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 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 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 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 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 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 x − y X RN Ω Nonempty open subset of ∂Ω Boundary of Ω ΩT ΩT := Ω × (0, T ) (., )X Inner product in the Hilbert space X x X Norm of x in the space X ut Partial derivative of u in variable t X∗ Dual space of the space X x ,x Duality pairing between x ∈ 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,λ (Ω) 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 T such that L∞ (0, T ; X) f (t) p X dt 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 −   t a( ∆u udx) Ω = f (u), x ∈ Ω, t > 0, 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 = 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 Thus, SK (t)u0 2L2 (Ω) u0 ∈ L (Ω) : B0K := ≤ 2c1 |Ω|(CK,s − µ) + g L2 (Ω) (CK,s − µ)2 +1 (4.41) is a bounded absorbing set in L2 (Ω) for the semigroup SK (t) Lemma 4.3.2 The semigroup {SK (t)}t≥0 generated by problem (4.5) has a bounded s,2 absorbing set B1K in WK (Ω)) Proof Using (4.10), (4.11), (4.16), the condition (F) and the Cauchy inequality, we deduce from (4.40) that d u(t) dt L2 (Ω) L2 (Ω) + (CK,s − µ) u(t) + EK (u(t), u(t)) L2 (Ω) g fK (u(t))u(t)dx ≤ +2 µ Ω Applying the Gronwall inequality, we obtain t u(t) L2 (Ω) e(CK,s −µ)(τ −t) + EK (u(τ ), u(τ )) + fK (u(τ ))u(τ )dx dτ Ω s (CK,s −µ)(s−t) L2 (Ω) e ≤ u(s) g + (4.42) L2 (Ω) µ(CK,s − µ) It follows from (4.41) and (4.42) that t EK (u(τ ), u(τ )) + f (u(τ ))u(τ )dx dτ ≤ C, t−1 (4.43) Ω for t large enough and some positive constant C := C(CK,s , µ, η, c1 , |Ω|, g L2 (Ω) ) Now, multiplying (4.5) by ut , we have ut where F (u) := L2 (Ω) + u f (s)ds t ut (τ ) d dt EK (u, u) + F (u)dx − Ω = 0, (4.44) Integrating over (s, t) with respect to time variable leads to + EK (u(t), u(t)) + ≤ EK (u(s), u(s)) + 2 L2 (Ω) dτ s gudx Ω F (u(t))dx − Ω gu(t)dx Ω F (u(s))dx − Ω (4.45) gu(s)dx Ω We infer from (4.45) that EK (u(t), u(t)) + t ≤ t−1 F (u(t))dx − Ω EK (u(s), u(s)) + gu(t)dx Ω (4.46) F (u(s))dx − Ω 92 gu(s)dx ds Ω On the other hand, it follows from the condition (F) that: If K ∈ {D, R, E}, then − + |f (0)|2 u − ≤ F (u) ≤ f (u)u + u2 , for all u ∈ R 2 (4.47) If K = N , then | − η| + |f (0)|2 | − η| (4.48) u − ≤ F (u) ≤ f (u)u + u , for all u ∈ R 2 Using the Cauchy inequality, we deduce from (4.41), (4.43), (4.46), (4.47) and (4.48) − that EK (u(t), u(t)) + F (u(t))dx − Ω gu(t)dx ≤ C, (4.49) Ω and so EK (u, u) ≤ C, for some positive constant C := C(CK,s , µ, η, c1 , , |Ω|, g L2 (Ω) ) and t large enough By s,2 Lemma 4.1.1, Lemma 4.3.1 and the compactness of the embedding WK (Ω) → L2 (Ω), there is a positive constant ρ1 := ρ1 (CK,s , µ, η, c1 , , |Ω|, g B1K := u0 ∈ L2 (Ω) : SK (t)u0 L2 (Ω) ) s,2 WK (Ω) such that ≤ ρ1 s,2 is a bounded absorbing set in WK (Ω) As a direct consequence of Lemma 4.3.2 and the compactness of the embedding s,2 WK (Ω) → L2 (Ω), we get the following result Theorem 4.3.3 ((L2 (Ω), L2 (Ω))-global attractor) Assume that the assumptions (F), (G), and (D) hold Then, for each K ∈ {D, N , R, E}, the semigroup SK (t) generated by problem (4.5) has a global attractor A1K in L2 (Ω) We now prove an uniform estimate of the derivatives of solutions in time and the s,2 asymptotic compactness in WK (Ω)) for the semigroup Lemma 4.3.4 Assume that the assumptions (F), (G), and (D) hold Then for every bounded subset B2 in L2 (Ω), there exists a constant T = T (B2 ) > such that ut (s) where ut (s) = L2 (Ω) d dt (SK (t)u0 )|t=s ≤ ρ2 for all u0 ∈ B2 , and s ≥ T, for each K ∈ {D, N , R, E} and ρ2 is a positive constant independent of B2 Proof We infer from (4.45) and (4.49) that t ut (τ ) L2 (Ω) dτ s 93 ≤ C, (4.50) for s, t large enough and C is a positive constant independent of B2 Differentiating the first equation in (4.5) with respect to t, then taking the duality with v := ut yields 1d v dt L2 (Ω) f (u)|v|2 dx = + EK (v, v) + Ω Using the condition (F) we have d v dt L2 (Ω) ≤ max{ ; | − η|} v L2 (Ω) (4.51) Combining (4.50) with (4.51) and using the uniform Gronwall inequality, the proof is finished Lemma 4.3.5 Assume that the assumptions (F), (G), and (D) hold Then, for each K ∈ {D, N , R, E}, the semigroup {SK (t)}t≥0 generated by problem (4.5) is s,2 (L2 (Ω), WK (Ω))-asymptotically compact Proof Let B3 be a bounded subset of L2 (Ω), we will show that for any {u0n } ⊂ B3 s,2 and tn → +∞, {un (tn )}∞ n=1 is precompact in WK (Ω), where un (tn ) = SK (tn )u0n s,2 To this, we will prove that {un (tn )} is a Cauchy sequence in WK (Ω) Thanks to Theorem 4.3.3, one can assume that {un (tn )} is a Cauchy sequence in L2 (Ω) By s,2 (Ω) → → L2 (Ω) → L1 (Ω), Lemma 4.1.1 and the compactness of the embedding WK we have If K ∈ {D, R, E}, then un (tn ) − um (tm ) s,2 WK (Ω) ≤ c3 EK (un (tn ) − um (tm ), un (tn ) − um (tm )) d d un (tn ) − f (un (tn )) + um (tm ) + f (um (tm )), un (tn ) − um (tm ) dt dt d d ≤ c3 un (tn ) − um (tm ) L2 (Ω) un (tn ) − um (tm ) L2 (Ω) dt dt + c3 un (tn ) − um (tm ) 2L2 (Ω) ≤ c3 − (4.52) If K = N , then un (tn ) − um (tm ) ≤ c5 s,2 WN (Ω) EN (un (tn ) − um (tm ), un (tn ) − um (tm )) + un (tn ) − um (tm ) L1 (Ω) d d un (tn ) − f (un (tn )) + um (tm ) + f (um (tm )), un (tn ) − um (tm ) dt dt + c5 un (tn ) − um (tm ) 2L1 (Ω) ≤ c5 − d d un (tn ) − um (tm ) L2 (Ω) un (tn ) − um (tm ) dt dt + (| − η| + |Ω|)c5 un (tn ) − um (tm ) 2L2 (Ω) ≤ c5 94 L2 (Ω) (4.53) It follows from Lemma 4.3.4, (4.52) and (4.53) that {un (tn )} is a Cauchy sequence in s,2 WK (Ω) This completes the proof The following result follows immediately from Lemma 4.3.2, Lemma 4.3.5 and Theorem 1.2.29 s,2 Theorem 4.3.6 ((L2 (Ω), WK (Ω))-global attractor) Assume that assumptions (F), (G), and (D) hold Then, for each K ∈ {D, N , R, E}, the semigroup {SK (t)}t≥0 s,2 generated by problem (4.5) has a (L2 (Ω), WK (Ω))-global attractor A2K Remark 4.3.7 The global attractors A1K and A2K obtained in Theorem 4.3.3 and Theorem 4.3.6 are of course the same object because the uniqueness of the global attractor of a semigroup and will be denoted by AK In particular, AK is a compact s,2 and connected set in WK (Ω) 4.4 Fractal dimension estimates of the global attractor In this section we will study the finiteness of the fractal dimension of the global attractor of the semigroup generated by problem (4.5) To this, we assume that the nonlinearity f and the external force g satisfy stronger conditions: (Fbis) f satisfies the condition (F) and there exists s0 > such that fK (s) ≥ g L∞ (Ω) if s ≥ s0 , fK (s) ≤ g L∞ (Ω) if s ≤ −s0 , where fK (s) = f (s) if K ∈ {D, R, E} and fK (s) = fN (s) if K = N (Gbis) g ∈ L∞ (Ω) For a measurable function u : Ω → R and k ≥ 0, we set u+ := max{u, 0}, uk := (|u| − k)+ sgn(u) The following result is deduced from [71, Lemma 2.6 and Lemma 7.4] s,2 Proposition 4.4.1 Let u ∈ WK (Ω), K ∈ {D, N , R, E}, and let k ≥ be a real s,2 number Then uk ∈ WK (Ω) and EK (uk , uk ) ≤ EK (u, uk ), ∀k ≥ We now prove the following important result 95 Lemma 4.4.2 Assume that the assumptions (Fbis), (Gbis), and (D) hold Then the global attractor AK associated to problem (4.5) is bounded in L∞ (Ω) Proof Suppose that u(t) ∈ AK Multiplying (4.5) by uk , we get 1d dt |uk |2 dx + EK (u, uk ) + Ωk f (u)uk dx = Ωk guk dx, Ωk where Ωk = {x ∈ Ω : |u(x)| − k ≥ 0} Using (4.10), (4.11), (4.16), the condition (F) and Proposition 4.4.1, we have 1d dt |uk |2 dx + CK,s Ωk |uk |2 dx ≤ Ωk (g − fK (u))uk dx Ωk If we choose k large enough, then it follows from (Fbis) and the above inequality that d dt |uk |2 dx + 2CK,s Ωk |uk |2 dx ≤ Ωk Using the Gronwall inequality, we have |uk |2 dx ≤ e−2CK,s t Ωk |(|u0 | − k)+ |2 dx → as t → ∞ Ωk Since AK is invariant, we get |(|u| − k)+ |2 dx = Ωk This implies that u L∞ (Ω) ≤ k, for all u ∈ AK To prove the finite fractal dimension of the global attractor, we will use Theorem 1.2.33 The following theorem is the main result of this section Theorem 4.4.3 Assume that (Fbis), (Gbis), and (D) hold Then, for each K ∈ {D, N , R, E}, the global attractor AK of problem (4.5) has a finite fractal dimension in L2 (Ω), namely, dimf AK ≤ q ln( 9e )[ln( )]−1 1−δ 1+δ Proof Let u01 , u02 ∈ AK Let u1 (t) = SK (t)u01 and u2 (t) = SK (t)u02 be solutions of problem (4.5) with the initial values u01 , u02 , respectively We set w(t) = u1 (t) − u2 (t) Assume that w(t) = w1 (t) + w2 (t), where w1 (t) is the projection onto Pq L2 (Ω) = s,2 K K K ∞ ∞ span{eK , e2 , , eq }, where {ej }j=1 ⊂ WK (Ω) ∩ L (Ω) are eigenfunctions corre∞ K ∞ sponding to the eigenvalues {λK j }j=1 of AK for K ∈ {D, N , R, E} Moreover, {ej }j=1 s,2 is an orthonormal basis of L2 (Ω) and orthogonal basis of WK (Ω) Therefore, w(t) L2 (Ω) = w1 (t) L2 (Ω) 96 + w2 (t) L2 (Ω) , (4.54) and s,2 WK (Ω) w(t) = w1 (t) s,2 WK (Ω) s,2 WK (Ω) + w2 (t) Performing the same arguments as in the proof of Theorem 4.2.2 we get from the equation (4.55) wt + AK w + f (u1 ) − f (u2 ) = that w(t) L2 (Ω) ≤ w(0) L2 (Ω) exp(2 t) w1 (t) L2 (Ω) ≤ w(0) L2 (Ω) exp(2 t) Hence, from (4.54) (4.56) On the other hand, we have f (u1 ) − f (u2 ) = (4.57) f (su1 (t) + (1 − s)u2 (t))ds (u1 (t) − u2 (t)) We infer from the condition (Fbis) and Lemma 4.4.2 that there exists a positive constant β such that f (su1 (t) + (1 − s)u2 (t))ds ≤ β sup x∈Ω,t≥0 (4.58) Multiplying (4.55) by w2 (t), then using the Hăolder inequality, (4.57) and (4.58), we have 1d w2 (t) dt L2 (Ω) + AK w2 (t), w2 (t) ≤ β w(t) L2 (Ω) w2 (t) (4.59) L2 (Ω) Since AK w2 (t), w2 (t) ≥ λK q w2 (t) L2 (Ω) , (4.60) by the Cauchy inequality, we deduce from (4.59) and (4.60) that d w2 (t) 2L2 (Ω) + (2λK q − 1) w2 (t) dt Using the Gronwall inequality, we get w2 (t) L2 (Ω) ≤ w2 (0) L2 (Ω) L2 (Ω) ≤ β w(t) L2 (Ω) exp(−(2λK q − 1)t) t + β exp(−(2λK q − 1)t) exp((2λK q − 1)s) w(s) L2 (Ω) ds Combining with (4.56), we get w2 (t) L2 (Ω) ≤ w2 (0) L2 (Ω) exp(−(2λK q − 1)t) t +β exp(−(2λK q exp((2λK q − + )s) w(0) − 1)t) L2 (Ω) ds ≤ β exp(2 t) exp(−(2λK − 1)t) + q 2λK q −1+2 w(0) L2 (Ω) (4.61) 97 From (4.56) and (4.61) we have w1 (1) L2 (Ω) ≤ e2 w(0) L2 (Ω) , w2 (1) L2 (Ω) ≤ δ w(0) L2 (Ω) , β exp(2 ) < whenever q is large enough 2λK q −1+2 Using Theorem 1.2.33, one obtains the bound given in the statement of the theorem where δ = exp(−(2λK q − 1)) + Remark 4.4.4 By repeating the similar arguments in Section 2.5, Chapter 2, we are able to obtain the existence and exponential stability of stationary solutions for these problems Remark 4.4.5 The fractional Laplacian and the regional fractional Laplacian are naturally nonlocal operators Along with lack of an upper growth restriction of the nonlinearity and various boundary type conditions, the technicalities are more involved and different from two previous chapters We have used the method of Dirichlet forms in nonsmooth domains to investigate our problems and the attained results have improved and extended the well-known results as follows: ❼ In case that the nonlinearity satisfies polynomial growth and using the above method, we recover the results by C.G Gal and M Warma [35] ❼ For the reaction-diffusion equations involving fractional Laplacian with extended Dirichlet boundary conditions and lack of an upper growth restriction of the nonlinearity, we have the results by C Zhang et al [76], but not use the method of Dirichlet forms in nonsmooth domains ❼ Our results have improved and extended the above results since the nonlinear term has no restriction on the upper growth and all boundary conditions are considered Conclusion of Chapter In this chapter, we have studied the global attractors for nonlocal parabolic equations involving the fractional Laplacian and the regional fractional Laplacian with a new class of nonlinearities We have achieved the following results: 1) Existence and uniqueness of weak solutions (Theorem 4.2.2); 2) Existence of a (L2 (Ω), L2 (Ω))-global attractor (Theorem 4.3.3); s,2 3) Existence of a (L2 (Ω), WK (Ω))-global attractor (Theorem 4.3.6); 4) The global attractor has a finite fractal dimension in L2 (Ω) (Theorem 4.4.3) 98 CONCLUSION AND FUTURE WORK Conclusion In this dissertation, a number of nonlocal parabolic problems have been investigated It has contributed some results on studying long-time behavior of solutions of these nonlocal parabolic problems via existence of its finite dimensional global attractors, and the existence and exponential stability of stationary solutions, namely: Existence and uniqueness of weak solutions, existence of connected compact global attractors, finite fractal dimensional estimate of global attractor, existence and exponential stability of stationary solutions for nonlocal parabolic equations involving Laplacian with a new class of nonlinearities Existence and uniqueness of weak solutions, existence of global attractors, existence and exponential stability of stationary solutions for a nonlocal parabolic equation involving p-Laplacian operators Existence and uniqueness of weak solutions, existence of connected compact global attractors, finite fractal dimensional estimate of global attractor for nonlocal parabolic equations involving the fractional Laplacian and the regional fractional Laplacian with a new class of nonlinearities The results obtained in the dissertation are meaningful contributions to the theory of infinite dynamical systems for nonlocal parabolic equations Future Work I continue my research in the classes of nonlocal equations Especially, I concern more about mathematical models in fluid mechanics 99 LIST OF PUBLICATIONS Accepted papers [CT1] C.T Anh, L.T Tinh and V.M Toi, Global attractors 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Differential Equations 223, 367-399 106 ... asymptotic behaviour of strong solutions for (1) with moving boundaries Moreover, they studied some properties of the solutions and implemented a numerical algorithm based on Moving Finite Element... many open parabolic problems involving nonlocal equations via studying the global attractors of the associated semigroups, for instance, ❼ The asymptotic behavior of solutions to nonlocal parabolic... at: ❼ Seminar of Division of Mathematical Analysis at Hanoi National University of Education ❼ Seminar at Vietnam Institute for Advanced Studies in Mathematics ❼ Seminar of Division of Mathematical