Về một số lớp phương trình parabolic không địa phương

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION LE TRAN TINH ON SOME CLASSES OF NONLOCAL PARABOLIC EQUATIONS DOCTORAL DISSERTATION OF MATHEMATICS Hanoi - 2020 MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION LE TRAN TINH ON SOME CLASSES OF NONLOCAL PARABOLIC EQUATIONS Speciality: Differential and Integral Equations Speciality Code: 9.46.01.03 DOCTORAL DISSERTATION OF MATHEMATICS Supervisor: PROF DR CUNG THE ANH Hanoi - 2020 DECLARATION I assure that my scientific results are completed under the guidance of Prof Dr Cung The Anh The results stated in the dissertation are completely honest and they have never been published in any scientific documents before All publications that work with other authors have been approved by them to include in the dissertation I take full responsibility for my research results in the dissertation February 5, 2020 Author Le Tran Tinh i ACKNOWLEDGEMENT This dissertation has been completed at Hanoi National University of Education under instruction of Prof Dr Cung The Anh, Faculty of Mathematics and Informatics, Hanoi National University of Education I wish to acknowledge my supervisor’s instruction with greatest appreciation and thanks I would like to thank all Professors and Assoc Professors who have taught me at Hanoi National University of Education and my friends for their help I also thank all the lecturers and PhD students at the seminar of Division of Mathematical Analysis for their encouragement and valuable comments I especially express my gratitude to my beloved parents, wife, brothers, and sons for their love and support Finally, my thanks go to Hong Duc University for financial support during my period of PhD study Hanoi, February 5, 2020 Le Tran Tinh ii CONTENTS i ii iii DECLARATION ACKNOWLEDGEMENT CONTENTS LIST OF SYMBOLS INTRODUCTION Chapter 1.1 1.2 1.3 1.4 13 PRELIMINARIES AND AUXILIARY RESULTS Function spaces 13 1.1.1 Banach and Hilbert spaces 13 1.1.2 The Lp spaces of Lebesgue integrable functions 14 1.1.3 Nonnegative integer order Sobolev spaces 15 1.1.4 Fractional order Sobolev spaces 17 1.1.5 Bochner spaces 20 Global attractors of partial differential equations 22 1.2.1 Existence of global attractors 22 1.2.2 Finite fractal dimension 30 1.2.3 A general diagram of studying global attractors for autonomous parabolic equations on bounded domains 31 Operators 31 1.3.1 Laplace and p-Laplace operators 31 1.3.2 Fractional Laplacian and regional fractional Laplacian 32 Some auxiliary results 36 Chapter GLOBAL ATTRACTORS FOR NONLOCAL PARABOLIC EQUATIONS WITH A NEW CLASS OF NONLINEARITIES 38 2.1 Problem setting 38 2.2 Existence and uniqueness of weak solutions 40 2.3 Existence of a global attractor 46 2.4 Fractal dimension estimates of the global attractor 51 2.5 Existence and exponential stability of stationary solutions 53 Chapter LONG-TIME BEHAVIOR OF SOLUTIONS TO A NONLOCAL QUASI- iii LINEAR PARABOLIC EQUATION 59 3.1 Problem setting 59 3.2 Existence and uniqueness of weak solutions 61 3.3 Existence of global attractors 68 3.3.1 The (L2 (Ω), L2 (Ω))-global attractor 68 3.3.2 The (L2 (Ω), Lq (Ω))-global attractor 70 3.3.3 The (L2 (Ω), W01,p (Ω) ∩ Lq (Ω))-global attractor 73 Existence and exponential stability of stationary solutions 76 3.4 Chapter GLOBAL ATTRACTORS FOR NONLOCAL PARABOLIC EQUATIONS INVOLVING THE FRACTIONAL LAPLACIAN AND THE REGIONAL FRACTIONAL LAPLACIAN WITH A NEW CLASS OF NONLINEARITIES 81 4.1 Problem setting 81 4.2 Existence and uniqueness of weak solutions 85 4.3 Existence of global attractors 93 4.4 Fractal dimension estimates of the global attractor 97 CONCLUSION AND FUTURE WORK LIST OF PUBLICATIONS 102 104 REFERENCES 105 iv LIST OF SYMBOLS R Set of real numbers R+ Set of positive real numbers RN N -dimensional Euclidean vector space A := B A is defined by B A⊂B A¯ A is a subset of B Closure of the set A dist(A, B) Hausdorff semidistance in two sets A and B,i.e., dist(A, B) = supx∈A inf y∈B kx − ykX Ω Nonempty open subset of RN ∂Ω Boundary of Ω ΩT ΩT := Ω × (0, T ) (., )X Inner product in the Hilbert space X kxkX Norm of x in the space X ut Partial derivative of u in variable t X∗ Dual space of the space X hx0 , xi Duality pairing between x0 ∈ X ∗ and x ∈ X X ,→ Y X is imbedded in Y X ,→,→ Y X is compactly imbedded in Y Lp (Ω) Space of p-integrable measurable functions on Ω Lp (∂Ω) Space of p-integrable measurable functions on ∂Ω L∞ (Ω) Space of essential bounded measurable functions on Ω C0∞ (Ω) ≡ D(Ω) Space of infinitely differentiable functions with compact support in Ω C(Ω) Space of continuous functions on C 0, () Space of Hăolder continuous functions of exponent on C 0, () Space of Hăolder continuous functions of exponent λ on ∂Ω C(X; Y ) Space of continuous functions from X to Y Lp (0, T ; X), < p < ∞ Space of functions f : [0, T ] → X such that L∞ (0, T ; X) RT kf (t)kpX dt < ∞ Space of functions f : [0, T ] → X such that kf (.)kX is almost everywhere bounded on [0, T ]    W s,p (Ω),       W s,p (∂Ω),       W s,p (Ω),   e s,p (Ω), W     e 0s,p (Ω),  W       H s (Ω),     H s (Ω) Sobolev spaces of nonnegative integer order or fractional order −m H (Ω) Dual space of H0m (Ω) H −s (Γ) Dual space of H s (Γ) x·y Scalar product between x, y ∈ RN ∇ ( ∂x∂ , ∂x∂ , · · · , ∂x∂ n ) ∆ Laplace operator ∆p p-Laplace operator (−∆)s Fractional Laplace operator AsΩ Regional fractional Laplace operator on Ω NΩs u Fractional normal derivative of the function u D(A) Domain of operator A {xk } Sequence of vectors xk xk → x xk converges strongly to x xk * x xk converges weakly to x xk *∗ x xk converges weakly-* to x i.e id est (that is) a.e Almost every P.V Cauchy principal value p 225 Page 225 The proof is complete 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 In the last couple of decades, 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 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 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 system The nonlocality in the equation can have different forms such as nonlocal source terms (see Y Chen and M Wang [22], P Souplet [68]), nonlocal boundary conditions (see C Mu, D Liu and S Zhou [55], Y Wang, C Mu and Z Xiang [75], H M Yin [80]), and nonlocal diffusion operators (see L Caffarelli [12], C.G Gal and M Warma [39], N Pan, B Zhang and J Cao [58], P Pucci, M Xiang and B Zhang [59], M Xiang, V D R˘adulescu and B Zhang [78]) They can be in space, time or both time and space The most common one is perhaps the nonlocal diffusion operators Nonlocal properties generates a lot of difficulties which we encounter in analysis of the problem, for instance, the uniqueness and regularity of weak solutions cannot be guaranteed Therefore, 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 nonlocal diffusion operators which are involving the content of my dissertation Let us first consider the class of nonlocal parabolic problems involving Laplace operator which is nonlocal in the sense that the diffusion coefficient is determined by a global quantity These problems arise in various physical situations such as migration of populations, heat propagation, propagation of mutant genes, epidemic theory or nonlinear vibration theory, etc We now list some results in recent years In 1997, M Chipot and B Lovat [27] studied the following nonlocal problem    u − a(l(u))∆u = f (t, x), x ∈ Ω, t > 0,   t x ∈ ∂Ω, t > 0, u(x, t) = 0,    u(x, 0) = u (x), (1) x ∈ Ω, where Ω is a bounded smooth domain in RN (N ≥ 1), l : L2 (Ω) → R is a continuous functional, a is continuous function from R to R+ , f ∈ L2loc ([0, ∞), H −1 (Ω)) Under specific conditions, they proved the existence and uniqueness of solutions for homogeneous or nonhomogeneous cases In 1999, M Chipot and B Lovat [26] studied system (1) with the mixed boundary condition in place of the Dirichlet boundary condition They investigated the existence and uniqueness of a weak solution and its asymptotic behaviour In 2000, A S Ackleh and L Ke [1] studied the following nonlocal problem    u − R1 ∆u = f (u), x ∈ Ω, t > 0,   t a( Ω udx) x ∈ ∂Ω, t > 0, u(x, t) = 0,    u(x, 0) = u (x), x ∈ Ω, where a is locally Lipschitz continuous such that a(s) > for all s 6= and a(0) ≥ 0, f is locally Lipschitz continuous satisfying f (0) = They proved the existence and uniqueness of strong solutions and investigated conditions on u0 for the extinction in finite time and for the persistence of solutions They also gave some numerical results in one dimension In 2004, F J S A Corrˆea et al [33] gave an extension of the result for system (1) obtained in M Chipot and B Lovat [27], considering a = a(l(u)) and f = f (x, u) continuous functions We see that the nonlinearity appears not only in the diffusive operator but also in source term Under regularity conditions on a, f and u0 , they considered both stationary and evolutionary problems In the evolution case, they investigated the existence, uniqueness and asymptotic behaviour of solutions In 2005, S Zheng and M Chipot [82] studied the asymptotic behaviour of solutions to the nonlinear parabolic equation (1) as time tends to infinity with two classes of nonlocal terms a = a(l(u)) and a = a(k∇uk2L2 (Ω) ), f ∈ L2 (Ω) and u0 ∈ H01 (Ω) for any v ∈ Lp (0, T ; W01,p (Ω)) We deduce that n   k∇un kpLp (Ω) −div a  p−2 |∇un | ∇un o 0 is bounded in Lp (0, T ; W −1,p (Ω)) On the other hand, it follows from (3.4) that |f (u)| ≤ C(|u|q−1 + 1) Using this together with the boundedness of {un } in Lq (ΩT ), one can shows that {f (un )} is bounded in Lq (ΩT ) We rewrite the equation as dun = g + div a k∇un kpLp (Ω) |∇un |p−2 ∇un − f (un ) dt     Therefore, dun dt   (3.16) is bounded in V ∗ In addition, we have the following chain of embeddings 0 W01,p (Ω) ,→,→ Lp (Ω) ,→ W −1,p (Ω) + Lq (Ω) Thanks to the Aubin-Lions lemma, we deduce that {un } is compact in Lp (0, T ; Lp (Ω)) Now applying the diagonalization procedure and using Lemma 1.3 in [50, p 12] and Theorem 1.1.8, we obtain (up to a subsequence) that un * u in Lp (0, T ; W01,p (Ω)), un → u in Lp (0, T ; Lp (Ω)), dun du * in V ∗ , dt dt un (T ) → u(T ) in L2 (Ω), and f (un ) * f (u)    in Lq (ΩT ),  (3.17) 0 −div a k∇un kpLp (Ω) |∇un |p−2 ∇un * −χ in Lp (0, T ; W −1,p (Ω)) (3.18) Now, passing to the limit in (3.16), one has in the distributional sense in ΩT ut − χ + f (u) = g (3.19) Integrating (3.11) from to T leads to Z T a  k∇un kpLp (Ω) Z p Z Z |∇un | dxdt = gun dxdt − Ω ΩT + 65 f (un )un dxdt ΩT kun (0)k2L2 (Ω) − kun (T )k2L2 (Ω) (3.20) Since lim kun (T )k2L2 (Ω) = ku(T )k2L2 (Ω) and lim kun (0)k2L2 (Ω) = ku0 k2L2 (Ω) , we deduce n→∞ n→∞ from (3.20) that Z T lim a n→∞  k∇un kpLp (Ω) Z Z Z p gudxdt − |∇un | dxdt = Ω f (u)udxdt ΩT ΩT + ku0 k2L2 (Ω) − ku(T )k2L2 (Ω) (3.21) On the other hand, from Lemma 3.2.3 we have Z   a ΩT k∇un kpLp (Ω)  p−2 |∇un | ∇un − a  k∇vkpLp (Ω)  p−2 |∇v|  ∇v ·∇(un −v)dxdt ≥ We derive by taking limit for any v ∈ Lp (0, T ; W01,p (Ω)) Z T lim n→∞ a Z0 − ΩT  k∇un kpLp (Ω)  Z Z p T |∇un | dxdt + Ω hχ, vi dt  a k∇vkpLp (Ω) |∇v|p−2 ∇v · ∇(u − v)dxdt ≥ Therefore, in view of (3.21) and the last inequality, we have Z Z gudxdt − ΩT f (u)udxdt + ku0 k2L2 (Ω) ΩT Z − a  ΩT − k∇vkpLp (Ω)  ku(T )k2L2 (Ω) Z T hχ, vi dt + |∇v|p−2 ∇v · ∇(u − v)dxdt ≥ (3.22) On the other hand, by integrating (3.19) from to T after taking inner product with u, we obtain Z T Z − hχ, ui dt = Z gudxdt − ΩT f (u)udxdt + ku0 k2L2 (Ω) ΩT − ku(T )k2L2 (Ω) (3.23) Combining (3.22) with (3.23), we have T Z D     E χ − div a k∇vkpLp (Ω) |∇v|p−2 ∇v , u − v dt ≤ 0, ∀v ∈ Lp (0, T ; W01,p (Ω)) Choosing v = u − δϕ, we see that T Z D     E     E χ − div a k∇(u − δϕ)kpLp (Ω) |∇(u − δϕ)|p−2 ∇(u − δϕ) , ϕ dt ≤ 0, if δ > and Z T D χ − div a k∇(u − δϕ)kpLp (Ω) |∇(u − δϕ)|p−2 ∇(u − δϕ) , ϕ dt ≥ 0, if δ < 0, for all ϕ ∈ Lp (0, T ; W01,p (Ω)) Letting δ → 0, we get Z T D     E χ − div a k∇ukpLp (Ω) |∇u|p−2 ∇u , ϕ dt = 0, ∀ϕ ∈ Lp (0, T ; W01,p (Ω)) 66     0 This implies that χ = div a k∇ukpLp (Ω) |∇u|p−2 ∇u in Lp (0, T ; W −1,p (Ω)) We now need to check that u(0) = u0 Choosing a test function ϕ ∈ C ([0, T ]; W01,p (Ω) ∩ Lq (Ω)) with ϕ(T ) = We see that ϕ ∈ Lp (0, T ; W01,p (Ω)) ∩ Lq (ΩT ) Taking integration by parts in the t variable, we have Z Z − Z un (0)ϕ(0)dx − un ϕ dxdt + Ω a ΩT  ΩT k∇un kpLp (Ω)  |∇un |p−2 ∇un · ∇ϕdxdt Z Z + f (un )ϕdxdt = ΩT gϕdxdt ΩT Doing the same in the Galerkin approximations and taking limit as n → ∞ we obtain Z Z − Z u0 ϕ(0)dx − uϕ dxdt + Ω a ΩT  ΩT k∇ukpLp (Ω)  |∇u|p−2 ∇u · ∇ϕdxdt Z Z gϕdxdt f (u)ϕdxdt = + (3.24) ΩT ΩT On the other hand, from (3.6), we have Z − Z u(0)ϕ(0)dx − Ω Z  uϕ dxdt + ΩT ΩT  a k∇ukpLp (Ω) |∇u|p−2 ∇u · ∇ϕdxdt Z Z gϕdxdt f (u)ϕdxdt = + Then, comparing (3.24) with (3.25), it holds that R (3.25) ΩT ΩT u ϕ(0)dx = Ω R Ω u(0)ϕ(0)dx This leads to u(0) = u0 This completes the proof of the existence of the weak solution Moreover, analogously to (3.15) we have ku(t)k2L2 (Ω) ≤ ku0 k2L2 (Ω) e−mλ1 t +
(c5 + c6 ) − e−mλ1 t mλ1 (3.26) This implies that the weak solution u exists globally in time ii) Uniqueness and continuous dependence on the initial data Let u, v be two weak solutions to (3.1) with initial data u0 , v0 ∈ L2 (Ω), respectively Taking w = u − v, and then the following equations are directly obtained from (3.1) by subtraction   wt −      div a k∇ukpLp (Ω)   +div a  k∇vkpLp (Ω) |∇u|p−2 ∇u   |∇v|p−2 ∇v    w(0) = u − v 0 67  + f (u) − f (v) = 0, (3.27) Multiplying (3.27) by w and integrating over Ω, one gets 1d kwk2L2 (Ω) dtZ    + ZΩ    a k∇ukpLp (Ω) |∇u|p−2 ∇u − a k∇vkpLp (Ω) |∇v|p−2 ∇v · ∇(u − v)dx (f (u) − f (v))(u − v)dx = + Ω It follows from (3.5) and Lemma 3.2.3 that d kwk2L2 (Ω) ≤ 2c3 kwk2L2 (Ω) dt Applying the Gronwall inequality, we obtain kwk2L2 (Ω) ≤ kw(0)k2L2 (Ω) e2c3 t This completes the proof 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 For the sake of brevity, in the following propositions, we just give some formal calculations, their rigorous proofs are done by use of Galerkin approximations and Lemma 11.2 in [64] √ We see from (3.26) that the ball B0 = B( ρ0 ) with ρ0 = (L2 (Ω), L2 (Ω))-bounded mλ1 (c5 + c6 ), is an absorbing set of {S(t)}t≥0 , i.e., for any bounded set B in L2 (Ω), there exists T0 = T0 (B) depending only on the L2 -norm of B such that kS(t)u0 k2L2 (Ω) ≤ ρ0 , for all t ≥ T0 , u0 ∈ B Proposition 3.3.1 The semigroup {S(t)}t≥0 has an (L2 (Ω), W01,p (Ω))-bounded absorbing set B1 Proof First, as in (3.12) we have d kuk2L2 (Ω) + c4 dt Z p Z |∇u| dx + Ω |u| dx Ω 68 q  ≤ c5 Integrating the above inequality from t to t + 1, for t ≥ T0 , and using u(t) ∈ B0 we have Z t t+1 k∇u(s)kpLp (Ω) ds ≤ c5 + ρ c4 (3.28) Now, multiplying the first equation in (3.1) by −∆p u, we get 1d k∇ukpLp (Ω) + k∇ukpLp (Ω) + a k∇ukpLp (Ω) k∆p uk2L2 (Ω) p dt p Z p = k∇ukLp (Ω) − f (u)|∇u|p dx − hg, ∆p ui p Ω   Putting this together with (3.2) and (3.5) leads to 1d 1 k∇ukpLp (Ω) + k∇ukpLp (Ω) + mk∆p uk2L2 (Ω) ≤ ( + c3 )k∇ukpLp (Ω) − hg, ∆p ui (3.29) p dt p p On the other hand, using the Cauchy inequality we have 1 ( + c3 )k∇ukpLp (Ω) − hg, ∆p ui = −( + c3 ) hu, ∆p ui − hg, ∆p ui p p (1/p + c3 )2 ≤ mk∆p uk2L2 (Ω) + kgk2L2 (Ω) + kuk2L2 (Ω) 2m 2m (3.30) In view of (3.29) and (3.30) with note that u(t) ∈ B0 for all t ≥ T0 , we have p (1 + pc3 )2 ρ0 d k∇ukpLp (Ω) + k∇ukpLp (Ω) ≤ R1 := kgk2L2 (Ω) + dt 2m 2m (3.31) Applying the uniform Gronwall inequality to (3.28) and (3.31) we have k∇u(t)kpLp (Ω) ≤ R1 + c5 + ρ , c4 ∀t ≥ T0 + This makes sure that the ball B1 = BW 1,p (Ω) (ρ−p ) with ρ1 = (1+1/λ1 ) R1 + a bounded absorbing set in set B in L2 (Ω), W01,p (Ω) c5 +ρ0 c4
, is for the semigroup {S(t)}t≥0 , i.e., for any bounded there exists T1 = T1 (B) := T0 + depending only on the L2 -norm of B such that kS(t)u0 kp W01,p (Ω) ≤ ρ1 , (3.32) for all t ≥ T1 , u0 ∈ B 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 69 The (L2 (Ω), Lq (Ω))-global attractor 3.3.2 In this and the next subsections, 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) First, we prove the existence of a bounded absorbing set in W01,p (Ω) ∩ Lq (Ω) for the semigroup {S(t)}t≥0 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 Z Z p |u|q dx ≤ ρ2 , |∇u| dx + Ω Ω for all t ≥ T2 and u0 ∈ B, where u is the unique weak solution of (3.1) with the initial datum u0 Proof Multiplying the first equation in (3.1) by u and integrating by parts, we have 1d kuk2L2 (Ω) + a k∇ukpLp (Ω) kukp 1,p + W0 (Ω) dt   Z Z f (u)udx = gudx Ω Ω Then integrating this inequality over [t, t + 1] with t ≥ T1 , we derive Z t+1 h a t  k∇ukpLp (Ω) Z Z p |∇u| dx + Z f (u)udx − Ω Ω for all t ≥ T1 We define i gudx ds ≤ Ω ρ0 (3.33) u Z F (u) = f (s)ds Due to (3.4) and (3.5), it fulfills the bounds for some positive constants c7 , c8 c7 |u|q − c8 ≤ F (u) ≤ uf (u) + c3 |u| (3.34) c3 ρ (3.35) Therefore, Z Z F (u)dx ≤ f (u)udx + Ω Ω We deduce from (3.33) and (3.35) that Z t t+1 h a k∇ukpLp (Ω) p  Z Ω p Z Z |∇u| dx + F (u)dx − Ω 70 i gudx ds ≤ Ω ρ0 (c3 + 1) (3.36) On the other hand, multiplying (3.1) by ut , we obtain kut k2L2 (Ω) + a  k∇ukpLp (Ω) Z |∇u|p−2 ∇u · ∇ut dx + Z Z f (u)ut dx − Ω Ω gut dx = Ω We can rewrite the last equality as kut k2L2 (Ω) Z Z Z d a k∇ukpLp (Ω) |∇u|p dx + F (u)dx − gudx + dt p Ω Ω Ω d = a0 (k∇ukpLp (Ω) ) k∇ukpLp (Ω) k∇ukpLp (Ω) p dt h  i (3.37) Setting L = sup |a0 (s)|, then from (3.31), (3.32) and (3.37), we have 0≤s≤ρ1 d a k∇ukpLp (Ω) dt p  h Z Z p |∇u| dx + Ω Z i F (u)dx − Ω gudx ≤ Ω LR12 p (3.38) Therefore, from (3.36) and (3.38), by using the uniform Gronwall inequality, we get a k∇ukpLp (Ω) p  Z |∇u|p dx + Ω Z Z F (u)dx − Ω gudx ≤ Ω ρ0 (c3 + 1) LR12 + p Using (3.2), (3.4) and the Cauchy inequality for the term R Ω (3.39) gudx, we deduce from (3.39) and (3.34) that for all t ≥ T2 = T1 + 1: Z p Z |∇u| dx + Ω q |u| dx ≤ ρ2 := c8 |Ω| + ρ0 (1 + c3 /2) + LR12 /p + kgk2L2 (Ω) /2 Ω n m p ; c7 o This ends the proof 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 Proof Choosing Y = L2 (Ω), X = W01,p (Ω)∩Lq (Ω), the conclusion follows immediately from Theorem 1.2.17 The set B2 obtained in Proposition 3.3.3 is also of course an (L2 (Ω), Lq (Ω))-bounded absorbing set for the semigroup {S(t)}t≥0 To prove the existence of a global attractor in Lq (Ω), we will use Theorem 1.2.31 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 Proof We know that {S(t)}t≥0 has an (L2 (Ω), Lq (Ω))-bounded absorbing set B2 and {S(t)}t≥0 has an (L2 (Ω), L2 (Ω))-global attractor By Lemma 1.2.31, it is sufficient to 71 prove that for any ε > and any bounded subset B ⊂ L2 (Ω), there exist two positive constants T = T (ε, B) and M = M (ε) such that Z |u|q < Cε, Ω(|u|≥M ) for all u0 ∈ B and t ≥ T , where the constant C is independent of ε and B, where Ω(u ≥ M ) := {x ∈ Ω : u(x) − M ≥ 0} It follows from Lemma 1.2.23 that for any fixed ε > 0, there exist δ > 0, T = T (B) and M = M (ε) such that the Lebesgue measure |Ω(|S(t)u0 | ≥ M )| ≤ δ for all u0 ∈ B and t ≥ T and Z |g|2 < ε (3.40) Ω(|S(t)u0 |≥M ) We now multiply the first equation in (3.1) by (u − M )q−1 + to get that ut (u − M )q−1 +   − div a k∇ukpLp (Ω) + f (u)(u − M )q−1 + =  p−2 |∇u|  ∇u (u − M )q−1 + (3.41) g(x)(u − M )q−1 + , where (u − M )+ denotes the positive part of (u − M ), that is, (u − M )+ =  u − M, if u ≥ M, 0, if u < M, and M is a positive constant We deduce from (3.4) that f (u) ≥ e c|u|q−1 with u ≥ M and M is large enough Thus f (u)(u − M )q−1 c|u|q−1 (u − M )q−1 + ≥e + e c q−1 e c (u − M )q−1 = |u|q−1 (u − M )q−1 + + |u| + 2 e c e c 2(q−1) ≥ (u − M )+ + |u|q−2 (u − M )q+ 2 e c e c 2(q−1) ≥ (u − M )+ + M q−2 (u − M )q+ 2 (3.42) In addition e c |g|2 2(q−1) (u − M ) g(u − M )q−1 ≤ + + + 2e c It follows from (3.41), (3.42) and (3.43) that 1d q dt Z (u − M )q+ dx + (q Z − 1) Ω(u≥M ) Ω(u≥M ) e c + M q−2 (3.43)   (u − M )q+ dx ≤ 2e c a k∇ukpLp (Ω) |∇u|p (u − M )q−2 + dx Z Ω(u≥M ) Z |g|2 dx, Ω(u≥M ) and then d dt Z Ω(u≥M ) (u − M )q+ dx + e c qM q−2 Z (u − M )q+ dx Ω(u≥M ) 72 q ≤ 2e c Z Ω(u≥M ) |g|2 dx By the Gronwall inequality, we have Z (u − M )q+ dx ≤e − 2ec qM q−2 t Z (u(0) − M )q+ dx Ω(u≥M ) Ω(u≥M ) e c q−2 − e− qM + e c2 M q−2 t Z |g|2 dx Ω(u≥M ) If we take M large enough, taking (3.40) into account, the last inequality leads to Z (u − M )q+ dx < ε (3.44) Ω(u≥M ) Repeating the same steps above, just taking (u + M )− instead of (u − M )+ where (u + M )− = we also obtain  u + M if u ≤ −M, 0 if u > −M, Z |(u + M )− |q dx < ε, (3.45) Ω(u≤−M ) where Ω(u ≤ −M ) := {x ∈ Ω : u(x) + M ≤ 0} In both cases, we deduce from (3.44) and (3.45) that Z (|u| − M )q dx < ε, Ω(|u|≥M ) for M large enough Therefore, Z q Z (|u| − M + M )q dx |u| dx = Ω(|u|≥2M ) Ω(|u|≥2M ) q Z q ≤2 (|u| − M ) dx + Ω(|u|≥2M ) ≤ 2q+1 q Z M q dx Ω(|u|≥2M ) Z (|u| − M )q dx Ω(|u|≥2M ) < Cε, for M large enough and C is independent of ε and B As a consequence, the semigroup {S(t)}t≥0 has an (L2 (Ω), Lq (Ω))-global attractor Aq 3.3.3 The (L2 (Ω), W01,p (Ω) ∩ Lq (Ω))-global attractor 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 kut (s)k2L2 (Ω) ≤ ρ3 , for all u0 ∈ B, and s ≥ T3 , where ut (s) = d (S(t)u0 )|t=s and ρ3 is a positive constant independent of u0 dt 73 Proof By differentiating the first equation in (3.1) in time and denoting v = ut , we get    vt − div a k∇ukpLp (Ω) |∇u|p−2 ∇v     − (p − 2)div a k∇ukpLp (Ω) |∇u|p−4 (∇u · ∇v)∇u  − p div a0 (k∇ukpLp (Ω) )   Z |∇u|p−2 (∇u · ∇v)dx|∇u|p−2 ∇u + f (u)v = Ω Multiplying the above equality by v, integrating over Ω and using (3.5), we have 1d kvk2L2 (Ω) + a k∇ukpLp (Ω) dt  Z  + (p − 2)a k∇ukpLp (Ω)  + pa0 k∇ukpLp (Ω) Z |∇u|p−2 |∇v|2 dx Ω |∇u|p−4 (∇u · ∇v)2 dx Ω  Z k∇ukp−2 (∇u · ∇v)dx (3.46) 2 Ω ≤ c3 kvk2L2 (Ω) Since a is nondecreasing, it follows from (3.46) that d kvk2L2 (Ω) ≤ 2c3 kvk2L2 (Ω) dt (3.47) On the other hand, we deduce from (3.31), (3.37), (3.38) and (3.39) that Z t+1 kut k2L2 (Ω) dx ≤ C, (3.48) t for some positive constant C and t ≥ T2 Combining (3.47) with (3.48) and using the uniform Gronwall inequality we obtain kut k2L2 (Ω) ≤ ρ3 , as t ≥ T3 = T2 + 1, and ρ3 is a some positive constant The proof is complete Lemma 3.3.7 Let p ≥ Then under the assumption (H1bis), we have for all u1 , u2 ∈ W01,p (Ω), that D     −div a(k∇u1 kpLp (Ω) )|∇u1 |p−2 ∇u1 + div a(k∇u2 kpLp (Ω) )|∇u2 |p−2 ∇u2 , u1 − u2 = Z  Ω a(k∇u1 kpLp (Ω) )|∇u1 |p−2 ∇u1 ≥ cp ku1 − u2 kp W01,p (Ω) − a(k∇u2 kpLp (Ω) )|∇u2 |p−2 ∇u2 ,  E · ∇(u1 − u2 )dx (3.49) where cp =  m if p = 2,  m if p > 8.3p/2 74 Proof One sees that (3.49) is equivalent to proving that for p ≥ 2, x, y ∈ RN , we have a(|x|p )|x|p−2 x − a(|y|p )|y|p−2 y, x − y ≥ cp |x − y|p (3.50) Here h., i be the standard scalar product in RN Following the ideas in [36, Lemma 4.4], we have I(p) = a(|x|p )|x|p−2 x − a(|y|p )|y|p−2 y, x − y Z = d a(|sx + (1 − s)y|p )|sx + (1 − s)y|p−2 (sx + (1 − s)y) ds, x − y ds h i  Z a(|sx + (1 − s)y|p )|sx + (1 − s)y|p−2 ds = |x − y| Z a(|sx + (1 − s)y|p )|sx + (1 − s)y|p−4 | h(sx + (1 − s)y), x − yi |2 ds + (p − 2) Z +p a0 (|sx + (1 − s)y|p )|sx + (1 − s)y|2p−4 | hsx + (1 − s)y, x − yi |2 ds Z |sx + (1 − s)y|p−2 ds ≥ m|x − y| • When p = then we get (3.50) from the above inequality with cp = m • Now, we consider the case p > If |x| ≥ |x − y|, we have |sx + (1 − s)y| = |x − (1 − s)(x − y)| ≥ ||x| − (1 − s)|x − y|| ≥ s|x − y| Therefore, Z p sp−2 ds = I(p) ≥ m|x − y| m |x − y|p p−1 If |x| < |x − y|, we have |sx + (1 − s)y| = |x + (1 − s)(y − x)| ≤ |x| + (1 − s)|x − y| < (2 − s)|x − y| Therefore, Z I(p) ≥ m|x − y| ≥ m Z p (|sx + (1 − s)y|2 ) ds (2 − s)|x − y|2 p (|sx + (1 − s)y|2 ) ds Z  p2 m |sx + (1 − s)y|2 ds ≥
p m = |x|2 + hx, yi + |y|2 p 32 m p ≥ p |x − y| 32 So we conclude for the case p > that I(p) ≥ cp |x − y|p with cp = 75 m 8.3p/2 To prove the existence of a global attractor in W01,p (Ω), we will use Theorem 1.2.29 We are now in the position to state the main result of this section 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 Proof By Theorem 1.2.29, Proposition 3.3.3 and Proposition 3.3.4, we only need to show that the semigroup {S(t)}t≥0 is (L2 (Ω), W01,p (Ω) ∩ Lq (Ω))-asymptotically compact This means that we take a bounded subset B of L2 (Ω), we have to show that 1,p q for any {u0n } ⊂ B and tn → +∞, {un (tn )}∞ n=1 is precompact in W0 (Ω) ∩ L (Ω), where un (tn ) = S(tn )u0n By Theorem 3.3.5, it is sufficient to verify that {un (tn )}∞ n=1 is precompact in W01,p (Ω) To this, we will prove that {un (tn )} is a Cauchy sequence in W01,p (Ω) Thanks to Theorem 3.3.2 and Theorem 3.3.5, one has that {un (tn )} is a Cauchy sequence in L2 (Ω) and in Lq (Ω) It follows from (3.49) that cp kun (tn ) − um (tm )kp W01,p (Ω)   d d ≤ − un (tn ) − f (un (tn )) + um (tm ) + f (um (tm )), un (tn ) − um (tm ) dt dt Z d d ≤ | un (tn ) − um (tm )| |un (tn ) − um (tm )|dx dt Ω dt Z |f (un (tn )) − f (um (tm ))| |un (tn ) − um (tm )|dx + Ω d d un (tn ) − um (tm )kL2 (Ω) kun (tn ) − um (tm )kL2 (Ω) dt dt + kf (un (tn )) − f (um (tm ))kLq0 (Ω) kun (tn ) − um (tm )kLq (Ω) ≤k It follows from Lemma 3.3.6 and the boundedness of {f (un (tn ))} in Lq (Ω) that {un (tn )} is a Cauchy sequence in W01,p (Ω) This completes the proof 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(k∇u∗ kpLp (Ω) ) Z |∇u∗ |p−2 ∇u∗ · ∇vdx + Ω Z Ω for all test functions v ∈ W01,p (Ω) ∩ Lq (Ω) 76 f (u∗ )vdx = Z gvdx, Ω (3.51) Theorem 3.4.1 Under the hypotheses (H1), (H2), and (H3), the problem (3.1) has at least one weak stationary solution u∗ satisfying ku∗ kp + ku∗ kqLq (Ω) ≤ `, W01,p (Ω) where p0 `= 2p0 c0 |Ω|(pmλ1 ) p + |Ω| (p−2)p0 2p (3.52) kgkpL2 (Ω) 2p p0 p 1, 2c m mp (pmλ1 )  Moreover, if f satisfies f (s) ≥ α > for all s ∈ R, (3.53) then for any solution u of (3.1), we have ku(t) − u∗ k2L2 (Ω) ≤ ku(0) − u∗ k2L2 (Ω) e−2αt for all t > (3.54) That is, the weak stationary solution of (3.1) is unique and exponentially stable Proof i) Existence We find an approximate stationary solution un by n X un = γnj ej , j=1 1,p q where {ej }∞ j=1 is a basis of W0 (Ω) ∩ L (Ω) For each n > 1, we denote Vn = span{e1 , e2 , , en } It follows from (3.51) that a(k∇un kpLp (Ω) ) Z p−2 |∇un | Z Z ∇un · ∇vdx + Ω f (un )vdx = Ω gvdx, (3.55) Ω for all test functions v ∈ Vn We construct the operator Rn : Vn → Vn by [Rn u, v] = a  k∇ukpLp (Ω) Z p−2 |∇u| Z Z ∇u · ∇vdx + f (u)vdx − Ω Ω gvdx, Ω for all u, v ∈ Vn Due to the Cauchy inequality and (3.8) with ε = m/2, it follows from (3.2) and (3.4) that [Rn u, u] = a  k∇ukpLp (Ω) Z p Z Z |∇u| dx + Ω f (u)udx − gudx Ω Ω (p−2)p0 |Ω| 2p m p0 kgk ≥ kukp 1,p + c1 kukqLq (Ω) − c0 |Ω| − p L2 (Ω) W0 (Ω) p0 (pmλ1 /2) p p0 = h m kukp W01,p (Ω) 2c1 m ≥ 1, m n + 2c1 kukqLq (Ω) − m oh kukp 1,p W0 (Ω) 2p0 c0 |Ω|(pmλ1 ) p + |Ω| + kukqLq (Ω) 77 (p−2)p0 2p 0 kgkpL2 (Ω) 2p i p0 mp0 (pmλ1 ) p i −` , (3.56) for all u ∈ Vn , where p0 2p0 c0 |Ω|(pmλ1 ) p + |Ω| `= 2c1 m  1, (p−2)p0 2p kgkpL2 (Ω) 2p p0 mp0 (pmλ1 ) p We deduce from (3.56) that [Rn u, u] ≥ for all u ∈ Vn satisfying kukp kukqLq (Ω) W01,p (Ω) + = ` Consequently, by Lemma 1.4.4, there exists un ∈ Vn such that Rn (un ) = with kun kp + kun kqLq (Ω) ≤ ` W01,p (Ω) W01,p (Ω) ∩ Therefore, {un } is bounded in (3.57) Lq (Ω) By the compactness of the injection W01,p (Ω) ∩ Lq (Ω) ,→ L2 (Ω), we can extract a subsequence of {un } (relabeled the same) that converges weakly in W01,p (Ω) ∩ Lq (Ω) and strongly in L2 (Ω) to an element u∗ ∈ W01,p (Ω) ∩ Lq (Ω) Thus, it has a.e convergent   subsequence  in Ω Moreover,  f (un ) is bounded in Lq (Ω), f ∈ C (R) and −div a k∇un kpLp (Ω) |∇un |p−2 ∇un is bounded in W −1,p (Ω) An application of diagonalization procedure and using [50, Lemma 1.3, p.12] and Theorem 1.1.8, it follows that (up to a subsequence) f (un ) * f (u∗ ) in Lq (Ω),     −div a k∇un kpLp (Ω) |∇un |p−2 ∇un * −χ in W −1,p (Ω) (3.58) Replacing v = un in (3.55) leads to a  k∇un kpLp (Ω) Z Z p Z |∇un | dx = gun dx − Ω f (un )un dx Ω Ω Hence lim a n→∞  k∇un kpLp (Ω) Z Z p ∗ |∇un | dx = Z f (u∗ )u∗ dx gu dx − Ω Ω (3.59) Ω Using (3.9), we have Z   Ω     a k∇un kpLp (Ω) |∇un |p−2 ∇un − a k∇vkpLp (Ω) |∇v|p−2 ∇v · ∇(un − v)dx ≥ 0, for all v ∈ W01,p (Ω) ∩ Lq (Ω) Therefore a  k∇un kpLp (Ω) Z Z p |∇un | dx − Ω a  Ω −a  k∇vkpLp (Ω) k∇un kpLp (Ω) Z  |∇un |p−2 ∇un · ∇vdx |∇v|p−2 ∇v · ∇(un − v)dx ≥ Ω We derive by taking limit for any v ∈ W01,p (Ω) ∩ Lq (Ω) that lim a n→∞  k∇un kpLp (Ω) Z |∇un |p dx Ω + hχ, vi − a  k∇vkpLp (Ω) Z 78 Ω |∇v|p−2 ∇v · ∇(u∗ − v)dx ≥ (3.60) In view of (3.59) and (3.60), one gets that Z Z ∗ ∗ gu dx − ∗ f (u )u dx + hχ, vi − a Ω  Ω k∇vkpLp (Ω) Z |∇v|p−2 ∇v · ∇(u∗ − v)dx ≥ Ω (3.61) In addition, we deduce the ’limit equation’ from (3.55) and (3.58) that −χ + f (u∗ ) = g This implies that Z ∗ Z ∗ − hχ, u i = f (u∗ )u∗ dx gu dx − Ω (3.62) Ω Putting (3.61) and (3.62) together, we obtain D   χ − div a k∇vkpLp (Ω)  p−2 |∇v|  E ∗ ∇v , u − v ≤ 0, for all v ∈ W01,p (Ω) ∩ Lq (Ω) Taking v = u∗ − δw, and then let δ → 0, we have   χ = div a k∇u∗ kpLp (Ω)  ∗ p−2 |∇u | ∗ ∇u  Taking everything into consideration, we infer that u∗ ∈ W01,p (Ω) ∩ Lq (Ω) is the weak stationary solution to problem (3.1) The inequality (3.52) is obtained directly from (3.57) as n tends to infinity ii) Uniqueness and exponential stability Denote w(t) = u(t) − u∗ , one gets Z wt vdx + Z   a Ω Ω k∇ukpLp (Ω)  p−2 |∇u| ∇u − a  k∇u∗ kpLp (Ω)  Z ∗ p−2 |∇u | ∗ ∇u  · ∇vdx (f (u) − f (u∗ ))vdx = 0, + Ω for all test functions v ∈ W01,p (Ω) ∩ Lq (Ω) In particular, choosing v = w, we have 1d kwk2L2 (Ω) + dt + Z   a Ω Z (f (u) − f (u∗ ))(u − u∗ )dx Ω k∇ukpLp (Ω)  |∇u|p−2 ∇u −a  k∇u∗ kpLp (Ω)  ∗ p−2 |∇u | ∇u ∗  · ∇(u − u∗ )dx = In view of (3.53) and Z   Ω     a k∇ukpLp (Ω) |∇u|p−2 ∇u − a k∇u∗ kpLp (Ω) |∇u∗ |p−2 ∇u∗ · ∇(u − u∗ )dx ≥ 0, we infer that d kwk2L2 (Ω) + 2αkwk2L2 (Ω) ≤ dt This concludes the proof by using the Gronwall inequality 79