Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.Dáng diệu tiệm cận nghiệm của một số phương trình và hệ phương trình Parapolic phi tuyến.
MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION ——————–o0o——————— TRAN THI QUYNH CHI ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF SOME NONLINEAR PARABOLIC EQUATIONS AND SYSTEMS Speciality: Differential and Integral Equations Code: 9460103 SUMMARY OF DOCTORAL DISSERTATION IN MATHEMATICS HA NOI-2023 The dissertation was written on the basis of the author’s research works carried at Hanoi National University of Education Supervisor: Prof PhD Cung The Anh Referee 1: Professor Nguyen Thieu Huy Hanoi University of Science and Technology Referee 2: Professor Le Van Hien Hanoi National University of Education Referee 3: Professor Vu Trong Luong University of Education - Vietnam National University, Hanoi This dissertation is presented to the examining committee at Hanoi National University of Education, 136 Xuan Thuy Road, Hanoi, Vietnam At the time of , 2023 Full-text of the dissertation is publicly available and can be accessed at: - The National Library of Vietnam - The Library of Hanoi National University of Education INTRODUCTION Motivation and history of the problem The parabolic partial differential equations appear in many physical and biological processes, such as in heat processes and diffusion processes, in population models in biology, etc (see [49, 56]) The study of these classes of equations has important meaning in science and technology That is why it has been attracting the attention of many scientists in the world After studying the correctness of the problem, it is very important to study the asymptotic behavior of the solution when the time goes out to infinity This is a practical undertaking since the solutions of partial differential equations often describe the states of real models Therefore, knowing the asymptotic behavior of the solutions allows us to understand and predict the development of the dynamics system in the future, then we can make appropriate assessments and adjustments to get the desired results To study the asymptotic behavior of the solution of parabolic partial differential equations, the corresponding dynamical systems are very complex because it is infinite-dimensional dynamical systems, they often uses the theory of the attractor The attractor of a dynamical system is a compact and invariant set that attracts all bounded sets and contains a lot of information about the asymptotic behavior of the dynamical system under consideration In recent years, the existence and properties of the attractor have been investigated for many classes of semi-linear parabolic equations, both in the non-degenerate and degenerate case However, most of the results on the attractor are obtained in the case of scalar parabolic equations (see [12, 17, 49, 56] and documents cited therein) The development of these results for the case of systems, which is much more difficult due to the interactions between the nonlinear terms in the system, has only made some progress in recent years (see [13, 17, 37, 38] ) This is a topical issue that has many meanings and attracts many attention of mathematicians around the world Beside studying the existence and properties of the attractor, the existence and stability of stationary solutions are also important issues The stationary solution of the system corresponds to the stationary state of the process under consideration and is therefore very important Therefore, the study of the existence and properties of the attractor for classes of nonlinear parabolic equations and systems still has many open problems and needs to be further studied That is the reason that we choose the above mentioned problems as the research content of the thesis "Asymptotic behavior of solution of some nonlinear parabolic equations and systems" Purpose of the thesis The thesis studies the wellposedness and the asymptotic behavior of solution for some classes of nonlinear parabolic equations and systems through the existence of the global attractors and pullback attractors In addition, the existence and exponential stability of stationary solutions in the case of system is also one of the main research purposes of the thesis Object and scope of the thesis • Research object: Studying some nonlinear parabolic equations and systems • Research scope: Studing the wellposedness and the asymptotic behavior when the time goes out to infinity for the three following equations: Content 1: The class of quasilinear degenerate parabolic equations with non-linearity of arbitrary order Content 2: The class of non-autonomous quasilinear degenerate parabolic equations with non-negative coefficients, varying on RN Content 3: The class of reaction-diffusion systems with exponential nonlinearities Research methods • To study the existence of solutions: Galerkin approximation method and compact method (see J.L Lions (2010), semigroups method (see A Pazy (1983)) • To study the existence of the attractors: Methods of the theory of infinitedimensional dynamical systems (see V.V Chepyzhov and M.I Vishik (2002); R Temam (1997)) Structure and results of the thesis The thesis consists of main chapters as follows: • Chapter presents some preliminaries • Chapter presents the wellposedness and the asymptotic behavior when the time goes out to infinity for a class of quasilinear degenerate parabolic equations containing p-Laplace weighted operator with nonlinearity of arbitrary order • Chapter presents the existence of the pullback attractor for a class of non-autonomous quasilinear degeneracy equations in the case lack of compactness of embedding with nonlinear polynominal growth • Chapter presents the wellposedness and the existence of the global attractors for a class of reaction-diffusion systems with exponential nonlinearity; the existence of a global attractor in the case of gradient system In addition, we also present the existence and exponential stability of stationary solutions to the problem Chapter PRELIMINARIES In this chapter, we present some preliminaries including: operators, some functional spaces, some additional results (the usual inequalities, the compactness methods, the weak form of the bounded convergent theorem) that is used to prove the main results of the thesis in the following chapters 1.1 Operators and function spaces 1.2 The weighed Sobolev spaces In the thesis, we used some Sobolev space involving weighted as follow: 1,p 1,p W0 (Ω, a), W0 (RN , σ) 1.3 Operators In this section, we introduce the operator classes studied in the problems of the thesis: The operator L p,a degenerates on a set of measure zero, operator L p,σ degenerates Cadiroli-Musina, Laplace operator 1.4 Summary of the global attractor In this section, we recall some of the concepts of dynamics, global attractors, and theorems about the existence of global attractors to be used in the thesis The content of this section is based on monographs by J C Robinson (2001) and R Temam (1997) 1.5 Summary of the pullback attractor In this section, we recall some of the concepts and results related to the pullback attractor theory based on the documents of: T Caraballo, G Łukasiewicz and J Real (2006); Y Li and C.K Zhong (2007); C.K Zhong, M.H Yang, and C.Y Sun (2006) 1.6 Some usual results In this section, we recall some of the primary but important inequalities as well as some important propositions and theorems that are frequently used in the thesis: Aubin-Lions-Simon compact lemma by F Boyer and P Fabrie (2013); Lemma 6.1 on the strong convergence of nonlinear functions by P G Geredeli (2015); Ehrling lemma by Robinson (2021), etc Chapter GLOBAL ATTRACTOR FOR THE CLASS OF QUASILINEAR DEGENERATE PARABOLIC EQUATION WITH NON-LINEARITY OF ARBITRARY ORDER In this paper we study the existence and long-time behavior of weak solutions to a class of quasilinear degenerate parabolic equations involving weighted p-Laplacian operators with a new class of nonlinearities First, we prove the existence and uniqueness of weak solutions by combining the compactness and monotone methods and the weak convergence techniques in Orlicz spaces Then, we prove the existence of global attractors by using the asymptotic a priori estimates method The contents of this chapter is written based on the paper [CT1] in the section of author’s works related to the thesis that has been published 2.1 Problem setting In this section, we consider the following quasilinear degenerate parabolic equations u t − div(a(x)|∇u| p−2 ∇u) + f (u) = g(x), u(x, t) = 0, u(x, 0) = u0 (x), x ∈ Ω, t > 0, x ∈ ∂ Ω, t > 0, (2.1) x ∈ Ω, where Ω is a bounded domain in RN (N ≥ 2) with smooth boundary ∂ Ω, ≤ p ≤ N , u0 ∈ L (Ω) given, the coefficient a(·), the nonlinearity f and the external force g satisfy the following conditions: (H1) The function a : Ω → R satisfies the following assumptions: a ∈ Lloc (Ω) and a(x) = for x ∈ Σ, and a(x) > for x ∈ Ω \ Σ, where Σ is a closed subset of Ω with meas(Σ) = Furthermore, we assume that Z d x < ∞ for some α ∈ (0, p); N α Ω [a(x)] (H2) f : R → R is a continuously differentiable function satisfying f (u)u ≥ −µu2 − c1 , (2.2) f (u) ≥ −`, (2.3) where c1 , `, µ are positive constants, and if p = then we assume furthermore that < µ < c0 with c0 is determined in (2.4) (H3) g ∈ L (Ω) 2.2 Some usual results In order to prove some results in this chapter, we use the following propositions Proposition 2.1 Assume that Ω is a bounded domain in RN (N ≥ 2) and a(·) satisfies (H1) Then the following embeddings hold: 1,p 1,β (i) W0 (Ω, a) ,→ W0 (Ω) continuously if ≤ β ≤ pN ; N +α 1,p (ii) W0 (Ω, a) ,→ L r (Ω) continuously if ≤ r ≤ pα∗ , where pα∗ = pN N −p+α 1,p (iii) W0 (Ω, a) ,→ L r (Ω) compactly if ≤ r < pα∗ Thanks to Proposition 2.1, there exists a best constant c0 such that c0 kuk2L (Ω) ≤ kuk2 1,p W0 (Ω,a) (2.4) Putting 1,p L p,a u = −div(a(x)|∇u| p−2 ∇u), u ∈ W0 (Ω, a) The following proposition, its proof is straightforward, gives some important properties of the operator L p,a 1,p Proposition 2.2 The operator L p,a maps W0 (Ω, a) into its dual W −1,q (Ω, a) Moreover, 1,p (i) L p,a is hemicontinuous, i.e., for all u, v, w ∈ W0 (Ω, a), the map λ 7→ 〈L p,a (u + λv), w〉 is continuous from R to R; (ii) L p,a is strongly monotone when p ≥ 2, i.e., 〈L p,a u − L p,a w, u − w〉 ≥ δku − wk 1,p p 1,p W0 (Ω,a) for all u, w ∈ W0 (Ω, a) Proposition 2.3 Assume that Ω is bounded domain in RN (N ≥ 2) and a(·) 1,p satisfies (H1) Then the embedding D(L p,a ) ,→ W0 (Ω, a) is compact 2.3 Existence and uniqueness of global weak solutions Denote Ω T = Ωx(0, T ) and let (p, p0 ) be conjugate, i.e., 1p + p0 = We give the definition of weak solutions to problem (2.1) Definition 2.1 A function u is called a weak solution of problem (2.1) on the interval (0, T ) if 1,p u ∈ L p (0, T ; W0 (Ω, a)) ∩ C([0, T ]; L (Ω)) du dt 0 ∈ L p (0, T ; W −1,p (Ω, a)) + L (Ω T ), u| t=0 = u0 a.e in Ω, f (u) ∈ L (Ω T ), and Z ΩT ∂u ∂t χ + a(x)|∇u| p−2 ∇u∇χ + f (u)χ − gχ d x d t = 0, 1,p for all test functions χ ∈ L p (0, T ; W0 (Ω, a) ∩ L ∞ (Ω)) The result of the existence and uniqueness of weak solutions to the problem (2.1) is presented in the following theorem Theorem 2.1 Under assumptions (H1) − (H3), for each u0 ∈ L (Ω) and T > given, problem (2.1) has a unique weak solution on (0, T ) Moreover, the mapping u0 7→ u(t) is continuous on L (Ω) 2.4 Existence of global attractors By Theorem 2.1, we can define a continuous nonlinear semigroup S(t) : L (Ω) → L (Ω), u0 7→ S(t)u0 := u(t), where u(t) is the unique weak solution to problem (2.1) with initial datum u0 The aim of this section is to prove the existence of global attractors in various spaces for the semigroup S(t) The rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in J.C Robinson (2001) Lemma 2.1 The semigroup {S(t)} t≥0 has a bounded absorbing set in L (Ω) 1,p Lemma 2.2 The semigroup {S(t)} t≥0 has a bounded absorbing set in W0 (Ω, a) 1,p From Lemma 2.1 and the compactness of the embedding W0 (Ω, a) ,→ L (Ω), we immediately obtain the following result Theorem 2.2 Assume that assumptions (H1) − (H3) are satisfied Then the semigroup {S(t)} t≥0 associated to (2.1) has a global attractor A L in L (Ω) Lemma 2.3 Assume that assumptions (H1) − (H3) hold Then there exists a bounded absorbing set for semigroup {S(t)} t≥0 in D(L p,a ) Theorem 2.3 Assume that assumptions (H1)−(H3) are satisfied Then the semi1,p group {S(t)} t≥0 associated to (2.1) has a global attractor A D(L p,a ) in W0 (Ω, a) Chapter PULLBACK ATTRACTOR FOR A CLASS OF NON-AUTONOMOUS QUASILINEAR DEGENERATE PARABOLIC EQUATIONS In this chapter, we study a class of non-autonomous quasilinear degenerate parabolic equations on the whole RN , N ≥ 3, under a new condition concerning a variable non-negative diffusivity σ(x) without restricting the limiting behavior of σ(·) at infinity We will prove the existence of pullback attractors in the 1,p space W0 (RN , σ) ∩ L q (RN ) ∩ L (RN ) for the process associated to problem The contents of this chapter is written based on the paper [CT2] in the section of author’s works related to the thesis that has been published 3.1 Problem setting In this chapter, we consider the following non-autonomous quasilinear degenerate parabolic equations with a variable, nonnegative coefficient in RN , N ≥ 3, ∂u ∂t − div(σ(x)|∇u| p−2 ∇u) + λ|u| p−2 u + f (u) = g(x, t), x ∈ RN , t > τ, u(x, τ) = uτ (x), x ∈ RN , (3.1) where λ > 0, ≤ p < N , uτ ∈ L (RN ) is given, the diffusion coefficient σ, the nonlinearity f , and the external force g satisfy some conditions specified later (F ) f : R → R is a C function satisfying α1 |u|q − β1 |u| p + γ1 |u|2 ≤ f (u)u ≤ α2 |u|q + β2 |u| p + γ2 |u|2 , f (u) ≥ −γ, 10 (3.2) (3.3) for some q ≥ 2, where αi , βi , γi , γ are positive constants with β1 < λ Rs Denote F (s) = f (τ)dτ, then α3 |u|q − β3 |u| p + γ3 |u|2 ≤ F (u) ≤ α4 |u|q + β4 |u| p + γ4 |u|2 , (3.4) where αi , βi , γi are positive constants (H ∞ ) σ is a non-negative measurable function such that σ ∈ L l1oc (RN ), and for some α ∈ (0, p), lim inf |x − z|−α σ(x) > for every z ∈ RN , x→z and σ satisfies one of the following two conditions: (i) there exists M0 > such that sup sup (ii) there exists M0 > such that sup R p m≥M0 m≤|x|≤ 2m m≥M0 σ(x) < ∞; 2p−2 p m≤|x|≤ 2m |σ(x)| p−2 d x < ∞ 1,2 (G ) g ∈ Wl oc (R; L (RN )) satisfies the following conditions for all t ∈ R: Zt eγ1 s kg(s)k2L (RN ) + kg (s)k2L (RN ) ds < ∞, (3.5) −∞ and Zt Z lim sup m→∞ eγ1 s |g(x, s)|2 d x ds = (3.6) −∞ |x|≥m 3.2 Existence of pullback attractors We first give the definition of a weak solution 1,p Definition 3.1 A function u : (τ, +∞) → W0 (RN , σ)∩L q (RN )∩L (RN ) is said 1,p to be a weak solution of (3.1) if u ∈ L p (τ, T ; W0 (RN , σ)) ∩ L q (τ, T ; L q (RN )) ∩ L ∞ (τ, T ; L (RN )) for all T > τ, and Z tZ (u(t), w) L (RN ) + + σ|∇u| τ ∇u∇wd x d t + λ Z tZ |u| p−2 uwd x d t τ RN Z tZ τ p−2 f (u)wd x d t = (uτ , w) L (RN ) + RN 11 Z RN t τ (g, w) L (RN ) d t, ∀t > τ, 1,p for all test functions w ∈ W0 (RN , σ) ∩ L q (RN ) ∩ L (RN ) Theorem 3.1 Under conditions (F ) − (H ∞ ) − (G ), for any τ ∈ R, T > τ, and uτ ∈ L (RN ) given, problem (3.1) has a unique weak solution u Moreover, the following estimate holds ku(t)k2L (RN ) ≤C e −γ1 (t−τ) |uτ |22 +e −γ1 t Z t e γ1 s |g(s)|22 ds (3.7) −∞ Due to Theorem 3.1, problem (3.1) defines a process 1,p U(t, τ) : L (RN ) → W0 (RN , σ) ∩ L q (RN ) ∩ L (RN ), where U(t, τ)uτ is the unique weak solution of (3.1) with the initial datum uτ Lemma 3.1 Suppose (F ) − (H ∞ ) − (G ) hold Then the process {U(t, τ)} associated to (3.1) has a family of bounded pullback absorbing sets Bˆ = {B(t) : 1,p t ∈ R} in W0 (RN , σ) ∩ L q (RN ) ∩ L (RN ), that is, for every t ∈ R and bounded subset B in L (RN ), there exists a number r0 = r0 (t) > and τ0 = τ0 (t, B) < t such that for all τ ≤ τ0 , uτ ∈ B, we have ku(t)k p q 1,p W0 (RN ,σ) + ku(t)k L q (RN ) + ku(t)k2L (RN ) ≤ r0 (t) We now derive uniform estimates of the derivatives of solutions in time Lemma 3.2 Suppose (F ) − (H ∞ ) − (G ) hold Then for any t ∈ R and any bounded subset B in L (RN ), there exist r1 = r1 (t) > and τ0 = τ0 (t, B) < t such that ku t (t)k2L (RN ) ≤ r1 (t), for all τ ≤ τ0 , uτ ∈ B Lemma 3.3 Suppose (F ) − (H ∞ ) − (G ) hold Then the process U(t, τ) has a family of bounded pullback absorbing sets in L 2p−2 (RN ), i.e., for every t ∈ R and any bounded subset B in L (RN ), there exist r2 (t) > and τ0 = τ0 (t, B) < t such that ku(t)k L 2p−2 (RN ) ≤ r2 (t), for all τ ≤ τ0 , uτ ∈ B 12 From Lemma 3.1 we see that the process {U(t, τ)} maps a compact set of 1,p 1,p W0 (RN , σ) ∩ L q (RN ) ∩ L (RN ) to be a bounded set of W0 (RN , σ) ∩ L q (RN ) ∩ L (RN ) for any t ≥ τ, and thus by Proposition 1.2, the process {U(t, τ)} is 1,p norm-to-weak continuous in W0 (RN , σ) ∩ L q (RN ) ∩ L (RN ) Since {U(t, τ)} 1,p has a family of pullback absorbing sets in W0 (RN , σ) ∩ L q (RN ) ∩ L (RN ), in order to prove the existence of a pullback attractor, we need only check that {U(t, τ)} is pullback asymptotically compact 3.2.1 Existence of a pullback attractor in L (RN ) Lemma 3.4 Suppose (F ) − (H ∞ ) − (G ) hold Then for any η > 0, t ∈ R, and any bounded subset B ⊂ L (RN ), there exist τ0 = τ0 (η, B, t) < t and M = M (η, B, t) > such that for all τ ≤ τ0 and m ≥ M , Z |u(x, t)|2 d x ≤ η, |x|≥m where u is the weak solution of (3.1) subject to the initial condition u(τ) = uτ ∈ B Lemma 3.5 Suppose (F ) − (H ∞ ) − (G ) hold Then the process {U(t, τ)} is pullback asymptotically compact in L (RN ), that is, for any t ∈ R, any bounded subset B ⊂ L (RN ), any sequences τn → −∞, and any sequence {x n } ⊂ B, the sequence {U(t, τn )x n } has a convergent subsequence in L (RN ) We are now ready to prove the existence of a pullback attractor for the process U(t, τ) in L (RN ) Theorem 3.2 Suppose (F ) − (H ∞ ) − (G ) hold Then the process {U(t, τ)} associated to problem (3.1) has a pullback attractor Aˆ2 in L (RN ) 13 3.2.2 Existence of a pullback attractor in L q (RN ) Lemma 3.6 Let {U(t, τ)} be a norm-to-weak continuous process in L q (RN ) and L (RN ), and let {U(t, τ)} satisfy the following two conditions: (i) {U(t, τ)} is pullback asymptotically compact in L (RN ); (ii) for any " > 0, any bounded subset B ∈ L (RN ), there exist constants L(", B) and τ0 (", B) ≤ t such that Z RN (|U(t,τ)uτ |≥L) |U(t, τ)uτ | q 1 q < ", for any τ ≤ τ0 and uτ ∈ B Then {U(t, τ)} is pullback asymptotically compact in L q (RN ) Theorem 3.3 Suppose (F ) − (H ∞ ) − (G ) hold Then the process {U(t, τ)} associated to problem (3.1) has a pullback attractor Aˆq in L q (RN ) 1,p 3.2.3 Existence of a pullback attractor in W0 (RN , σ) 1,p Proposition 3.1 The operator L p,σ maps W0 (RN , σ) into its dual W −1,p (RN , σ) Moreover, 1,p L p,σ is hemicontinuous, i.e., for all u, v, w ∈ W0 (RN , σ), the map λ 7→ 〈L p,σ (u + λv), w〉 is continuous from R to R L p,σ is strongly monotone when p ≥ 2, that is, there exists δ > such that: 〈L p,σ u − L p,σ w, u − w〉 ≥ δku − wk 1,p p 1,p W0 (RN ,σ) , for all u, w ∈ W0 (RN , σ) Lemma 3.7 Suppose (F ) − (H ∞ ) − (G ) hold Then the process {U(t, τ)} is 1,p pullback asymptotically compact in W0 (RN , σ) From Lemmas 3.1 and 3.7, we immediately get the following main theorem Theorem 3.4 Suppose (F ) − (H ∞ ) − (G ) hold Then the process {U(t, τ)} 1,p associated to problem (3.1) has a pullback attractor AˆW 1,p in W0 (RN , σ) 14 Chapter EXISTENCE AND ASYMTOTIC BEHAVIOR OF SOLUTIONS TO A CLASS OF REACTION-DIFFUSION SYSTEMS WITH EXPONENTIAL NONLINEARITIES In this chapter, we study the wellposedness and the asymptotic behavior for a class of reaction-diffusion systems in the case of bounded domains and exponential nonlinearities We prove the existence of a global attractor for the 2 semigroup associated to the above system in the space L (Ω) and prove the 2 existence of a global attractor in the space H01 (Ω) in the case of gradient system Moreover, we also prove the existence and exponential stability of stationary solution to the problem The contents of this chapter is written based on the paper [CT3] in the section of author’s works related to the thesis that has been published 4.1 Problem setting and statement of the main result In this chapter, we study the asymptotic behavior of the solutions for the following reaction-diffusion system in a bounded domain Ω ⊂ Rn , n ≥ 1, with smooth boundary ∂ Ω: ∂ u − ∆u + f (u, v) = g1 (x), ∂t ∂v − ∆v + h(u, v) = g2 (x), ∂t u(x, t) = 0, v(x, t) = 0, u(x, 0) = u0 , v(x, 0) = v0 , x ∈ Ω, t > 0, x ∈ Ω, t > 0, (4.1) x ∈ ∂ Ω, t > 0, x ∈ Ω We assume that nonlinearities f , h and external forces g1 , g2 satisfy the following conditions: 15 (H1) f , h : R2 → R are continuous functions satisfying f (u, v) + h(u, v) (u + v) ≥ −µ(u + v)2 − C0 , f (u, v) − h(u, v) (u − v) ≥ −µ(u − v)2 − C0 , (4.2) (4.3) f (u, v)h(u, v)uv ≥ 0, ` w1 + w22 , fu0 (u, v)w1 + f v0 (u, v)w2 w1 ≥ − ` h0u (u, v)w1 + h0v (u, v)w2 w2 ≥ − w1 + w22 , (4.4) (4.5) (4.6) for all u, v, w1 , w2 ∈ R Here, ` and C0 are positive constants, < µ < λ1 , with λ1 is the first eigenvalue of the operator Au = −∆u in H01 (Ω) (H2) g1 ∈ L (Ω), g2 ∈ L (Ω) The aim of this chapter is to prove the following result 4.2 Existence and uniqueness of weak solutions Definition 4.1 The pair of functions (u(t), v(t)) is called a weak solution of problem (4.1) on [0, T ] if u, v ∈ C([0, T ]; H) ∩ L (0, T ; V ), du d v , ∈ L (0, T ; H −1 (Ω) + L (Ω)), dt dt u(x, 0) = u0 (x), v(x, 0) = v0 (x) for a.e x ∈ Ω, and ZZ u t ϕd x d t + QT ZZ QT ZZ ∇u.∇ϕd x d t + ZZ QT vt ϕd x d t + ZZ f (u, v)ϕd x d t = ZZ QT ∇v.∇ϕd x d t + ZZ QT QT ∞ g1 ϕd x d t, QT h(u, v)ϕd x d t = ZZ g2 ϕd x d t, QT for all test functions ϕ ∈ C0∞ ([0, T ]; V ∩ L (Ω)) Theorem 4.1 Assume that the conditions (H1)-(H2) hold Then for any u0 , v0 ∈ H and T > given, problem (4.1) has a unique global weak solution (u, v) on [0, T ], and the weak solution depends continuously on the initial data 16 4.3 Existence of a global attractor in (L (Ω))2 By Theorem 4.1, we can define the semigroup S(t) : (L (Ω))2 → (L (Ω))2 associated with problem (4.1) as follows S(t)(u0 , v0 ) = (u(t), v(t)), where (u, v) is the unique weak solution of (4.1) with initial datum (u0 , v0 ) Proposition 4.1 The semigroup {S(t)} t≥0 has a bounded absorbing set in (L (Ω))2 Proposition 4.2 The semigroup {S(t)} t≥0 has a bounded absorbing set in (H01 (Ω))2 Because the embedding H01 (Ω) ,→,→ L (Ω) is compact Therefore, we have the following important theorem Theorem 4.2 Let {S(t)} t≥0 be the semigroup in (L (Ω))2 associated with problem (4.1) Then under assumptions (H1)-(H2), {S(t)} t≥0 has a global attractor in (L (Ω))2 4.4 Global attractor for the gradient system In this section, we study the system (4.1) in the case of gradient system We assume that f and h satisfy (H1bis) f and h satisfy (H1) and moreover, there exists µ > such that f (u, v)u + h(u, v)v ≥ F (u, v) − µ (u2 + v ) − µ C0 F (u, v) ≥ − (u2 + v ) − , 2 C0 , (4.7) (4.8) where F (u, v) is the function satisfying Fu = f , F v = h By the above conditions, the system (4.1) becomes a gradient system Then we get a global attractor in the space (H01 (Ω))2 17 Proposition 4.3 Under assumptions (H1bis)-(H2), the semigroup {S(t)} t≥0 has a bounded absorbing set in (H (Ω) ∩ H01 (Ω))2 Combining Propositions 4.2 and 4.3, by the embebding H (Ω)∩H01 (Ω) ,→,→ H01 (Ω) is compact, we get the following result immediately Theorem 4.3 Let {S(t)} t≥0 be a semigroup in (L (Ω))2 associated with problem (4.1) Then under assumptions (H1bis)-(H2), {S(t)} t≥0 has a global attractor in (H01 (Ω))2 4.5 Existence and exponential stability of stationary solutions A weak stationary solution to problem (4.1) is an element (u∞ , v∞ ) ∈ (H01 (Ω))2 such that Z Ω and Z Ω ∇u∞ · ∇ϕd x + ∇v∞ · ∇ψd x + Z Ω Z Ω f (u∞ , v∞ )ϕd x = h(u∞ , v∞ )ψd x = Z g1 ϕd x, Ω Z g2 ψd x, Ω for all test functions ϕ, ψ ∈ H01 (Ω) ∩ L ∞ (Ω) Theorem 4.4 Under hypotheses (H1)- (H2) Then the problem (4.1) has at least one weak stationary solution (u∞ , v∞ ) satisfying ku∞ k2L (Ω) + kv∞ k2L (Ω) ≤ η(τ0 ), where η(τ0 ) = λ1 (4k1 τ0 + k2 ) 4τ0 (k3 − τ0 ) (4.9) , with p τ0 = k22 + 4k1 k2 k3 − k2 4k1 for k1 = C0 |Ω|, k2 = kg1 k2L (Ω) +kg2 k2L (Ω) , k3 = λ1 −µ Moreover, if the following condition holds ` < λ1 , 18 (4.10) then the stationary solution (u∞ , v∞ ) of (4.1) is unique and exponentially stable, i.e for any weak solution (u, v) of (4.1), we have ku(t) − u∞ k2L (Ω) + kv(t) − v∞ k2L (Ω) ≤ ku(0) − u∞ k2L (Ω) + kv(0) − v∞ k2L (Ω) e−2(λ1 −`)t , (4.11) for all t ≥ 4.6 Examples In this section, we give some examples of the nonlinearities f , h which satisfy conditions (H1)-(H2) in this paper Example 4.1 Consider the following system in the domain Ω × R+ , where Ω is a bounded domain in Rn ∂u − ∆u + u e2u + ln(1 + v ) = g1 (x), ∂t ∂ v − ∆v + v e2v + ln(1 + u2 ) = g2 (x), ∂t where f (u, v) = u e2u + ln(1 + v ) and h(u, v) = v e2v + ln(1 + u2 ) , the functions g1 and g2 are in L (Ω) Example 4.2 Consider the following system in the domain Ω × R+ ∂u − ∆u + u(e2u + |v|α ) = g1 (x), ∂t ∂ v − ∆v + v(e2v + |u|α ) = g2 (x), ∂t where f (u, v) = u(e2u + |v|α ) and h(u, v) = v(e2v + |u|α ), with ≤ α < 2, the 2 functions g1 and g2 are in L (Ω) 19 CONCLUSION AND RECOMMENDATION Results of the thesis The results of the thesis include: • Proving the wellposedness and the existence of the global attractor for a class of quasilinear degenerative parabolic equation containing p-Laplace weighted operator with nonlinearity of arbitrary order • Proving the existence of the pullback attractor for a class of non-autonomous quasilinear degenerate parabolic equation in the non-compact case with the polynomial growth of the nonlinear function • Proving the wellposedness, the existence of the global attractor, the existence and exponential stability of stationary solutions for reaction-diffusion systems of with exponential nonlinear function; in particular, in gradient system, the smoothness of the global attractor has been shown Recommendations for future works Some open issues that need to be further studied are: • Extending the results of Chapter in case the domain Ω is not bounded In this case, the problem will become much more complicated because of the lack of compactness of Sobolev type embedding • Studying the important properties of the pullback attractor obtained in Chapter 3, such as finiteness of its fractal dimension, the continuous dependence on the parameters, for example, on the diffusion coefficient σ or on the exponent p (as p → 2), Another interesting question is to investigate the existence of solutions and of pullback attractors for nonautonomous quasilinear degenerate parabolic equations in unbounded domains with nonlinearity of arbitrary • Researching and developing the results of Chapter for degenerate systems of Caldiroli and Musina type or containing a degenerate diffusion coefficient a(x) on sets with zero measure 21 AUTHOR’S WORKS RELATED TO THE THESIS THAT HAVE BEEN PUBLISHED [CT1].Tran Thi Quynh Chi, Le Thi Thuy and Nguyen Xuan Tu (2021), Global attractor for a class of quasilinear degenerate parabolic equations with non-linearity of arbitrary order, Communications of the Korean Mathematical Society, Vol 36, No 3, 447-463 [CT2].Tran Thi Quynh Chi, Le Thi Thuy and Nguyen Xuan Tu (2021), Dynamics of non-autonomous quasilinear degenerate parabolic equations: the non-compact case, Acta Mathematica Vietnamica, Vol 46, 579-598 [CT3].Cung The Anh, Tran Thi Quynh Chi and Vu Manh Toi (2022), Existence and asymptotic behavior of solutions to a class of reactiondifusion systems with exponential nonliniearities, submitted Results of the thesis have been reported at: Seminar of Department of Analysis, Faculty of Mathematics, Hanoi National University of Eduacation