∥p−1 Lp ≤Lg∥θ(t−ρ(t)) −θ∗∥Lp∥w(t)∥p−1 Lp ≤Lg p∥w(t−ρ(t))∥p Lp+λ−α 1Lg(p−1) p∥|w(t)|p 2∥2 Hα,(4.20) where we have used the following embedding inequality: ∥w(t)∥p Lp=∥|w(t)|p 2∥2 L2≤λ−α 1∥|w(t)|p 2∥2 Hα.(4.21) Note that under the assumptions of Theorem 3, the solution to Eq. (4.1) satisfies ∥θ∗∥Hs+α≤β1(see (4.15)), then we conclude from (4.17)-(4.20) that d dt∥w(t)∥p Lp≤(−4κ(p−1) p+pKβ1+λ−α 1Lg(p−1))∥|w(t)|p 2∥2 Hα +Lg∥w(t−ρ(t))∥p Lp.(4.22) Taking σ=r−ρ(r), we infer that Lg∫t 0∥w(r−ρ(r))∥p Lpdr =Lg∫t−ρ(t) −ρ(0) ∥w(σ)∥p Lp 1 1−ρ′dσ ≤Lg 1−ρ∗∫t −ρ(0) ∥w(σ)∥p Lpdσ. 24 Liang, Wang and Caraballo For the case ρ(0) >0, using the embedding inequality (4.21) again and integrating (4.22) over [0, t], we have ∥w(t)∥p Lp≤ ∥w(0)∥p Lp +(−4κ(p−1) p+pKβ1+λ−α 1Lg(p−1) + λ−α 1Lg 1−ρ∗)∫t 0∥|w(σ)|p 2∥2 Hαdσ +Lg 1−ρ∗∫0 −ρ(0) ∥w(σ)∥p Lpdσ, (4.23) which, together with (4.16), implies that ∥w(t)∥p Lp≤ ∥w(0)∥p Lp+Lg 1−ρ∗∫0 −ρ(0) ∥w(σ)∥p Lpdσ. For another case ρ(0) = 0, the assertion of this theorem follows by neglecting the last term in (4.23), and thus the proof is complete.  4.3 Local stability: Construction method of Lyapunov functionals In the previous subsection we have showed the stability of stationary solutions to Eq. (1.1) by the Lyapunov function method. However, sometimes we can construct Lyapunov functionals rather than Lyapunov functions. In this paragraph we will analyze the stability of stationary solutions to Eq. (1.1) by constructing appropriate Lyapunov functionals. Theorem 5 Fix α∈(1 2,1),s≥2−2αand p > 2 2α−1. Consider f∈Hs−α∩Lp and G∈Lip0(Hs, Hs)∩Lip0(Lp, Lp). Let the delay forcing term gbe given by g(t, θt) = G(θ(t−ρ(t))), with ρ∈C1([0,∞),R+)such that ρ∗= supt≥0ρ′(t)< 1. If the conditions (4.2) and 4κ(p−1) p≥pKβ1+2Lg λα 1 (p−1)p−1 p (1 −ρ∗)1 p (4.24) hold true, where β1and Kare as shown in Theorem 4, then the stationary solution to Eq. (1.1) is stable, i.e., for any φ∈C(Hs)∩C(Lp), the solution θ(·;φ)to Eq. (1.1) satisfies that for all t≥0, ∥θ(t)−θ∗∥p Lp≤γ∥φ−i(θ∗)∥p C(Lp),(4.25) where c=Lg((1 −ρ∗)(p−1))p−1 pand γ= 1 + cρ(0) 1−ρ∗. Proof We construct Θ: [0,∞)×C(Lp)→R+in the form Θ(t, φ) = ∥φ(0)∥p Lp+c 1−ρ∗∫0 −ρ(t)∥φ(σ)∥p Lpdσ, Stability of quasi-geostrophic equation 25 where c > 0 is a constant to be determined later on, such that Θis a Lyapunov functional. Denoting by Θ(t) = Θ(t, θt(·;φ)−θ∗), where θt(·;φ) is the solution to Eq. (1.1) with initial datum φand θ∗is the stationary solution to Eq. (1.1), we have Θ(t) = ∥θ(t)−θ∗∥p Lp+c 1−ρ∗∫t t−ρ(t)∥θ(σ)−θ∗∥p Lpdσ. (4.26) By using (1.1) we deduce from (4.26) that d dtΘ(t) = p∫T2 d dt(θ(t)−θ∗)|θ(t)−θ∗|p−1sgn(θ(t)−θ∗)dx +c 1−ρ∗∥θ(t)−θ∗∥p Lp−c(1 −ρ′(t)) 1−ρ∗∥θ(t−ρ(t)) −θ∗∥p Lp =−pκ ∫T2 (−∆)α(θ(t)−θ∗)|θ(t)−θ∗|p−1sgn(θ(t)−θ∗)dx −p∫T2 (R⊥θ(t)·∇θ(t)−R⊥θ∗·∇θ∗)|θ(t)−θ∗|p−1sgn(θ(t)−θ∗)dx +p∫T2 (G(θ(t−ρ(t))) −G(θ∗))|θ(t)−θ∗|p−1sgn(θ(t)−θ∗)dx +c 1−ρ∗∥θ(t)−θ∗∥p Lp−c(1 −ρ′(t)) 1−ρ∗∥θ(t−ρ(t)) −θ∗∥p Lp. Arguing as in (4.17)-(4.20), we conclude that d dtΘ(t)≤(−4κ(p−1) p+pKβ1)∥|θ(t)−θ∗|p 2∥2 Hα +pLg∥θ(t−ρ(t)) −θ∗∥Lp∥θ(t)−θ∗∥p−1 Lp +c 1−ρ∗∥θ(t)−θ∗∥p Lp−c∥θ(t−ρ(t)) −θ∗∥p Lp ≤(−4κ(p−1) p+pKβ1+λ−α 1((p−1) (Lg c1/p )p p−1 +c 1−ρ∗)) ×∥|θ(t)−θ∗|p 2∥2 Hα, where we have used the Young inequality and the following embedding inequality: ∥θ(t)−θ∗∥p Lp=∥|θ(t)−θ∗|p 2∥2 L2≤λ−α 1∥|θ(t)−θ∗|p 2∥2 Hα.(4.27) Minimizing the coefficient in brackets in the right-hand side, which is attained for c=Lg((1 −ρ∗)(p−1))p−1 p, we deduce that d dtΘ(t)≤(−4κ(p−1) p+pKβ1+2Lg λα 1 (p−1)p−1 p (1 −ρ∗)1 p)∥|θ(t)−θ∗|p 2∥2 Hα, 32 Liang, Wang and Caraballo 27. P. Mar´ın-Rubio, A.M. M´arquez-Dur´an, J. Real, Asymptotic behavior of solutions for a three dimensional system of globally modified Navier-Stokes equations with a locally Lipschitz delay term, Nonlinear Anal., 79, 68-79 (2013) 28. T.T. Medjo, Attractors for the multilayer quasi-geostrophic equations of the ocean with delays, Appl. Anal., 87(3), 325-347 (2008) 29. S. Murakami, Stability in functional-differential equations with infinite delay, Tohoku Math. J. (2), 37(4), 561-570 (1985) 30. C.J. Niche, G. Planas, Existence and decay of solutions to the dissipative quasigeostrophic equation with delays, Nonlinear Anal., 75(9), 3936-3950 (2012) 31. J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York (1987) 32. J. Ren, X. Ma, L2stability for solutions of subcritical quasi-geostrophic equation with large perturbations, Appl. Math. Comput., 218(9), 5337-5345 (2012) 33. J.C. Robinson, Infinite-Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge (2001) 34. M.E. Schonbek, T.P. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. Math. Anal., 35(2), 357-375 (2003) 35. E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, NJ: Princeton University Press (1970) 36. L. Wan, J. Duan, Exponential stability of the multi-layer quasi-geostrophic ocean model with delays, Nonlinear Anal., 71(3-4), 799-811 (2009)