phương trình vi phân và tích phân tt tiếng anh

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION LE TRAN TINH ON SOME CLASSES OF NONLOCAL PARABOLIC EQUATIONS SUMMARY OF DOCTORAL DISSERTATION IN MATHEMATICS Speciality: Speciality Code: Differential and Integral Equations 46 01 03 HA NOI, 2020 This dissertation has been written at Hanoi National University of Education Supervisor: Prof Dr Cung The Anh Referee 1: Prof Dr Sc Nguyen Minh Tri Institute of Mathematics, Vietnam Academy of Science and Technology Referee 2: Assoc Prof Dr Nguyen Sinh Bay Thuongmai University Referee 3: Assoc Prof Dr Le Van Hien Hanoi National University of Education The dissertation shall be defended at the University level Thesis Assessment Council at Hanoi National University of Education on This dissertation can be found in: - The National Library of Vietnam; - Library of Hanoi National University of Education INTRODUCTION Motivation and overview of researching issues Analysis of diffusion phenomena appearing naturally in different domains such as physics, biology, economics, engineering etc, leads us to study partial differential equations The last twenty years we have seen great developments in the theory of local and nonlocal equations, especially, the theory of parabolic equations Local equations are relations between the values of an unknown function and its derivatives of different orders and in order to check it holds at a particular point, one needs to known only the values of the function in an arbitrarily small neighborhood, so that all derivatives can be computed whereas nonlocal equations are a relation for which the opposite happens In order to check whether a nonlocal equation holds at a point, information about the values of the function far from that point is needed Roughly speaking, we can understand a nonlocal equation whose output or value depends on the whole domain of the input or argument This characteristic is usually translated in the applications as phenomena that involve, for instance, the interaction of bacteria, economic agents, layered materials and so forth, whose individual or local reaction to an external force depends on the reaction of all the other components of the system The nonlocality in the equation can have different forms such as nonlocal source terms (see Y Chen and M Wang (2009), P Souplet (1998)), nonlocal boundary conditions (see C Mu et al (2007, 2010), H M Yin (2004)), and nonlocal diffusions (see L Caffarelli (2012), C G Gal and M Warma (2016), N Pan et al (2017), P Pucci et al (2017), M Xiang et al (2018)) They can be in space or in time or in both time and space The most common one is perhaps the nonlocal diffusions Because of nonlocal properties which generates a lot of difficulties, for instance, the uniqueness and regularity of weak solutions cannot be guaranteed, and we will encounter in analysis of the problem So, we need to have powerful methods That is why studying nonlocal parabolic equations is a topical issue We now recall some recent important results related to the existence and qualitative properties of solutions to nonlocal parabolic problems with the nonlocal diffusions which are involving the content of my dissertation First, we consider the class of nonlocal parabolic problems involving Laplacian operator which is nonlocal in the sense that the diffusion coefficient is determined by a global quantity These problems arise in various physical situations For instance, when we study questions related with a culture of bacterias, it could describe the population of these bacterias subject to spreading, where the diffusion coefficient is supposed to depend on the entire population in the domain rather than on the local density, that is, the measurement are not made at a point but represent an average in a neighbourhood of a point This equation also appears in the study of heat propagation or propagation of mutant genes or in epidemic theory or in mechanics with nonlinear vibrations of beams We can list some results in recent years as M Chipot and B Lovat (1997, 1999), A S Ackleh and L Ke (2000), F J S A Corrˆea et al (2004), S Zheng and M Chipot (2005), S B de Menezes (2006), C A Raposo et al (2008), A A Ovono (2010), J Simsen and J Ferreira (2014), R J Robalo et al (2014), T Caraballo et al.(2015, 2016), R M P Almeida et al (2016), Y Han and Q Li (2018) The second class of nonlocal parabolic problems got a lot of attention is nonlocal p-Laplace equations This type of nonlocal problems is also wide applications in both physics and biology For example, these problems can be used to describe the motion of a nonstationary fluid or gas in a nonhomogeneous and anisotropic medium, and the nonlocal term a has a meaning as a possible change in the global state of the fluid or gas caused by its motion in the considered medium They can be regarded as a general form of population model or a nonstationary model of fluid We can list some results related to this class in recent years as M Chipot and T Savitska (2014, 2015), T Caraballo et al (2017, 2018), J Li and Y Han (2019), Y Fu and M Xiang (2019) Finally, we consider some classes of nonlocal problems involving fractional diffusion operators In the last couple of decades, the fractional differential models have become a powerful tool for modeling challenging phenomena including anomalous transport, long-range interactions, or from local to nonlocal dynamics, which cannot be described properly by integer-order partial differential equations The interest in studying these nonlocal problems relies not only on mathematical purposes but also on their significance in real models (see L Caffarelli (2012), S Duo et al (2019), N Laskin (2000)) Importantly, fractional diffusion operators appear in the treatment of reaction diffusion equations in C.G Gal and M Warma (2016) which get more our interest We can list some results in recent years as K Bogdan et al (2003), Z Q Chen et al (2003), M Warma et al (2015, 2016), C Zhang et al (2018), B Zhang et al (2017, 2018) As we know that understanding the asymptotic behavior of dynamical systems is one of the most important problems of modern mathematics One way to attack this problem for dissipative dynamical systems is to consider its global attractors A first question is to study the existence of a global attractor Once a global attractor is obtained, a next natural question is to study the most important properties of the global attractor, such as dimension, dependence on parameters, regularity of the attractor, determining modes, etc For the above analysis, we see that there are many open problems involving nonlocal equations via studying the global attractors of associated semigroups, for instance, ❼ The asymptotic behavior of solutions to nonlocal parabolic equations with differ- ent diffusive operators and more general nonlinear terms ❼ The asymptotic behavior of solutions to nonlocal nonlinear coupled systems with different diffusive operators and more general nonlinear terms This is our motivation to choose the topic “On some classes of nonlocal parabolic equations” as my dissertation Objects and Objectives In this dissertation, we will study the following problems: (P1) Global attractors for nonlocal parabolic equations with a new class of nonlinearities (P2) Long-time behavior of solutions to a nonlocal quasilinear parabolic equation (P3) Global attractors for nonlocal parabolic equations involving the fractional Laplacian and the regional fractional Laplacian with a new class of nonlinearities The objectives of this dissertation are to study the asymptotic behavior of solutions of these nonlocal problems via existence of (its finite dimensional) global attractors, and the existence and exponential stability of stationary solutions 3 The structure and results of the dissertation Beside introduction, conclusion, author’s works related to the dissertation and references, the dissertation includes chapters: Chapter is devoted to recall some basic concepts and recapitulate some results about function spaces, existence of global attractors in partial differential equations, operators, and some auxiliary results Chapter is devoted to study the global attractors for nonlocal parabolic equations with a new class of nonlinearities Chapter is devoted to investigate the long-time behavior of solutions to a nonlocal p-Laplacian equation Chapter is devoted to study the global attractors for nonlocal parabolic equations involving the fractional Laplacian and the regional fractional Laplacian with a new class of nonlinearities The results obtained in Chapters 2, and are researching results for the problems (P1), (P2) and (P3) respectively Chapter and Chapter are based on the papers [CT1], [CT2] in the List of Publications which were published in the journals Journal of the Korean Mathematical Society and Communications of the Korean Mathematical Society, respectively The results of Chapter is the content of the work [CT3] in the List of Submitted Papers Chapter PRELIMINARIES AND AUXILIARY RESULTS In this chapter, we recall some basic concepts and recapitulate some results about function spaces, existence of global attractors in partial differential equations, operators, and some auxiliary results Function spaces ❼ Banach and Hilbert spaces, some types of convergence ❼ Lp , nonnegative integer order Sobolev spaces, fractional order Sobolev spaces Especially, continuous and compact imbeddings, and inequalities ❼ Bochner spaces Global attractors in partial differential equations: Review general theory about the existence of (finite dimensional) global attractors for semigroups generated from partial differential equations Operators: Laplace operator, p-Laplace operator, fractional Laplacian operator, regional Laplacian operator and their spectrum Some auxiliary results: Young’s inequality with , Gronwall’s inequality, Uniform Gronwall’s inequality Chapter GLOBAL ATTRACTORS FOR NONLOCAL PARABOLIC EQUATIONS WITH A NEW CLASS OF NONLINEARITIES In this chapter, we will consider a class of nonlocal parabolic equations in bounded domains with Dirichlet boundary conditions and a new class of nonlinearity which is no restriction imposed on the upper growth of the nonlinearities We first prove the existence and uniqueness of weak solutions by using the compactness method Then we study the existence and fractal dimension estimates of the global attractor for the associated continuous semigroup generated We also prove the existence of stationary solutions and give a sufficient condition for the uniqueness and global exponential stability of the stationary solution The content of this chapter is based on the work [CT1] in the List of Publications 2.1 Setting problem Let Ω be a bounded smooth domain in RN (N ≥ 1) We consider the following nonlocal nonlinear parabolic equation     ut − a(l(u))∆u + f (u) = g(x), x ∈ Ω, t > 0,    u(x, t) = 0, x ∈ ∂Ω, t > 0,      u(x, 0) = u0 (x), x ∈ Ω (2.1) The nonlinearity f , the external force g and the diffusion coefficient a satisfy (H1) a ∈ C(R, R+ ) is Lipschitz continuous, i.e., there exists a positive constant L such that |a(t) − a(s)| ≤ L|t − s|, ∀t, s ∈ R, and a(·) is bounded, i.e., there are two positive constants m, M such that < m ≤ a(t) ≤ M, ∀t ∈ R Moreover, suppose that a depends upon a continuous linear functional l(u) on L2 (Ω), i.e., a = a(l(u)), with l : L2 (Ω) → R is defined by l(u) = φ(x)u(x)dx, Ω where φ(·) is a given function in L2 (Ω) (H2) f : R → R is a continuously differentiable function satisfying f (u)u ≥ −µu2 − c1 , f (u) ≥ −α, where c1 , α are two positive constants, < µ < mλ1 with λ1 > is the first eigenvalue of the operator (−∆, H01 (Ω)) (H3) g ∈ L2 (Ω) 2.2 Existence and uniqueness of weak solutions Definition 2.2.1 A weak solution to (2.1) on the interval (0, T ) is a function u ∈ L2 (0, T ; H01 (Ω)) ∩ C([0, T ]; L2 (Ω)) such that du dt ∈ L2 (0, T ; H −1 (Ω)) + L1 (ΩT ), f (u) ∈ L1 (ΩT ), u(0) = u0 , and d u, v + a(l(u))((u, v)) + f (u), v = (g, v), dt for all v ∈ H01 (Ω) ∩ L∞ (Ω) and for a.e t ∈ (0, T ), where ΩT = Ω × (0, T ) We have the existence and uniqueness of weak solutions as the following theorem Theorem 2.2.2 Let u0 ∈ L2 (Ω) and T > be given Assume (H1), (H2) and (H3) hold Then problem (2.1) has a unique weak solution u on the interval (0, T ) Moreover, the mapping u0 → u(t) is continuous on L2 (Ω), that is, the solution depends continuously on the initial data 2.3 Existence of a global attractor Thanks to Theorem 2.2.2, we can define a continuous (nonlinear) semigroup S(t) : L2 (Ω) → L2 (Ω) associated to problem (2.1) as follows S(t)u0 := u(t), where u(·) is the unique global weak solution of (2.1) with the initial datum u0 We will prove that the semigroup S(t) has a compact global attractor A Lemma 2.3.1 The semigroup {S(t)}t≥0 has a bounded absorbing set in L2 (Ω) Lemma 2.3.2 The semigroup {S(t)}t≥0 has a bounded absorbing set in H01 (Ω) As a direct consequence of Lemma 2.3.2 and the compactness of the embedding H01 (Ω) → L2 (Ω), we get the following result Theorem 2.3.3 Suppose that the hypotheses (H1), (H2) and (H3) hold Then the semigroup S(t) generated by problem (2.1) has a connected compact global attractor A in L2 (Ω) To study regularity of the global attractor, we suppose (H1bis) a ∈ C(R, R+ ) is continuously differentiable and satisfies condition (H1) Lemma 2.3.4 Under the hypotheses (H1bis), (H2) and (H3), the semigroup {S(t)}t≥0 has a bounded absorbing set in H (Ω) ∩ H01 (Ω) Due to the compactness of the embedding H (Ω) → H01 (Ω), we get Theorem 2.3.5 Suppose that the hypotheses (H1bis), (H2) and (H3) hold Then the semigroup S(t) generated by problem (2.1) has a connected compact global attractor A in H01 (Ω) 2.4 Fractal dimension estimates of the global attractor We assume the nonlinearity f and the external force g satisfy stronger conditions: (H2bis) f satisfies the condition (H2) and there exists s0 > such that f (s) ≥ g L∞ (Ω) if s ≥ s0 , f (s) ≤ g L∞ (Ω) if s ≤ −s0 (H3bis) g ∈ L∞ (Ω) Lemma 2.4.1 Assume that (H1), (H2bis), and (H3bis) hold Then the global attractor A of problem (2.1) is bounded in L∞ (Ω) The following theorem is the main result of this section Theorem 2.4.2 Assume that (H1), (H2bis), and (H3bis) hold Then the global attractor A of problem (2.1) has a finite fractal dimension in L2 (Ω), namely, 9eC 1−δ dimf A ≤ q ln 2.5 ln 1+δ −1 Existence and exponential stability of stationary solutions A weak stationary solution to problem (2.1) is an element u∗ ∈ H01 (Ω) such that ∇u∗ · ∇vdx + a(l(u∗ )) Ω f (u∗ )vdx = Ω gvdx, Ω for all test functions v ∈ H01 (Ω) ∩ L∞ (Ω) Theorem 2.5.1 Under hypotheses (H1), (H2) and (H3), problem (2.1) has at least one weak stationary solution u∗ satisfying u∗ ≤ (τ0 ), where (τ ) = λ1 (4k1 τ + k2 ) , 4τ (k3 − τ ) with τ0 = k22 + 4k1 k2 k3 − k2 , for k1 = c1 |Ω|, k2 = |g|22 , k3 = mλ1 − µ 4k1 Moreover, if the following condition holds mλ1 > α + L2 |φ|22 (τ0 ) λ1 , then for any weak solution u of (2.1), we have |u(t) − u∗ |22 ≤ |u(0) − u∗ |22 e−2Γ0 t for all t > 0, where Γ0 = mλ1 − α − L2 |φ|22 (τ0 )λ1 > That is, the weak stationary solution of (2.1) is unique and exponentially stable Remark 2.5.2 The absence of the upper growth condition on the nonlinearity and the nonlocality cause a lot of difficulties in our proofs However, our results have improved and extended some results as follows: ❼ In case of a ≡ and the nonlinear term satisfies sublinear or polynomial growth, we receive the well-known results of the reaction-diffusion equations with the Dirichlet boundary condition ❼ With nonlinear term satisfying polynomial growth and without source term, we recover the results of J Simsen and J Ferreira (2014) ❼ Our results have improved and extended the above results since this class of non- linear term covers both the above classes of nonlocalities and even the exponential nonlinearities 10 Chapter LONG-TIME BEHAVIOR OF SOLUTIONS TO A NONLOCAL QUASILINEAR PARABOLIC EQUATION In this chapter, we consider a class of nonlinear nonlocal parabolic equations involving p-Laplacian operator where the nonlocal quantity is present in the diffusion coefficient which depends on Lp -norm of the gradient and the nonlinear term is of polynomial type We first prove the existence and uniqueness of weak solutions by combining the compactness method and the monotonicity method Then we study the existence of global attractors in various spaces for the associated continuous semigroup Finally, we investigate the existence and exponential stability of the weak stationary solutions The content of this chapter is based on the work [CT2] in the List of Publications 3.1 Setting problem Let Ω ⊂ RN be a bounded open set with Lipschitz boundary ∂Ω and p ≥ We study the following nonlinear parabolic equation with a nonlocal diffusion term     ut − div a ∇u pLp (Ω) |∇u|p−2 ∇u + f (u) = g(x), x ∈ Ω, t > 0,    (3.1) u(x, t) = 0, x ∈ ∂Ω, t > 0,      u(x, 0) = u0 (x), x ∈ Ω The nonlinearity f , the diffusion coefficient a and the external force g satisfy (H1) a ∈ C(R, R+ ) and there are two positive constants m and M such that < m ≤ a(s) ≤ M, ∀s ∈ R Moreover, the mapping a is such that s → a(sp )sp−1 is nondecreasing (H2) f : R → R is a continuously differentiable function satisfying c1 |u|q − c0 ≤ f (u)u ≤ c2 |u|q + c0 , 11 f (u) ≥ −c3 , for some q ≥ 2, where c0 , c1 , c2 , c3 are positive constants (H3) g ∈ L2 (Ω) 3.2 Existence and uniqueness of weak solutions Denote ΩT := Ω × (0, T ), V := Lp (0, T ; W01,p (Ω)) ∩ Lq (ΩT ), V ∗ := Lp (0, T ; W −1,p (Ω)) + Lq (ΩT ), where 1/p + 1/p = and 1/q + 1/q = Definition 3.2.1 Let u0 ∈ L2 (Ω) A function u is called a weak solution of problem (3.1) on the interval (0, T ) if and only if u ∈ V, ut ∈ V ∗ , u|t=0 = u0 a.e in Ω, and ut v + a ∇u ΩT p Lp (Ω) |∇u|p−2 ∇u · ∇v + f (u)v − gv dxdt = 0, for all test functions v ∈ V The following result makes the initial condition in problem (3.1) meaningful Lemma 3.2.2 If u ∈ V and ut ∈ V ∗ , then u ∈ C([0, T ]; L2 (Ω)) Lemma 3.2.3 Under the assumption (H1), −div a( ∇u p p−2 ∇u Lp (Ω) )|∇u| is a monotone operator in W01,p (Ω) Theorem 3.2.4 Under the assumptions (H1), (H2), and (H3), for each u0 ∈ L2 (Ω) given, problem (3.1) has a unique weak solution u(·) satisfying u ∈ C([0, ∞); L2 (Ω)) ∩ Lploc (0, ∞; W01,p (Ω)) ∩ Lqloc (0, ∞; Lq (Ω)), ut ∈ Lploc (0, ∞; W −1,p (Ω)) + Lqloc (0, ∞; Lq (Ω)) Moreover, the mapping u0 → u(t) is (L2 (Ω), L2 (Ω))-continuous 12 3.3 Existence of global attractors 3.3.1 The (L2 (Ω), L2 (Ω))-global attractor Theorem 3.2.4 allows us to construct a continuous (nonlinear) semigroup S(t) : L2 (Ω) → L2 (Ω) associated to problem (3.1) as follows S(t)u0 := u(t), where u(·) is the unique global weak solution of (3.1) with the initial datum u0 We deduce from the proof of Theorem 3.2.4 that there exists an (L2 (Ω), L2 (Ω))bounded absorbing set B0 of {S(t)}t≥0 Proposition 3.3.1 The semigroup {S(t)}t≥0 has an (L2 (Ω), W01,p (Ω))-bounded absorbing set B1 As a direct result of Proposition 3.3.1 and the compactness of the embedding W01,p (Ω) → L2 (Ω), we get the following result Theorem 3.3.2 Assume that the hypotheses (H1), (H2), and (H3) are satisfied Then the semigroup {S(t)}t≥0 generated by problem (3.1) has an (L2 (Ω), L2 (Ω))-global attractor A2 3.3.2 The (L2 (Ω), Lq (Ω))-global attractor We will prove the existence of an (L2 (Ω), Lq (Ω))- and (L2 (Ω), W01,p (Ω) ∩ Lq (Ω))global attractors, respectively To this, we assume furthermore that (H1bis) a is continuously differentiable, nondecreasing and satisfies condition (H1) Proposition 3.3.3 Assume that the assumptions (H1bis), (H2), and (H3) hold Then the semigroup {S(t)}t≥0 has an (L2 (Ω), W01,p (Ω) ∩ Lq (Ω))-bounded absorbing set B2 , that is, there is a positive constant ρ2 such that for any bounded subset B in L2 (Ω), there is a positive constant T2 depending only on L2 -norm of B such that |∇u|p dx + Ω |u|q dx ≤ ρ2 , Ω for all t ≥ T2 and u0 ∈ B, where u is the unique weak solution of (3.1) with the initial datum u0 13 Proposition 3.3.4 The semigroup {S(t)}t≥0 is norm-to-weak continuous on S(B2 ), where B2 is the (L2 (Ω), W01,p (Ω)∩Lq (Ω))-bounded absorbing set obtained in Proposition 3.3.3 Theorem 3.3.5 Assume that the hypotheses (H1bis), (H2), and (H3) are satisfied Then the semigroup S(t) associated to (3.1) has an (L2 (Ω), Lq (Ω))-global attractor Aq The (L2 (Ω), W01,p (Ω) ∩ Lq (Ω))-global attractor 3.3.3 Lemma 3.3.6 Assume that the assumptions (H1bis), (H2), and (H3) hold Then for any bounded subset B in L2 (Ω), there exists a positive constant T3 = T3 (B) such that ut (s) where ut (s) = L2 (Ω) ≤ ρ3 , for all u0 ∈ B, and s ≥ T3 , d (S(t)u0 )|t=s and ρ3 is a positive constant independent of u0 dt Lemma 3.3.7 Let p ≥ Then under the assumption (H1bis), we have for all u1 , u2 ∈ W01,p (Ω), that p p−2 ∇u1 Lp (Ω) )|∇u1 | −div a( ∇u1 a( ∇u1 = Ω ≥ cp u1 − u2 p p−2 ∇u1 Lp (Ω) )|∇u1 | + div a( ∇u2 − a( ∇u2 p p−2 ∇u2 Lp (Ω) )|∇u2 | p p−2 ∇u2 Lp (Ω) )|∇u2 | , u1 − u2 · ∇(u1 − u2 )dx p , W01,p (Ω) where cp =   m if p = 2,   m if p > 8.3p/2 Theorem 3.3.8 Assume that the assumptions (H1bis), (H2), and (H3) are satisfied Then the semigroup {S(t)}t≥0 associated to (3.1) has an (L2 (Ω), W01,p (Ω)∩Lq (Ω))global attractor A 3.4 Existence and exponential stability of stationary solutions An element u∗ ∈ W01,p (Ω)∩Lq (Ω) is said to be a weak stationary solution to problem (3.1) if a( ∇u∗ p Lp (Ω) ) |∇u∗ |p−2 ∇u∗ · ∇vdx + Ω f (u∗ )vdx = Ω for all test functions v ∈ W01,p (Ω) ∩ Lq (Ω) 14 gvdx, Ω Theorem 3.4.1 Under the hypotheses (H1), (H2), and (H3), the problem (3.1) has at least one weak stationary solution u∗ satisfying u∗ p W01,p (Ω) q Lq (Ω) + u∗ ≤ , where p = 2p c0 |Ω|(pmλ1 ) p + |Ω| (p−2)p 2p g p p L2 (Ω) p 1, 2cm1 mp (pmλ1 ) p Moreover, if f is strictly increasing, i.e f (s) ≥ α > for all s ∈ R, then for any solution u of (3.1), we have u(t) − u∗ L2 (Ω) ≤ u(0) − u∗ −2αt L2 (Ω) e for all t > That is, the weak stationary solution of (3.1) is unique and exponentially stable Remark 3.4.2 If p = 2, a satisfies (H1bis) and mλ1 > c3 , it is easily verified that the weak stationary solution u∗ is unique and exponentially stable Moreover, for any solution u to (3.1), we have u(t) − u∗ L2 (Ω) ≤ u(0) − u∗ −2(mλ1 −c3 )t L2 (Ω) e for all t > Remark 3.4.3 In case of p > 2, p-Laplacian operators are nonlinear Therefore, the technicalities in this chapter are different from that one in previous chapeter Moreover, we can not examine all conditions of Lemma 1.2.33 hold because of nonlinearity of pLaplacian operator and nonlocality, and so the finite fractal dimension estimate of the global attractor of our problem (3.1) is still an open issue May be we need to use another approaches Remark 3.4.4 The nonlocal quantity which depends on Lp -norm of the gradient cause a lot of difficulties in analysis of our problem This leads that the technicalities in this chapter are more different from the previous one Our results have improved and extended the well-known results as follows: ❼ In case of a ≡ 1, we recover some well-known results by Babin and Vishik (1992), Temam (1997), Geredeli et al (2013, 2015) (see also the results by Carvalho et al (1999, 2001, 2003) with monotone operators) 15 ❼ In the case f = 0, g ∈ W −1,p (Ω) and u0 ∈ W01,p (Ω) ∩ L2 (Ω) with < p < ∞, p + p , we have had the results by M Chipot and T Savitska (2014), but say nothing about the global attractor ❼ Our results are the first work proving the existence of the global attractors for this nonlocal equation 16 Chapter GLOBAL ATTRACTORS FOR NONLOCAL PARABOLIC EQUATIONS INVOLVING THE FRACTIONAL LAPLACIAN AND THE REGIONAL FRACTIONAL LAPLACIAN WITH A NEW CLASS OF NONLINEARITIES In this chapter, we will use the method of Dirichlet forms in nonsmooth domains to investigate a class of nonlocal parabolic equations involving the fractional Laplacian and the regional fractional Laplacian with various boundary conditions and a new class of nonlinearities We first prove the existence and uniqueness of weak solutions by using the compactness method and the weak convergence techniques in Orlicz spaces Then we study the existence and fractal dimension estimates of the global attractor for the associated continuous semigroup The main novelty of the obtained results is that no restriction is imposed on the upper growth of the nonlinearities The content of this chapter is based on the work [CT3] in the List of Publications 4.1 Setting problem We will consider nonlocal diffusion equations involving the fractional Laplacian (−∆)s with extended Dirichlet boundary condition given by    ut + (−∆)s u + f (u) = g in Ω × (0, ∞),    u=0 on (RN \Ω) × (0, ∞),     u(x, 0) = u0 (x) in Ω, (4.1) and nonlocal diffusion equations involving the regional fractional Laplacian AsΩ with Dirichlet, fractional Neumann and fractional Robin type boundary conditions, respectively, described as follows    ut + AsΩ u + f (u) = g    u=0     u(x, 0) = u0 (x) 17 in Ω × (0, ∞), on ∂Ω × (0, ∞), in Ω, (4.2) and    ut + AsΩ u + f (u) = g    N 2−2s u =     u(x, 0) = u0 (x) and in Ω × (0, ∞), on ∂Ω × (0, ∞), (4.3) in Ω,    ut + AsΩ u + f (u) = g in Ω × (0, ∞),    BN,s N 2−2s u + γu = on ∂Ω × (0, ∞),     u(x, 0) = u0 (x) in Ω, (4.4) where N 2−2s u denotes the fractional normal derivative of the function u, BN,s is a normalized constant, γ is a positive constant We can rewrite problems (4.1)-(4.4) in the unified form  ut + AK u + f (u) = g in Ω × (0, ∞), u(x, 0) = u (x) in Ω (4.5) To study problem (4.5), we suppose that the following assumptions hold: (F) f : R → R is a continuously differentiable function satisfying: (i) If K ∈ {D, R, E}, we have f (u)u ≥ −µu2 − c1 , f (u) ≥ − , (4.6) (4.7) where c1 , are two positive constants and < µ < CK,s (ii) If K = N , we require fN (u) = f (u) − ηu satisfying (4.6)-(4.7) with some η ≥ |Ω| and < µ < CN ,s (G) g ∈ L2 (Ω) (D) The domain Ω and s satisfy the following conditions: (i) If K ∈ {D, E}, then Ω is an arbitrary bounded open set and < s < 1, (ii) If K ∈ {N , R}, then Ω is a bounded set with Lipschitz continuous boundary and 4.2 < s < Existence and uniqueness of weak solutions We denote WKs,2 (Ω), K ∈ {D, N , R, E}, for each situation as follows: WDs,2 (Ω) := W0s,2 (Ω), WEs,2 (Ω) := W0s,2 (Ω) = {u ∈ W s,2 (RN ), u = on RN \Ω}, WNs,2 (Ω) ≡ WRs,2 (Ω) := W s,2 (Ω) Moreover, WK−s,2 (Ω) := (WKs,2 (Ω))∗ is the dual space of WKs,2 (Ω) 18 Definition 4.2.1 A weak solution to problem (4.5) on the interval (0, T ) is a function u ∈ L2 (0, T ; WKs,2 (Ω)) ∩ C([0, T ]; L2 (Ω)) satisfying du dt ∈ L2 (0, T ; WK−s,2 (Ω)) + L1 (ΩT ), f (u) ∈ L1 (ΩT ), u(0) = u0 almost everywhere and the equation d u, v + EK (u, v) + f (u), v = g, v , dt holds for all test functions v ∈ WKs,2 (Ω) ∩ L∞ (Ω), K ∈ {D, N , R, E} and for a.e t ∈ (0, T ), where ΩT = Ω × (0, T ) Theorem 4.2.2 Let u0 ∈ L2 (Ω) and T > be given Assume (F), (G), and (D) hold Then problem (4.5) has a unique weak solution u on the interval (0, T ) Moreover, the mapping u0 → u(t) is continuous on L2 (Ω), that is, the solution depends continuously on the initial data 4.3 Existence of global attractors Thanks to Theorem 4.2.2, for each K ∈ {D, N , R, E}, we can define a continuous (nonlinear) semigroup SK (t) : L2 (Ω) → L2 (Ω) associated to problem (4.5) as follows SK (t)u0 := u(t), where u(·) is the unique global weak solution of (4.5) with the initial datum u0 We will prove that the semigroup SK (t) has a compact global attractor AK Lemma 4.3.1 The semigroup {SK (t)}t≥0 generated by problem (4.5) has a bounded absorbing set B0K in L2 (Ω) Lemma 4.3.2 The semigroup {SK (t)}t≥0 generated by problem (4.5) has a bounded absorbing set B1K in WKs,2 (Ω)) As a direct consequence of Lemma 4.3.2 and the compactness of the embedding WKs,2 (Ω) → L2 (Ω), we get the following result Theorem 4.3.3 ((L2 (Ω), L2 (Ω))-global attractor) Assume that the assumptions (F), (G), and (D) hold Then, for each K ∈ {D, N , R, E}, the semigroup SK (t) generated by problem (4.5) has a global attractor A1K in L2 (Ω) Lemma 4.3.4 Assume that the assumptions (F), (G), and (D) hold Then for every bounded subset B2 in L2 (Ω), there exists a constant T = T (B2 ) > such that ut (s) where ut (s) = L2 (Ω) d (SK (t)u0 )|t=s dt ≤ ρ2 for all u0 ∈ B2 , and s ≥ T, for each K ∈ {D, N , R, E} and ρ2 is a positive constant independent of B2 19 Lemma 4.3.5 Assume that the assumptions (F), (G), and (D) hold Then, for each K ∈ {D, N , R, E}, the semigroup {SK (t)}t≥0 generated by problem (4.5) is (L2 (Ω), WKs,2 (Ω))asymptotically compact The following result follows immediately from Lemma 4.3.2, Lemma 4.3.5 Theorem 4.3.6 ((L2 (Ω), WKs,2 (Ω))-global attractor) Assume that assumptions (F), (G), and (D) hold Then, for each K ∈ {D, N , R, E}, the semigroup {SK (t)}t≥0 generated by problem (4.5) has a (L2 (Ω), WKs,2 (Ω))-global attractor A2K Remark 4.3.7 The global attractors A1K and A2K obtained in Theorem 4.3.3 and Theorem 4.3.6 are of course the same object because the uniqueness of the global attractor of a semigroup and will be denoted by AK In particular, AK is a compact and connected set in WKs,2 (Ω) 4.4 Fractal dimension estimates of the global attractor We assume the nonlinearity f and the external force g satisfy stronger conditions: (Fbis) f satisfies the condition (F) and there exists s0 > such that fK (s) ≥ g L∞ (Ω) if s ≥ s0 , fK (s) ≤ g L∞ (Ω) if s ≤ −s0 , where fK (s) = f (s) if K ∈ {D, R, E} and fK (s) = fN (s) if K = N (Gbis) g ∈ L∞ (Ω) Proposition 4.4.1 Let u ∈ WKs,2 (Ω), K ∈ {D, N , R, E}, and let k ≥ be a real number Then uk ∈ WKs,2 (Ω) and EK (uk , uk ) ≤ EK (u, uk ), ∀k ≥ Lemma 4.4.2 Assume that the assumptions (Fbis), (Gbis), and (D) hold Then the global attractor AK associated to problem (4.5) is bounded in L∞ (Ω) The following theorem is the main result of this section Theorem 4.4.3 Assume that (Fbis), (Gbis), and (D) hold Then, for each K ∈ {D, N , R, E}, the global attractor AK of problem (4.5) has a finite fractal dimension in L2 (Ω), namely, dimf AK ≤ q ln( 9e )[ln( )]−1 1−δ 1+δ 20 Remark 4.4.4 By repeating the similar arguments in Section 2.5, Chapter 2, we are able to obtain the existence and exponential stability of stationary solutions for these problems Remark 4.4.5 The fractional Laplacian and the regional fractional Laplacian are naturally nonlocal operators Along with lack of an upper growth restriction of the nonlinearity and various boundary type conditions, the technicalities are more involved and different from two previous chapters We have used the method of Dirichlet forms in nonsmooth domains to investigate our problems and the attained results have improved and extended the well-known results as follows: ❼ In case that the nonlinearity satisfies polynomial growth and using the above method, we recover the results by C.G Gal and M Warma (2016) ❼ For the reaction-diffusion equations involving fractional Laplacian with extended Dirichlet boundary conditions and lack of an upper growth restriction of the nonlinearity, we have the results by C Zhang et al (2018), but not use the method of Dirichlet forms in nonsmooth domains ❼ Our results have improved and extended the above results since the nonlinear term has no restriction on the upper growth and all boundary conditions are considered 21 CONCLUSION AND FUTURE WORK Conclusion In this dissertation, a number of nonlocal problems have been investigated It has contributed some results on studying long-time behavior of solutions of these nonlocal parabolic problems via existence of its finite dimensional global attractors, and the existence and exponential stability of stationary solutions, namely: Existence and uniqueness of weak solutions, existence of connected compact global attractors, finite fractal dimensional estimate of global attractor, existence and exponential stability of stationary solutions for nonlocal parabolic equations involving Laplacian with a new class of nonlinearities Existence and uniqueness of weak solutions, existence of global attractors, existence and exponential stability of stationary solutions for a nonlocal parabolic equation involving p-Laplacian operators Existence and uniqueness of weak solutions, existence of connected compact global attractors, finite fractal dimensional estimate of global attractor for nonlocal parabolic equations involving the fractional Laplacian and the regional fractional Laplacian with a new class of nonlinearities The results obtained in the dissertation are meaningful contributions to the theory of infinite dynamical systems for nonlocal parabolic equations Future Work I continue my research in the classes of nonlocal equations Especially, I concern more about mathematical models in fluid mechanics 22 LIST OF PUBLICATIONS Accepted papers [CT1] C.T Anh, L.T Tinh and V.M Toi, Global attractors for nonlocal parabolic equations with a new class of nonlinearities, J Korean Math Soc 55 (2018), no 3, 531-551 (SCIE) [CT2] L.T Thuy and L.T Tinh, Long-time behavior of solutions to a nonlocal quasilinear parabolic equation, Commun Korean Math Soc 34 (2019), no 4, 1365-1388 (Scopus) Submitted papers [CT3] C.T Anh and L.T Tinh, Global attractors for nonlocal parabolic equations involving the regional fractional Laplacian with a new class of nonlinearities, submitted (2019) 23 ... T Savitska (2014), but say nothing about the global attractor ❼ Our results are the first work proving the existence of the global attractors for this nonlocal equation 16 Chapter GLOBAL ATTRACTORS... regularity of the attractor, determining modes, etc For the above analysis, we see that there are many open problems involving nonlocal equations via studying the global attractors of associated... dissertation has been written at Hanoi National University of Education Supervisor: Prof Dr Cung The Anh Referee 1: Prof Dr Sc Nguyen Minh Tri Institute of Mathematics, Vietnam Academy of Science