MINISTY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION ——————— * ——————— NGUYEN THI VAN ANH BEHAVIOUR OF SOLUTIONS TO DIFFERENTIAL VARIATIONAL INEQUALITIES Speciality: Integral and Differential Equations Code: 46 01 03 SUMMARY OF DOCTORAL THESIS IN MATHEMATICS Hanoi - 2019 This thesis has been completed at the Hanoi National University of Education Scientific Advisor: Assoc.Prof PhD Tran Dinh Ke Referee 1: Prof Dr Sci Nguyen Minh Tri, Vietnam Academy of Science and Technology, Institute of Mathematics Referee 2: Assoc.Prof Dr Nguyen Xuan Thao, Hanoi University of Science and Technology Referee 3: Assoc.Prof Dr Nguyen Sinh Bay, Vietnam University of Commerce The thesis shall be defended before the University level Thesis Assessment Council at on The thesis can be found in the National Library and the Library of Hanoi National University of Education INTRODUCTION Motivation and outline Quantitative theory of ordinary differential equation (ODE) is one of the basic theories of mathematics which has existed for centuries and gives us many models describing mechanics of motion in nature and engineering In the several decades by the end of the 20th century, ODE has generalized as differential algebraic equation (DAE) and then it has been extensively studied Therefore, various problems of engineering and science such as in flexible mechanics, electrical circuit design and chemical process control, etc have been mathematical modeling based on the DAE However, as evidenced by the growing literature that has surfaced in recent years on multi-rigid-body dynamics with frictional contacts and on hybrid engineering systems, ODE and DAE are vastly inadequate to deal with many naturally occurring engineering problems that contain inequalities (for modeling unilateral constraints) Hence, in order to research the differential model with unilateral constraints, that satisfies the demands mentioned above, mathematicians need to investigate a larger problem:Differential variational inequalities, which includes differential complementarity problem The notion of differential variational inequality was firstly used by Aubin and Cellina in 1984 In their book the authors considered the problem:    ∀t ≥ 0, x(t) ∈ K, supy∈K x (t) − f (x(t)), x(t) − y = 0,   x(0) = x , (1) where K is a convex closed subset in Rn By using concept about normal cone of subset K, the problem was replaced with the differential inclusion of the form: f (t) ∈ F (x(t)), x(0) = x0 Then the solvability of (1) can be studied by the topological tools of multivalued analysis After this work, the theory of DVIs was considered and expanded in the work of Avgerinous and Papageorgiou in 1997 Moreover, Avgerinous and Papageorgiou studied the periodic solutions to the DVI of the form −x (t) ∈ NK(t) (x(t)) + F (t, x(t)), a.e t ∈ [0, b], x(0) = x(b) where NK(t) (x(t)) denotes the normal cone of the convex closed set K(t) at the point x(t) However, DVIs were first systematically studied by Pang and Stewart in 2008 Differential variational inequality is the problem to find an absolutely continuous function x : [0, T ] → Rn and an integrable function u : [0, T ] → Rm such that for almost all t ∈ [0, T ], one has x (t) = f (t, x(t), u(t)), v − u(t), F (t, x(t), u(t) ≥ 0, a.e t ∈ [0, T ]; ∀v ∈ K u(t) ∈ K (2) (3) (4) The name DVI is based on the fact that an ordinary differential equation is linked together with an algebraic constraint represented by the inequality By the mean of Pang and Stewart, the derivative of u(·) does not appear in (3), therefore u is called an algebraic variable On the other hand, x is called a differential variable Denote SOL(K, φ) is solution set of the variational inequality v − u, φ(u) ≥ 0, ∀v ∈ K Then u(t) solves variational inequality (3) with φ = F (t, x(t), ·) Therefore, the properties of the solution mapping (t, x) ⇒ SOL(K, F (t, x(t), ·)) will play a key role in our consideration We convert the problem to a differential inclusion x (t) = f (t, x(t), SOL(K, F (x(t), ·))), The general boundary condition is considered by Γ(x(0), x(T )) = 0, (5) In the stated form, the problem is a two-point boundary-value problem (BVP) in the sense that, linked by the abstract function Γ, both the initial state x(0) and the terminal state x(T ) are unknown variables to be computed; in particular, the former variable x(0) is not completely given The initial-value (IVP) version of the problem corresponds to the special case where Γ(x, y) ≡ x − x0 , with x0 being given One of the important problem generated by DVIs is differential complementarity problem (DCP), where K is a cone C x (t) = f (t, x(t), u(t)), C u(t) ⊥ F (t, x(t), u(t)) ∈ C ∗ In turn, a proper specialization of the latter DCP yields the linear complementarity system, which is studied extensively in many previous papers In the paper of Pand and Stewart, DVIs comes from many applications including linear complementarity systems, differential complementarity problems, and variational inequalities of evolution After the work of Pang and Stewart, more and more scholars are attracted to boost the development of theory and applications for (DVIs) For instance, Chen - Wang (2014), Liu et al in 2013 studied the existence and global bifurcation problems for periodic solutions to a class of differential variational inequalities in finite dimensional spaces by using the topological methods from the theory of multivalued maps and some versions of the method of guiding functions, Gwinner in 2013 obtained a stability result of a new class of differential variational inequalities by using the monotonicity method and the technique of the Mosco convergence, and Chen-Wang in 2014 used the idea of (DVIs) to investigate a dynamic Nash equilibrium problem of multiple players with shared constraints and dynamic decision processes On the other hand, many applied problems of DVIs in engineering, operations research, economical dynamics, and physical sciences, etc., are more precisely described by partial differential equations Based on this motivation, recently, Liu–Zeng–Motreanu in 2016 and Liu et al in 2017 proved the existence of solutions for a class of differential mixed variational inequalities in Banach spaces through applying the theory of semigroups, the Filippov implicit function lemma and fixed point theorems for condensing set-valued operators However, until now, only one reference, Liu et al , considered a differential hemivariational inequality in Banach spaces which is constituted by a nonlinear evolution equation and a hemivariational inequality of elliptic type rather than of parabolic type Also, in the paper, the authors required that the constraint set K is bounded, the nonlinear function u → f (t, x, u) maps convex subsets of K to convex sets and the C0 -semigroup eAt is compact The thesis concerns one of the important problem related to the dynamic system linked to variational inequalities That is, we study the behaviour of solution of differential variational inequalities when time tends to infinity The recent results of DVI in finite spaces have investigated in some works, including Liu (2013), Loi (2015) There are many open questions concerning to study the DVIs, such as the stability in the sense of Lyapunov, the exsitence of decay solution and periodic solution, the existence of global attractors of m-semiflow generated by differential variational inequalities In addition, DVI in Banach space is also attractive and has some open problem The main difficulty in research the infinite system is the fact that we not what exactly the solvability and properties of solutions of linked variational inequalities If the solution map of variational inequality is not suiable regularity, it is very hard to study the behaviour of solution of DVI via diffential inclusion theory Purpose, objects and scope of the thesis 2.1 Purpose: The thesis focus on studying qualtitive behavior of solutions to differential variational inequalities in finite dimensional space and infinite dimensional space 2.2 Objects In this thesis, we consider three problems as follows: 1) Establish sufficient conditions ensuring the existence of mila solutions of multi - valued dynamical systems generated by differential variational inequalities, and then prove the existence of global attractors 2) The existence of decay solutions for a class differential variational inequalities 2.2 Objects: In the thesis, we consider two types of nonautonomous semilinear differential inclusions in Banach spaces: ∗ The first type: Differential variational inequalities in finite dimensional space; ∗ The second one: Differential variational inequalities of parabolic-elliptic type in infinite spaces; ∗ The thirst one: Differential variational inequalities of parabolic-parabolic type in infinite spaces; 2.3 Scope: The scope of the thesis is defined by the following contents • Content 1: Study the solvability of differential variational inequalities; • Content 2: Study the existence of decay solutions for these differential variational inequalities; • Content 3: Study the existence global attractors for differential variational inequalities Research Methods The thesis use the tools of multivalued analysis, fixed point theorem, oneparameter semigroup theory to study the contents Moreover, we use some special technique to get our purpose: ◦ To prove the existence of solutions to differential variational inequalities, we employ the semigroup theory (see MNC’s estimates (see Bothe (1998) or Kamenskii et al (2001)) ◦ To prove the existence of solutions to differential variational inequalities: we use fixed point theory for condensing multimaps (see Kamenskii et al (2001)) ◦ In order to research on the existence of global attractor for multi-valued dynamical systems, we use the frame work proposed by Melnik and Valero (1998) In which, we estimated measure of noncompactness to get the asymptotically compactness of the process generated by differential inclusions Structure and Results Together with the Introduction, Conclusion, Author’s works related to the thesis that have been published and References, the thesis includes four chapters: Chapter is devoted to present some preliminaries In Chapter 2, we present the solvability and the existence of global attractor for a class of differential variational inequality in finite dimensional space Chapter presents a sufficient condition ensuring the existence of attractor for differential variational of parabolicelliptic type in Banach space Chapter presents a sufficient condition ensuring the existence of attractor for differential variational of parabolic-parabolic type in infinite dimensional spaces Chapter PRELIMINARIES In this chapter, we present some preliminaries including: some functional spaces; measure of noncompactness; multi-valued calculus and fixed point principles; global attractor for multi-valued autonomous dynamical systems, some auxiliary results related to some inequalities and theorems 1.1 ONE PARAMETER SEMIGROUP In this section, we present the basic knowledge about semigroup theory, including linear and nonlinear semigroup 1.1.1 Linear semigroup 1.1.2 Nonlinear semigroup 1.2 MEASURE OF COMPACTNESS (MNC) AND MNC ESTIMATES In this section, we recall some notions and facts related to measure of noncompactness (MNC) and Hausdorff MNC, followed by some MNC estimate which is necessary for the next chapters 1.3 MULTIVALUED CALCULUS AND SOME FIXED POINT THEOREMS 1.3.1 Multivalued calculus In this subsection, we present some definitions and results in multivalued calculus, including concept of a selector and the existence of a selection function 1.3.2 Condensing map and some fixed point theorems In this section, we recall some notions of set-valued analysis and condensing map, then introduce some fixed point theorem for multivalued maps 1.4 GLOBAL ATTRACTORS FOR MULTIVALUED SEMIFLOWS In this section, we present some definitions and results on global attractors for multivalued semiflows developed by Menik and Valero (1998) and the frame work for the existence of a compact global attractor for m-semiflows generated by a differential inclusion 1.5 SOME AUXILIARY RESULTS In this section, we recall some notions and facts related to wellknown inequalities, consist of Gronwall inequality, Hanalay inequality and some theorems such as Mazur Lemma, Arzela Ascoli theorem In addition, we hightlight some essential functional spaces which are used in this thesis 1.5.1 Some auxiliary inequalites 1.5.2 Some auxiliary theorems 1.5.3 Some functional spaces Chapter DIFFERENTIAL VARIATIONAL INEQUALITY IN FINITE SPACE In this chapter, we study behaviour of solutions of differential variational inequalities in finite space with delay Our purpose was to give the sufficient coditions to ensure the existence of solution and the stability of DVIs Thus, the existence of decay solution and a global attractor for m-semiflow generated by DVIs were proved The content of this chapter is written based on the paper [1] in the author’s works related to the thesis that has been published 2.1 Problem setting We consider the following problem: x (t) = Ax(t) + h(x(t)) + B(x(t), xt )u(t), t ∈ J = [0, T ], v − u(t), F (x(t)) + G(u(t)) ≥ 0, ∀v ∈ K, for a.e t ∈ J, x(s) = ϕ(s), s ∈ [−τ, 0], (2.1) (2.2) (2.3) where x(t) ∈ Rn , u(t) ∈ K with K being a closed convex subset in Rm , xt stands for the history of the state function up to time t, i.e ut (s) = u(t + s) for s ∈ [−h, 0]; A, B, F, G and h are given maps which will be specified in the next section 2.2 Solvability In this section, we will show the global existence of integral solution to problem (2.1)-(2.2) on J = [τ, T ] under the following assumptions: Put J = [0, T ], CT = C([0, T ]; Rn ), Cτ = C([−τ, 0]; Rn ), C = C([−τ, T ]; Rn ) In what follows, we use the assumptions that: (H1) A is a linear operator on Rn (H2) B : Rn ×Cτ → Rn×m is a continuous map such that there exist positive constants ηB , ζB verifying that: B(v, w) ≤ ηB ( v + w Cτ ) + ζB , for all v ∈ Rn , w ∈ Cτ (H3) The function F : Rn → Rm is continuous and there is a positive number ηF such that F (v) ≤ ηF for all v ∈ Rn (H4) G : K → Rm is a continuous function such that 1) G is monotone on K , i.e u − v, G(u) − G(v) ≥ 0, ∀u, v ∈ K; 2) there exists v0 ∈ K such that lim v∈K, v →∞ v − v0 , G(v) > v (H5) h : Rn → Rn is continuous such that there are positive constants ηh , ζh verifying h(u) ≤ ηh u + ζh , ∀u ∈ Rn We have the following definition of integral solution to (2.1)-(2.2) Definition 2.1 A pair of functions (x, u), where x : [−τ, T ] → Rn is continuous and u : [0, T → K] is integrable, called a solution of (2.1) − (2.3) iff the following equalities hold t x(t) = etA ϕ(0) + t e(t−s)A B(x(s), xs )u(s)ds + e(t−s)A h(x(s))ds, t ∈ J, v − u(t), F (x(t)) + G(u(t)) ≥ 0, for a.e t ∈ J, ∀v ∈ K, x(s) = ϕ(s), s ∈ [−τ, 0] We denote SOL(K, Q) = {v ∈ K : w − v, Q(v) ≥ 0, ∀w ∈ K}, (2.4) where Q : Rm → Rm is a given mapping Lemma 2.1 Suppose that (H4) holds Then, for every z ∈ Rm , the solution set SOL(K, z + G(·)) of 2.4 is nonempty, convex and compact Moreover, there exists a number ηG > such that v ≤ ηG (1 + z ), ∀v ∈ SOL(K, z + G(·)) (2.5) Denote U (z) = SOL(K, z + G(·)), z ∈ Rm By Lemma 2.1, the operator U : Rm → P(Rm ) has compact, convex values and U is upper semicontinuous Now we define Φ : Rn × Cτ → P(Rn ) as follows Φ(v, w) = {B(v, w)y + h(v) : y ∈ U (F (v))} (2.6) Then the composition multimap Φ is upper semicontinuous By above setting, the differential variational inequality (2.1)-(2.3) is converted to the differential inclusion as follows x (t) ∈ Ax(t) + Φ(x(t), xt ), t ∈ J, x(t) = ϕ(t), t ∈ [−τ, 0] (2.7) (2.8) Denote PΦ (x) = {f ∈ L1 (J; Rn ) : f (t) ∈ Φ(x(t), xt )}, with x ∈ C Thanks to Lemma 2.1, we have Φ(v, w) ≤ ηG (1 + ηF )[ηB ( v + w Cτ ) + ζB ] + η h v + ζh (2.9) 11 Chapter DIFFERENTIAL VARIATIONAL INEQUALITY OF PARABOLIC-ELLIPTIC TYPE In this chapter we consider a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic variational inequality The existence of global solution and global attractor for the semiflow governed by our system is proved by using measure of noncompactness The content of this chapter is written based on the paper [2] in the author’s works related to the thesis that has been published 3.1 Setting problem Let (X, · ) be a Banach space and (U, · U ) be another reflexive Banach space with the dual U ∗ , we consider the following problem: x (t) − Ax(t) ∈ F (x(t), u(t)), x(t) ∈ X, t ≥ 0, B(u(t)) + ∂φ(u(t)) g(x(t), u(t)), u(t) ∈ U, t ≥ 0, x(0) = ξ, (3.1) (3.2) (3.3) where x is the state function with values in X, u is a control function taking values in U , φ : U → R is a proper, convex, lower semicontinuous function with the subdifferential ∂φ ⊂ U × U ∗ By PF we will denote the set of Bochner integrable selections of F (·, ·), that means PF : C(J; X) × L1 (J; U ) → P(L1 (J; X)), PF (x, u) = {f ∈ L1 (J; X) : f (t) ∈ F (x(t), u(t)) for a.e t ∈ J} (3.4) We mention here the definition of mild solution of the problem (3.1)− (3.3) Definition 3.1 A pair of continuous functions (x, u), where x : [0, T ] → X, u : [0, T ] → U , is a mild solution of (3.1) − (3.3) iff there exists a selection f ∈ PF (x, u) such that t S(t − s)f (s)ds, t ∈ J, x(t) = S(t)ξ + Bu(t) + ∂φ(u(t)) 3.2 g(x(t), u(t)), t ∈ J Solvability We consider the problem (3.1)-(3.3) with the following assumptions (A) A is a closed linear operator generating a C0 −semigroup (S(t))t≥0 (F) F : X × U → Pc (X) is u.s.c with weakly compact and convex values and 12 (1) χ(F (C, D)) ≤ pχ(C)+qU(D) for all bounded set C ⊂ X and D ⊂ U , where p, q are positive constants; here χ and U stand for the Hausdorff MNC in the spaces X and H, respectively (2) F (x, u) := sup{ ξ X : ξ ∈ F (x, u)} ≤ a x X + b u U + c, for all x ∈ X , y ∈ U , where a, b, c are nonnegative constants (B) B is a linear continuous operator from U to U ∗ −the dual of U such that B is defined by the equation u, Bv = b(u, v), ∀u, v ∈ U, where b : U × U → R is a bilinear continuous function on U × U and there exists a positive number ηB satisfying b(u, u) ≥ ηB u U , ∀u ∈ U (G) g : X × U → U ∗ is Lipschitzian, i.e there exist two positive constants η1 and η2 such that g(x, u) − g(x , u ) U∗ ≤ η1 x − x X + η2 u − u U We consider the solution set of the elliptic variational inequality S(z) = {u ∈ U : Bu + ∂φ(u) z} By Corrolary 2.9 (V.Barbu), we have the following lemma Lemma 3.1 Suppose that the hypothesis (B) holds Then for each z ∈ U ∗ , the solution set S(z) is single-valued Moreover, the corresponding z → S(z) is Lipchizian from U ∗ to U In fact, the variational inequality (3.2) Bu + ∂φ(u) g(y, u), for each y (3.5) We consider the elliptic variational inequality Bu + ∂φ(u) g(y, u), for given y (3.6) We have the result related to the solution of elliptic variational inequality (3.3) as follows Lemma 3.2 Suppose that (B) and (G) hold In addition, we assume that η2 < ηB Then for each y ∈ X, there exists u ∈ U of (3.6) Moreover, the solution map V:X→U y → u, is Lipschitzian, i.e V(y1 ) − V(y2 ) U ≤ η1 y − y2 ηB − η2 X , ∀y1 , y2 ∈ X (3.7) 13 Consider the multivalued map G(y) := F (y, V(y)), y ∈ X Then we have χ(G(B)) = χ(F (B, V(B))) ≤ pχ(B) + qU(V(B)) qη1 ≤ (p + )χ(B) ηB − η2 (3.8) and G(y) := sup{ z X , z ∈ G(y)} ≤ a y X + b V(y) U + c bη1 ≤a y X+ y X + V(0) ηB − η2 bη1 ) y X + d ≤ (a + ηB − η2 U +c (3.9) We convert the DVI (3.1)-(3.3) to differential inclusion x (t) − Ax(t) ∈ G(x(t)), t ∈ [0, T ], x(0) = ξ (3.10) (3.11) We define RG : C([0, T ]; X) → P(L1 (0, T ; X)), RG (x) = {f ∈ L1 (0, T ; X) : f (t) ∈ G(x(t)), a.e t ∈ [0, T ]} Proposition 3.1 Under assumptions (B), (F) and (G), the map RG is weakly upper semicontinuous with convex and weakly compact values Consider Cauchy operator W : L1 (0, T ; X) → C([0, T ]; X) t W(f )(t) = S(t − s)f (s)ds Proposition 3.2 Suppose that (A) satisfies If the set D ⊂ L1 (0, T ; X) is semicompact then W(D) is relatively compact in C([0, T ]; X) In particular, if the sequence {fn } is semicompact and fn f ∗ in L1 (0, T ; X) then W(fn ) → W(f ∗ ) in C([0, T ]; X) Theorem 3.1 Let the hypotheses (A), (B), (F) and (G) hold Then the problem (3.1)− (3.3) has at least one mild solution for each initial datum ξ ∈ X Let πT , T > 0, be the truncation operator to [0, T ] imposed on C([0, +∞); X), that is, for z ∈ C([0, +∞); X), πT (z) is the restriction of z on the interval [0, T ] Denote Σ(ξ) = {x ∈ C([0, +∞); X) : x(0) = ξ, x is a mild solution of (3.1)-(3.2) on [0, T ] for any T > 0} 14 Obviously, πT ◦ Σ(ξ) ⊂ S(·)ξ + W ◦ PG (πT ◦ Σ(ξ)), (3.12) for all T > and πT ◦ Σ(ξ) =Fix(F), the fixed points set of the solution operator F of (3.1) - (3.3) in Cξ Lemma 3.3 Under the assumptions (A), (B), (F) and (G), πT ◦ Σ({ξn }) is relatively compact in C([0, T ]; X), where {ξn } ⊂ X is a convergent sequence In particular, πT ◦ Σ(ξ) is a compact set for each ξ ∈ X 3.3 Existence of global attractor The m-semiflow governed by (3.1)-(3.3) is defined as follows G : R+ × X → P(X), G(t, ξ) = {x(t) : x is a mild solution of (3.1) − (3.3), x(0) = ξ} In this section, we need an additional assumption as following (A∗ ) {S(t)}t≥0 is exponentially stable with exponent α, and is χ-decreasing with exponent β , that is S(t) L(X) where α, β > 0, N, P ≥ 1, ≤ N e−αt , S(t) · χ χ ≤ P e−βt , ∀t > 0, is the χ-norm Lemma 3.4 Under the assumptions (A), (B), (F) and (G), G(t, ·) is u.s.c with compact values for each t > Lemma 3.5 Let the hypotheses (A∗ ), (B), (F) and (G) hold If β − 4N (p + qη1 ) > 0, then there exist a number T0 > and a number ζ ∈ [0, 1) such that ηB −η2 for all T ≥ T0 we have χ(GT (B)) ≤ ζ · χ(B), for every bounded set B ⊂ X Lemma 3.6 Assume that (A∗ ), (B), (F) and (G) hold Then G has an absorbing set, provided that α > N (a + ηBbη−η ) Lemma 3.7 Let (A∗ ), (B), (F) and (G) hold If β − 4P (p + is asymptotically upper semicompact qη1 ) ηB −η2 > 0, then G Theorem 3.2 Let the hypotheses (A∗ ), (B), (F) and (G) hold Then the msemiflow G generated by the system (3.1)-(3.3) admits a compact global attractor provided that min{α − N (a + bη1 qη1 ), β − 4P (p + )} > ηB − η2 ηB − η2 15 3.4 Application Let Ω be a bounded domain in Rn with C boundary We consider the following system: ∂Z (t, x) − ∆x Z(t, x) = f (t, x), t ≥ 0, x ∈ Ω, ∂t f (t, x) ∈ [f1 (x, Z(t, x), u(t, x)), f2 (x, Z(t, x), u(t, x))], t > 0, x ∈ Ω, ∆x u(t, y) + β(u(t, x) − ψ(x)) g¯(x, Z(t, x), u(t, x)), t ≥ 0, x ∈ Ω, Z(t, x) = u(t, x) = 0, t ≥ 0, x ∈ ∂Ω, Z(0, x) = ϕ(x), x ∈ Ω, (3.13) (3.14) (3.15) (3.16) (3.17) where f1 , f2 , g : Ω × R × R → R are continuous functions, ψ ∈ H (Ω) and β : R → 2R is a maximal monotone graph   0 β(r) = R−  ∅ if r > 0, if r = 0, if r < This system describes a diffusion process subject to an elliptic VI that models a free boundary problem [20] Let X = L2 (Ω), U = V = H01 (Ω), H = L2 (Ω), V = H −1 (Ω) The norm in X and U are defined by |u(y)|2 dy, ∀u ∈ L2 (Ω), |u|2 = Ω |∇u(y)|2 dy, ∀u ∈ H01 (Ω), u = Ω We define the multivalued map F : X × U → P(X), ¯ u¯)(x) = {λf1 (x, Z(x), ¯ ¯ F (Z, u¯(x)) + (1 − λ)f2 (x, Z(x), u¯(x)) : λ ∈ [0, 1]} The problem (3.13)-(3.14) is rewritten by Z (t) − AZ(t) ∈ F (Z(t), u(t)), t ≥ 0, where A = ∆, D(A) = H (Ω) ∩ H01 (Ω), Z(t) ∈ X, u(t) ∈ U such that Z(t)(x) = Z(t, x), u(t)(x) = u(t, x) By Theorem 7.2.5 and Theorem 7.2.8 in Vrabie, the semigroup S(t) = etA generated by the operator A is compact and exponentially stable, that is S(t) L(X) ≤ e−λ1 t , where λ1 := inf{ ∇u 2X : u ∈ H01 (Ω), u X = 1} The assumption (A∗ ) is satisfied Suppose that there exist nonnegative functions a1 , a2 , b1 , b2 ∈ L∞ (Ω), c1 , c2 ∈ L2 (Ω) such that |f1 (x, p, q)| ≤ a1 (x)|p| + b1 (x)|q| + c1 (x), |f2 (x, p, q)| ≤ a2 (x)|p| + b2 (x)|q| + c2 (x), ∀x ∈ Ω, p, q ∈ R, 16 Because f1 and f2 are continuous, F has closed graph In addition, if {Z¯n } ⊂ X and {¯ un } ⊂ U then we can find a sequence fn ∈ F (Z¯n , u¯n ) converging to X Therefore, F is upper semicontinuous Thus, the condition (F) is testified We consider the elliptic variational inequality (3.15) Let B := −∆ : V → V , where −∆ is Laplace operator ∇u∇vdy, for each u, v ∈ H01 (Ω) u, −∆v := Ω It is easy to see that u, Bu = u 2H (Ω) ≥ λ1 u 2X Thus, assumption (B) hold with ηB = λ1 In terms of g, suppose that there exist nonnegative functions η1 , η2 ∈ L∞ (Ω) such that |g(x, p, q) − g(x, p , q )| ≤ η1 (x) p − p + η2 (x)|q − q |, ∀x ∈ Ω, p, q, p , q ∈ R Then we arrive at the main theorem in this problem Theorem 3.3 If η2 ∞ λ1 > max{ a1 < λ1 and ∞, a2 ∞} + max{ b1 ∞, b2 ∞ }√ η1 ∞ λ1 − η2 ∞ then there exists a global attractor for m-semiflow G governed by (3.13)− (3.17) 17 Chapter DIFFERENTIAL VARIATIONAL INEQUALITY OF PARABOLIC-PARABOLIC TYPE In this chapter, we consider a coupled parabolic-parabolic model formulated by a parabolic differential inclusion and a parabolic variational inequality We study the existence of solution for this problem in infinite spaces In addition, the existence of global attractor for the m-semiflow generated by our system is given via the techniques of measure of noncompactness The content of this chapter is written based on the paper [3] in the author’s works related to the thesis that has been submitted 4.1 Setting problem Let (X, · ) be a Banach space, U and H are real Hilbert spaces such that U is dense in H and U ⊂ H ⊂ U algebraically and topologically We denote by | · | and · U the norms of H and U , respectively, and by ·, · the scalar product in H and the pairing between U and its dual U The space H is identified with its own dual We consider the following problem x (t) ∈ Ax(t) + F (x(t), u(t)), u (t) + Bu(t) + ∂φ(u(t)) h(x(t)), x(0) = x0 v`a u(0) = u0 , (4.1) (4.2) (4.3) where φ : H → R is a proper, convex, lower semicontinuous function By the definition of subdifferential ∂φ, evolution inclusion (4.2) can be rewritten in a form of inequality as follow u (t) + Bu(t) − h(x(t)), u(t) − v + φ(u(t)) − φ(v), ∀v ∈ H Denote BH is a truncated operator of B as follows BH : D(BH ) ⊂ U → H, BH u = Bu, Bu ∈ H, D(BH ) = {u ∈ U : Bu ∈ H} We need the assumptions for the problem (4.1)-(4.3) as following: (A) A generates a C0 - semigroup {S(t)}t≥0 (B) B : U → U is a linear, symmetric, continuous operator satisfying (B1) corecive condition Bu, u ≥ ω u U for some ω > ; (B2) U ∩ D(φ) = ∅ and there exists h ∈ H such that φ((I + λBH )−1 (x + λh)) ≤ φ(x) + Cλ(1 + φ(x)), ∀x ∈ D(φ), λ > 0, 18 (F ) F : X × H → P(X) is u.s.c multimap with compact, convex values and sastisfying (F 1) There exist η1F > 0, η2F > 0, a ≥ 0, such that F (x) := sup{ ξ X : ξ ∈ F (x)} ≤ η1F x X + η2F |u| + a, (F 2) there exist p > 0, q > such that χ(F (B, D)) ≤ pχ(B) + qϑ(D), ∀B ∈ Pb (X), D ∈ Pb (H), where χ and ϑ is MNC in X and H, responsibility (H) h : X → H is a continuous satisfying h(0) ∈ ∂φ(0) and there exist ηh > 0, b ≥ such that h(x) H ≤ ηh x X + b Consider the corresponding PF as follows PF : C([0, T ]; X) × C([0, T ]; H) → P(L1 (0, T ; X)), PF (x, u) = {f ∈ L1 (0, T ; X) : f (t) ∈ F (x(t), u(t)), a.e t ∈ [0, T ]} We mention here the definition of solution of the problem (4.1)-(4.3) Definition 4.1 A pair of functions (x, u) where x ∈ C([0, T ]; X) and u ∈ L2 (0, T ; U ) ∩ W 1,2 (0, T, H) ∩ C([0, T ]; H), Bu(t) ∈ H almost everywhere t ∈ [0, T ] is a solution of the problem if there exist a selection f ∈ PF (x, u) such that t S(t − s)f (s)ds, t ∈ [0, T ], x(t) = S(t)x0 + u (t) + Bu(t) + ∂φ(u(t)) u(0) = u0 h(x(t)), ∀z ∈ U, a.e t, Denote V I(u0 , f ) is the following problem: For each u0 , f given, find u such that u (t) + Bu(t) + ∂φ(u(t)) f (t), u(0) = u0 Take φ0 : H → (−∞, ∞] such that φ0 (u) = Bu, u , u ∈ U, +∞, otherwise We call that ν is a positive number suct that |u|2 ≤ ν u U for all u ∈ U Proposition 4.1 We have the statements (a) BH is a m-accretive set in H × H (maximal monotone in H × H) Moreover, BH is ω ν − m-accretive in H × H and BH = ∂φ0 ; (b) BH + ∂φ is maximal monotone in H × H, then it implies BH + ∂φ is accretive set in H × H with domain D(BH + ∂φ) = D(BH ) ∩ D(φ); ω ν − m- 19 (c) BH + ∂φ = ∂ψ, where ψ = φ0 + φ1 , ψ : H → (−∞, ∞] and ψ(u) = Bu, u + φ(u), u ∈ U, +∞, otherwise (d) −(BH +∂φ) generates a semigroup S1 (t) which is equicontinuous in D(BH + ∂φ) (e) −(BH + ∂φ) generates a semigroup S1 (t) which is compact in D(BH + ∂φ) (f) S1 (t) is the semigroup ω ν exponential stability in D(BH + ∂φ) Proposition 4.2 Let (B) be hold Take u0 ∈ D(φ) ∩ U and f ∈ L2 (0, T, H) Then the problem u (t) + Bu(t) + ∂φ(u(t)) f (t), has a unique solution u such that u(t) ∈ D(BH ) a.e t ∈ [0, T ] and u ∈ L2 (0, T ; U )∩W 1,2 (0, T ; H)∩C([0, T ]; H) Moreover, the corresponding (u0 , f ) → u is Lipschitzian from H × L2 (0, T, H) to C([0, T ]; H) ∩ L2 (0, T ; H) Lemma 4.1 Suppose that (B) and (H) satisfy Then, for each x(·) ∈ C([0, T ]; X) and u0 ∈ D(φ) ∩ U , the problem u (t) + Bu(t) + ∂φ(u(t)) h(x(t)), u(0) = u0 , has a solution u(·) such that u(t) ∈ D(BH ) a.e t ∈ [0, T ] and u ∈ L2 (0, T ; U ) ∩ W 1,2 (0, T ; H) ∩ C([0, T ]; H) Lemma 4.2 For each x ∈ C([0, T ]; H) and B, h satisfy (B) and (H), parabolic variational inequality u (t) + Bu(t) + ∂φ(u(t)) h(x(t)), u(0) = u0 , (4.4) has a solution u(·) and we have the following estimate: ω |u(t)| ≤ e 4.2 −ω νt − e− ν t bν + ηh |u0 | + ω t −ω ν (t−s) e x(s) X ds (4.5) THE SOLVABILITY Let CxX0 = {x ∈ C([0, T ]; X) : x(0) = x0 }; CuH0 = {u ∈ C([0, T ]; H) : u(0) = u0 } Proposition 4.3 Suppose that the hypothesis (F ) satisfies Then SF (x, u) = ∅ for each x ∈ CxX0 , u ∈ CuH0 Moreover, the multimap SF is weakly u.s.c and has convex, weakly compact values We introduce the solution operator for given (x0 , u0 ) Φ : C([0, T ]; X) × C([0, T ]; H) → P(C([0, T ]; X) × C([0, T ]; H)), Φ(x, u) := S(·)x0 + t S(t − s)f (s)ds, f ∈ SF (x, u) W ◦ h(x(·)) 20 where W : L1 (0, T, X) → C([0, T ]; X), W (g)(t) = u(t, u0 , g), u is a unique strong solution of V I(u0 , g) Proposition 4.4 (1) If Ω ⊂ L1 (0, T ; H) is integral bounded then W (Ω) is equacontinuous in C([0, T ]; H) t ϑ({W (fn (t))}) ≤ ϑ{fn (s)}ds (2) W is a compact operator Consider Cauchy operator Q : L1 (0, T, X) → C([0, T ]; X), T Q(f )(t) = S(t − s)f (s)ds The solution map Φ is rewritten by Φ(x, u) := S(·)x0 + Q ◦ SF (x, u) W ◦ h(x(·)) We have the following result related to the operator Q Proposition 4.5 Let (A) hold If D ⊂ L1 (0, T ; X) is semicompact, then Q(D) is relatively compact in C(J; X) In particular, if sequence {fn } is semicompact and fn f ∗ in L1 (0, T ; X) then Q(fn ) → Q(f ∗ ) in C([0, T ]; X) Theorem 4.1 Suppose that (A), (B), (F ) and (H) hold Then the DVIs (4.1)(4.3) has a solution (x(·), u(·)) for each x0 ∈ X, u0 ∈ D(φ) ∩ U x −A x g(x, u) + F (x) Let Y = u , Y0 = u0 , A = B , F(Y ) = h(x) + ∂φ(u) , we convert the DVIs to the following system dY + AY ∈ F(Y ) dt Y (0) = Y0 , (4.6) (4.7) We give here the universal space to study behaviour of (4.6)-(4.7) as follows X := X × D(φ) ∩ U The m-semiflow generated by (4.6)-(4.7) is given G : R+ × X → X , G(t, x0 , u0 ) = {(x(t), u(t)) : Y is a solution of (4.6) − (4.7), x(0) = x0 , u(0) = u0 } 21 Denote that Σ(x0 , u0 , T ) is a set which contain all solution in [0, T ] of the problem with the initial condition (x0 , u0 ) and let Σ(x0 , u0 ) = ∪T >0 Σ(x0 , u0 , T ) We have G(t, x0 , u0 ) = {(x(t), u(t)) : (x(·), u(·)) ∈ Σ(x0 , u0 ), t ≥ 0, x0 ∈ X, u0 ∈ D(φ) ∩ U } Proposition 4.6 For each {(ξn , ηn )} ⊂ X such that ξn → ξ, ηn → η w.r.t in X and H Then Σ({(ξn , ηn , T )}) is a relatively compact in C([0, T ]; X) × C([0, T ]; H) In particular, Σ(ξ, η, T ) is a compact subset for each (ξ, η) ∈ X Corollary 4.1 The multimap G has compact values in X × H Lemma 4.3 G is a strict m-semiflow 4.3 Global attractor We need extra assumption: (A∗ ) The semigroup {S(t)}t≥0 is exponential stability with rate α, and χ-decreasing with exponential coefficient β, i.e S(t) L(X) ≤ N e−αt , S(t) where α, β > 0, N, P ≥ 1, norm · χ χ ≤ P e−βt , ∀t > 0, is norm of operator in the sense of MNC Lemma 4.4 Suppose that (A∗ ), (B), (F ) and (H) hold If we have β > 4P p, then there exist positive numbers T0 > and ζ ∈ [0, 1) such that for all T ≥ T0 , we have χ∗ (G(T, C, D)) ≤ ζχ(C), for all bounded subset (C, D) ∈ X Thus, G is asymptotically upper semicompact Lemma 4.5 For each t > 0, the m-semiflow G(t, ·, ·) is u.s.c Lemma 4.6 Ther exist an absorbing set of Gif the coefficients α, η1F , η2F , ηh , ω satisfying ω min{ , α} > max{η1F + ηh , η2F } ν Theorem 4.2 There exists a global attractor of m-semiflow G if the conditions of Lemma 4.4, 4.5 and 4.6 hold 22 4.4 Application Let Ω ⊂ Rn be bounded domain with C boundary Consider the following problem ∂Z (t, x) − ∆x Z(t, x) = f (t, x), ∂t f (t, x) ∈ [f1 (x, Z(t, x), u(t, x)), f2 (x, Z(t, x), u(t, x))], ∂u (t, x) − ∆x u(t, x) + β(u(t, x) − ψ(x)) h(x, Z(t, x)), ∂t Z(t, x) = u(t, x) = 0, x ∈ ∂Ω, t ≥ 0, Z(0, x) = Z0 (x), u(0, x) = u0 (x) (4.8) (4.9) (4.10) (4.11) (4.12) where f1 , f2 , h : Ω × R → R are continuous functions, the functions ψ is in H (Ω), ψ ≤ in ∂Ω and β : R → 2R is a maximal monotone graph   0 β(r) = R−  ∅ if r > 0, if r = 0, if r < Note that, parabolic variational inequality (4.10) can be read as follows ∂u (t, x) − ∆x u(t, x) = h(x, Z(t, x)) in {(t, x) ∈ Q := (0, T ) × Ω : u(t, x) ≥ ψ(x)}, ∂t ∂u (t, x) − ∆x u(t, x) ≥ h(x, Z(t, x)), in Q, ∂t u(t, x) ≥ ψ(x), ∀(t, x) ∈ Q which represents a rigorous and efficient way to treat dynamic diffusion problems with a free or moving boundary This model is called the obstacle parabolic problem (see V.Barbu(2010)) Let X = H = L2 (Ω), U = H01 (Ω) and U = H −1 (Ω), the norm in X and H is given by |u|2 = u2 (x)dx, u ∈ L2 (Ω) Ω The norm in H01 (Ω) is given by u |∇u(x)|2 dx = Ω By Poincar´e inequality, we get that u Define the multi-valued function −1/2 U ≤ λ1 |u| with inf{ ∇u X : u X F : X × H → P(X) F (¯ u, y¯) = {λf1 (x, u¯(x), y¯(x)) + (1 − λ)f2 (x, u¯(x), y¯(x)) : λ ∈ [0, 1]} and the operator A = ∆ : U → U ; D(A) = {H (Ω) ∩ H01 (Ω)} = 1} 23 Then (4.8)-(4.9) can be reformulated as Z (t) − AZ(t) ∈ F (Z(t), u(t)) where Z(t) ∈ X, u(t) ∈ H such that Z(t)(x) = Z(t, x) and u(t)(x) = u(t, x) It is known that in Vrabie (1987), S(t) generated by A is compact and exponentially stable, that is, S(t) L(X) ≤ e−λ1 t , The assumption (A∗ ) is satisfied We assume, in addition, that there exist nonnegative functions a1 , a2 ∈ L∞ (Ω) such that |f1 (x, p, q)| ≤ a1 (x)|p| + b1 (x)|q| + c1 (x), |f2 (x, p, q)| ≤ a2 (x)|q| + b2 (x)|q| + c2 (x), ∀x ∈ Ω, p ∈ R It is easy to see that F is a multimap with closed convex and compact values Moreover, ¯ u¯) ≤ max{ a1 ∞ , a2 ∞ } Z¯ X + max{ b1 ∞ , b2 ∞ } u¯ H F (Z, + max{|c1 |, |c2 |}, which implies (F ) Consider the parabolic variational inequality (4.10), putting B = −∆, where −∆ is Laplace operator u, −∆v := ∇u(x)∇v(x)dx, Ω then Bu, u ≥ λ1 u 2X So, the assumption (B) is testified with ω = λ1 The map h : Ω × R → R satisfies h(x, 0) = 0, ∀x ∈ Ω and |h(x, p1 )| ≤ b(x)|p1 | + c(x), ∀x ∈ Ω, p ∈ R, where b, c is nonnegative functions in L∞ (Ω) ¯ ¯ Let h : X → H, h(Z)(x) = h(x, Z(x)), we obtain ¯ H≤ b |h(Z)| ∞ Z¯ X + |c| Then the PVI (4.10) can be read as u (t) + Bu(t) + ∂IK (u(t)) h(Z(t)), where K = {u ∈ L2 (Ω) : u(y) ≥ ψ(x), for a.e x ∈ Ω}, ∂IK (u) = u ∈ L2 (Ω) : u(x)(v(x) − z(x))dx ≥ 0, ∀z ∈ K , Ω = {u ∈ L2 (Ω) : u(x) ∈ β(v(x) − ψ(x)), for a.e x ∈ Ω} It is easy to see that h(0) = ∈ ∂IK (0) and we have (H) We have the following result due to Theorem 4.2 Theorem 4.3 The m-semiflow generated by (4.8)-(4.12) admits a compact global attractor in L2 (Ω) × K provided that λ1 > max b ∞ + max{ a1 ∞, a2 ∞ }; max{ b1 ∞, b2 ∞} 24 Conclusions and recommendations Obtained results of the thesis The PhD thesis Behaviour of solutions to differential variational inequalities research on the stability of solutions for some classes differential variational inequalities This thesis consists of two essential differential variational inequalities: finite-dimensional and infinite-dimensional space problems In this thesis, we focused on three problems and obtained following results: • The existence of global solutions of differeantial variational inequalities • Behaviour of solutions to finite-dimensional differential variational inequalities with delay The existence of a decay solution with exponential rate and a global attractor for m-semigroup generated by DVI were proved • The existence of solutions and the existence of global attractors for m-semigroup generated by differential variational inequalities of parabolic-elliptic and parabolicparabolic types in infinite-dimensional spaces were found out Recommendations For future researches about differential variational inequalities, we propose the following recommendations: • Study the stability and the finite-time attractive for solutions of differential variational inequalities • In the case infinite-dimensional space, investigate the differential variational inequalities with multivalued solution set of constraints • Study the solvability and stability of differential variational inequalities with fraction derivative or neutral dynamics AUTHOR’S WORKS RELATED TO THE THESIS THAT HAVE BEEN PUBLISHED N.T.V Anh, T.D Ke (2015), ”Asymptotic behavior of solutions to a class of differential variational inequalities”, Annales Polonici Mathematici, 114.2, 147-164 (SCIE) N.T.V Anh, T.D Ke (2017), ”On the differential variational inequalities of parabolic-elliptic type”, Mathematical Methods in the Applied Sciences, 40(13), 4683–4695 (SCIE) N.T.V Anh, T.D Ke, ”On the differential variational inequalities of parabolicparabolic type”, submitted Results of the thesis have been reported at: • Seminar of Labour of Analysis, Department of Mathematics and Informatics, Hanoi National University of Education; • Conference for PHD students, Department of Mathematics and Informatics, Hanoi National University of Education, 2017; • Lecture scientific research conference of the Department of Mathematics and Informatics, Hanoi National University of Education, 2019 • Mini-workshop ”PDE 2019 Analysis and Numerics”, VIASM, Hanoi 9/2019