Tài liệu hạn chế xem trước, để xem đầy đủ mời bạn chọn Tải xuống
1
/ 112 trang
THÔNG TIN TÀI LIỆU
Nội dung
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 luan an 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 luan an COMMITTAL IN THE DISSERTATION 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 I published 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 luan an ACKNOWLEDGEMENTS This dissertation has completed at Hanoi National University of Education under instruction of Prof Dr Cung The Anh, Department of Mathematics and Informatics, Hanoi National University of Education who is my supervisor 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 parents, my wife, my brothers, and my beloved 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 luan an CONTENTS i ii iii COMMITTAL IN THE DISSERTATION ACKNOWLEDGEMENTS CONTENTS LIST OF SYMBOLS INTRODUCTION Chapter 1.1 1.2 1.3 1.4 12 PRELIMINARIES AND AUXILIARY RESULTS Function spaces 12 1.1.1 Banach and Hilbert spaces 12 1.1.2 The Lp spaces of Lebesgue integrable functions 13 1.1.3 Nonnegative integer order Sobolev spaces 14 1.1.4 Fractional order Sobolev spaces 16 1.1.5 Bochner spaces 19 Global attractors in partial differential equations 21 1.2.1 Existence of global attractors 21 1.2.2 Finite fractal dimension 29 Operators 30 1.3.1 Laplace and p-Laplace operators 30 1.3.2 Fractional Laplacian and regional fractional Laplacian operators 30 Some auxiliary results 35 Chapter GLOBAL ATTRACTORS FOR NONLOCAL PARABOLIC EQUATIONS WITH A NEW CLASS OF NONLINEARITIES 36 2.1 Problem setting 36 2.2 Existence and uniqueness of weak solutions 38 2.3 Existence of a global attractor 44 2.4 Fractal dimension estimates of the global attractor 48 2.5 Existence and exponential stability of stationary solutions 51 Chapter LONG-TIME BEHAVIOR OF SOLUTIONS TO A NONLOCAL QUASI- LINEAR PARABOLIC EQUATION 57 3.1 57 Problem setting iii luan an 3.2 Existence and uniqueness of weak solutions 59 3.3 Existence of global attractors 66 3.3.1 The (L2 (Ω), L2 (Ω))-global attractor 66 3.3.2 The (L2 (Ω), Lq (Ω))-global attractor 67 3.3.3 The (L2 (Ω), W01,p (Ω) ∩ Lq (Ω))-global attractor 71 Existence and exponential stability of stationary solutions 74 3.4 Chapter GLOBAL ATTRACTORS FOR NONLOCAL PARABOLIC EQUATIONS INVOLVING THE FRACTIONAL LAPLACIAN AND THE REGIONAL FRACTIONAL LAPLACIAN WITH A NEW CLASS OF NONLINEARITIES 79 4.1 Problem setting 79 4.2 Existence and uniqueness of weak solutions 83 4.3 Existence of global attractors 91 4.4 Fractal dimension estimates of the global attractor 95 99 CONCLUSION AND FUTURE WORK LIST OF PUBLICATIONS 100 REFERENCES 101 iv luan an 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 Ω 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 ] luan an 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 luan an 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 [21], P Souplet [63]), nonlocal boundary conditions (see C Mu, D Liu and S Zhou [50], Y Wang, C Mu and Z Xiang [70], H M Yin [75]), and nonlocal diffusions (see L Caffarelli [12], C.G Gal and M Warma [35], N Pan, B Zhang and J Cao [53], P Pucci, M Xiang and B Zhang [54], M Xiang, V D R˘adulescu and B Zhang [73]) 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 Let us first consider the class of nonlocal parabolic problems involving Laplacian luan an 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 means that the nonlocal terms may allow to give more accurate results 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 now list some results in recent years In 1997, M Chipot and B Lovat [26] 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 the solutions for homogeneous or nonhomogeneous cases In 1999, M Chipot and B Lovat [25] studied system (1) with the mixed boundary condition in place of the Dirichlet boundary condition They investigated the existence and uniqueness of the 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 [29] gave an extension of the result for system (1) obtained in M Chipot and B Lovat [26], considering a = a(l(u)) and f = f (x, u) continuous functions We see that the nonlinearity appears not only in the diffusive luan an W0 (Ω) p (3.8) p (pελ1 ) Ω We need the following lemma Lemma 3.2.3 [24] Under the assumption (H1), −div a(k∇ukpLp (Ω) )|∇u|p−2 ∇u is a monotone operator in W01,p (Ω), i.e., D E −div a k∇ukpLp (Ω) |∇u|p−2 ∇u + div a k∇vkpLp (Ω) |∇v|p−2 ∇v , u − v ≥ (3.9) for all u, v ∈ W01,p (Ω) 60 luan an We are now ready to prove the following theorem 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)), 0 0 ut ∈ Lploc (0, ∞; W −1,p (Ω)) + Lqloc (0, ∞; Lq (Ω)) Moreover, the mapping u0 7→ u(t) is (L2 (Ω), L2 (Ω))-continuous Proof i) Existence We consider approximate solutions un (t) in the form un (t) = n X unk (t)ek , k=1 1,p q where {ej }∞ j=1 is a basis of W0 (Ω) ∩ L (Ω) consisting of eigenvalues of the operator ∆p u Without loss of generality, one can suppose that {ej }∞ j=1 is orthonormal in L (Ω) Therefore, un can be obtained by solving the following problem R R dun p ek + a k∇un kLp (Ω) |∇un |p−2 ∇un · ∇ek + f (un )ek dx = g(x)ek dx, Ω n P dt Ω unk (0)ek → u0 strongly in L2 (Ω) as n → ∞ k=1 (3.10) Since f ∈ C (R), it follows from the Peano theorem that the Cauchy problem (3.10) possesses a unique (local) solution We now establish some a priori estimates for un Multiplying (3.10) by unk (t) and summing these relations from k = to n, we obtain 1d kun k2L2 (Ω) + a k∇un kpLp (Ω) dt Z Z p Z |∇un | dx + Ω f (un )un dx = Ω gun dx (3.11) Ω In view of (3.2) and (3.4), we have 1d kun k2L2 (Ω) + m dt Z Z p |∇un | dx + c1 Ω Z q |un | dx ≤ c0 |Ω| + Ω gun dx Ω Hence, it follows from (3.8) with ε = m/2 that d kun k2L2 (Ω) + c4 dt Z Z p |∇un | dx + Ω q |un | dx ≤ c5 , (3.12) Ω where c4 = min{m, 2c1 } and c5 = |Ω| (p−2)p0 2p p0 (pmλ1 /2) p0 p kgkpL2 (Ω) + 2c0 |Ω| Using the inequality (3.7), we get from (3.12) that d kun k2L2 (Ω) + mλ1 kun kpLp (Ω) ≤ c5 dt 61 luan an (3.13) Noting that for p ≥ 2, there exists a constant c6 > such that mλ1 kun kpLp (Ω) ≥ mλ1 kun k2L2 (Ω) − c6 Hence (3.13) becomes d kun k2L2 (Ω) + mλ1 kun k2L2 (Ω) ≤ c5 + c6 dt (3.14) Using the Gronwall inequality to (3.14), we obtain (c5 + c6 ) − e−mλ1 t mλ1 kun (t)k2L2 (Ω) ≤ kun (0)k2L2 (Ω) e−mλ1 t + (3.15) This estimate implies that the solution un (t) of (3.10) can be extended to [0, +∞) Let T be an arbitrary positive number, integrating both sides of (3.12) from to T , we obtain kun (T )k2L2 (Ω) Z Z p |∇un | dxdt + + c4 q |un | dxdt ΩT ≤ kun (0)k2L2 (Ω) + T c5 ΩT This inequality yields {un } is bounded in L∞ (0, T ; L2 (Ω)), {un } is bounded in Lp (0, T ; W01,p (Ω)), {un } is bounded in Lq (ΩT ) Notice that −div a k∇un kpLp (Ω) |∇un |p−2 ∇un defines an element of W −1,p (Ω) given by the duality D −div a k∇un kpLp (Ω) p−2 |∇un | E ∇un , v = a k∇un kpLp (Ω) Z |∇un |p−2 ∇un ·∇vdx, Ω for all v ∈ W01,p (Ω) By using (3.2) and the boundedness of {un } in Lp (0, T ; W01,p (Ω)), we have T Z | Z0 D =| ΩT E −div a k∇un kpLp (Ω) |∇un |p−2 ∇un , v dt| a k∇un kpLp (Ω) |∇un |p−2 ∇un · ∇vdxdt| Z |∇un |p−1 |∇v|dxdt ≤M ΩT ≤ M kun k p/p0 kvkLp (0,T ;W 1,p (Ω)) , Lp (0,T ;W01,p (Ω)) for any v ∈ Lp (0, T ; W01,p (Ω)) We deduce that n −div a k∇un kpLp (Ω) p−2 |∇un | ∇un o 0 is bounded in Lp (0, T ; W −1,p (Ω)) 62 luan an 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 [46, 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)), du dun * in V ∗ , dt dt un (T ) → u(T ) in L2 (Ω), and f (un ) * f (u) Lq (ΩT ), in (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 Z p Z |∇un | dxdt = gun dxdt − Ω f (un )un dxdt ΩT + 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 lim n→∞ T a k∇un kpLp (Ω) Z Z p Z |∇un | dxdt = Ω gudxdt − ΩT + 63 luan an f (u)udxdt 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 − a ΩT k∇un kpLp (Ω) k∇vkpLp (Ω) Z Z p T |∇un | dxdt + Ω hχ, vi dt |∇v|p−2 ∇v · ∇(u − v)dxdt ≥ Therefore, in view of (3.21) and the last inequality, we have Z ku0 k2L2 (Ω) Z gudxdt − f (u)udxdt + ΩT 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 Z − gudxdt − hχ, ui dt = f (u)udxdt + ku0 k2L2 (Ω) − ΩT ΩT ku(T )k2L2 (Ω) (3.23) Combining (3.22) with (3.23), we have T Z D χ − div a k∇vkpLp (Ω) p−2 |∇v| E ∇v , u − v dt ≤ 0, ∀v ∈ Lp (0, T ; W01,p (Ω)) Choosing v = u − δϕ, we see that T Z D k∇(u − δϕ)kpLp (Ω) k∇(u − δϕ)kpLp (Ω) χ − div a p−2 |∇(u − δϕ)| E E ∇(u − δϕ) , ϕ dt ≤ 0, if δ > and Z T D χ − div a p−2 |∇(u − δϕ)| ∇(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 χ − div a k∇ukpLp (Ω) p−2 |∇u| E ∇u , ϕ dt = 0, ∀ϕ ∈ Lp (0, T ; W01,p (Ω)) 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 ) = 64 luan an 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 ΩT ΩT a 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 ΩT uϕ dxdt + 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 (3.25) ΩT ΩT R Ω u0 ϕ(0)dx = 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) |∇u|p−2 ∇u +div a k∇vkpLp (Ω) |∇v|p−2 ∇v + f (u) − f (v) = 0, (3.27) w(0) = u − v 0 Multiplying (3.27) by w and integrating over Ω, one gets 1d kwk2L2 (Ω) dtZ + ZΩ (f (u) − f (v))(u − v)dx = + a k∇ukpLp (Ω) |∇u|p−2 ∇u − a k∇vkpLp (Ω) |∇v|p−2 ∇v · ∇(u − v)dx Ω 65 luan an 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 [59] √ We see from (3.26) that the ball B0 = B( ρ0 ) with ρ0 = mλ1 (c5 + c6 ), is an (L2 (Ω), L2 (Ω))-bounded 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 q |u| dx ≤ c5 Ω 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 ≤ 66 luan an 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 f (u)|∇u|p dx − hg, ∆p ui = k∇ukLp (Ω) − 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 kgk2L2 (Ω) + kuk2L2 (Ω) ≤ mk∆p uk2L2 (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 d (1 + pc3 )2 ρ0 p 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 3.3.2 The (L2 (Ω), Lq (Ω))-global attractor 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 67 luan an (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 gudx ds ≤ Ω for all t ≥ T1 We define i 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 Z p Z F (u)dx − |∇u| dx + Ω i Ω 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 p−2 |∇u| Z Z ∇u · ∇ut dx + Ω f (u)ut dx − Ω gut dx = Ω We can rewrite the last equality as kut k2L2 (Ω) Z Z Z d gudx + a k∇ukpLp (Ω) |∇u|p dx + F (u)dx − dt p Ω Ω Ω d = a0 (k∇ukpLp (Ω) ) k∇ukpLp (Ω) k∇ukpLp (Ω) p dt h 68 luan an i (3.37) Setting L = sup |a0 (s)|, then from (3.31), (3.32) and (3.37), we have 0≤s≤ρ1 h d dt p a k∇ukpLp (Ω) Z Z p |∇u| dx + Z i F (u)dx − Ω 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 ρ0 (c3 + 1) LR12 |∇u| dx + F (u)dx − gudx ≤ + p Ω Ω Ω Z Z p Z 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 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 Ω(|S(t)u0 |≥M ) 69 luan an (3.40) 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) p−2 |∇u| ∇u (u − M )q−1 + (3.41) q−1 + f (u)(u − M )q−1 + = g(x)(u − M )+ , 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 |g|2 e c 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) a Ω(u≥M ) Ω(u≥M ) e c + M q−2 Z k∇ukpLp (Ω) (u − M )q+ dx Ω(u≥M ) (3.43) |∇u|p (u − M )q−2 + dx ≤ 2e c Z |g|2 dx, Ω(u≥M ) and then d dt Z (u − M )q+ dx + Ω(u≥M ) e c qM q−2 Z (u − M )q+ dx Ω(u≥M ) q ≤ 2e c Z |g|2 dx Ω(u≥M ) By the Gronwall inequality, we have Z (u − M )q+ dx ≤e − 2ec qM q−2 t Z Ω(u≥M ) (u(0) − M )q+ dx 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 Ω(u≥M ) 70 luan an (3.44) 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 |u|q dx = Z Ω(|u|≥2M ) (|u| − M + M )q dx Ω(|u|≥2M ) q Z q ≤2 (|u| − M ) dx + q Ω(|u|≥2M ) ≤ 2q+1 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 Proof By differentiating the first equation in (3.1) in time and denoting v = ut , we get vt − div a k∇ukpLp (Ω) p−2 |∇u| ∇v − (p − 2)div a k∇ukpLp (Ω) |∇u|p−4 (∇u · ∇v)∇u − p div a (k∇ukpLp (Ω) ) Z p−2 |∇u| p−2 (∇u · ∇v)dx|∇u| Ω 71 luan an ∇u + f (u)v = 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 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 Here h., i be the standard scalar product in RN 72 luan an (3.50) Following the ideas in [32, 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 = 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 73 luan an 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 ∗ p−2 |∇u | Z ∗ ∇u · ∇vdx + Ω ∗ Z f (u )vdx = Ω gvdx, (3.51) Ω for all test functions v ∈ W01,p (Ω) ∩ Lq (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 ku∗ kp W01,p (Ω) + ku∗ kqLq (Ω) ≤ `, 74 luan an (3.52)