phương trình vi phân và tích phân

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 for nonlocal parabolic equations with a new class of nonlinearities, J Korean Math Soc 55 (2018), no 3, 531-551 (SCIE) [CT2] L.T Thuy and L.T Tinh, Long-time behavior of solutions to a nonlocal quasilinear parabolic equation, Commun Korean Math Soc 34 (2019), no 4, 1365-1388 (Scopus) Submitted papers [CT3] C.T Anh and L.T Tinh, Global attractors for nonlocal parabolic equations involving the regional fractional Laplacian with a new class of nonlinearities, submitted (2019) 100 REFERENCES [1] A.S Ackleh and L Ke (2000), Existence-uniqueness and long time behavior for a class of nonlocal nonlinear parabolic evolution equations, Proc Amer Math Soc 128, 3483-3492 [2] D.R Adams and L.I Hedberg, Function Spaces and Potential Theory, SpringerVerlag, Berlin, 1996 [3] R.A Adams and J.J.F Fournier, Sobolev Spaces, Volume 140, 2nd Edition, Elsevier 2003 [4] B Ahmad, A Alsaedi and M Kirane (2016), On a reaction diffusion equation with nonlinear time-nonlocal source term, Math Methods Appl Sci 39, no 2, 236-244 [5] R.M.P Almeida, S.N Antontsev and J.C.M Duque (2017), On the finite element method for a nonlocal degenerate parabolic problem, Comput Math Appl 73, no 8, 1724-1740 [6] R.M.P Almeida, S.N Antontsev and J.C.M Duque (2016), On a nonlocal degenerate parabolic problem, Nonlinear Anal Real World Appl 27, 146-157 [7] R.M.P Almeida, S.N Antontsev, J.C.M Duque and J Ferreira (2016), A reactiondiffusion model for the non-local coupled system: existence, uniqueness, long-time behaviour and localization properties of solutions, IMA J Appl Math 81, 344-364 [8] S Antontsev and S Shmarev (2009), Anisotropic parabolic equations with variable nonlinearity, Publ Mat 53, 355-399 [9] G Astarita and G Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, NewYork, 1974 [10] A.V Babin and M.I Vishik, Attractor of Evolution Equations, North-Holland, Amsterdam, 1992 [11] K Bogdan, K Burdzy and Z Chen (2003), Censored stable processes, Probab Theory Related Fields 127, 89-152 [12] L Caffarelli (2012), Non-local diffusions, drifts and games, Nonlinear partial differential equations, 37-52 [13] T Caraballo, M Herrera-Cobos and P Mar´ın-Rubio (2018), Asymptotic behaviour of nonlocal p-Laplacian reaction-diffusion problems, J Math Anal Appl 459, 997-1015 101 [14] T Caraballo, M Herrera-Cobos and P Mar´ın-Rubio (2017), Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution, Discrete Contin Dyn Syst Ser B 22, 1801-1816 [15] T Caraballo, M Herrera-Cobos and P Mar´ın-Rubio (2016), Robustness of nonautonomous attractors for a family of nonlocal reaction diffusion equations without uniqueness, Nonlinear Dynam 84, 35-50 [16] T Caraballo, M Herrera-Cobos and P Mar´ın-Rubio (2015), Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms, Nonlinear Anal 121, 3-18 [17] A.N Carvalho and C.B Gentile (2003), Asymptotic behaviour of nonlinear parabolic equations with monotone principal part, J Math Anal Appl 280, 252272 [18] A.N Carvalho and C.B Gentile (2001), Comparision results for nonlinear parabolic equations with monotone principal part, J Math Anal Appl 259, 319337 [19] A.N Carvalho, J.W Cholewa and T Dlotko (1999), Global attractors for problems with monotone operators, Boll Unione Math Ital Sez B Artic Ric Mat 2, 693706 [20] Z.Q Chen and T Kumagai (2003), Heat kernel estimates for stable-like processes on d-sets, Stoch Process Appl 108, 27-62 [21] Y Chen and M Wang (2009), Blow-up problems for partial differential equations and systems of parabolic type, Recent progress on reaction-diffusion systems and viscosity solutions, 245-281, World Sci Publ., Hackensack, NJ [22] V.V Chepyzhov and M.I Vishik, Attractors for Equations of Mathematical Physics, Amer Math Soc., Providence, RI, 2002 [23] M Chipot and T Savitska (2015), Asymptotic behaviour of the solutions of nonlocal p-Laplace equations depending on the Lp norm of the gradient, J Elliptic Parabol Equ 1, 63-74 [24] M Chipot and T Savitska (2014), Nonlocal p-Laplace equations depending on the Lp -norm of the gradient, Adv Diff Equa 19, 997-1020 [25] M Chipot and B Lovat (1999), On the asymptotic behaviour of some nonlocal problems, Positivity 3, no 1, 65-81 [26] M Chipot and B Lovat (1997), Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal 30, 4619-4627 102 [27] J.W Cholewa and T Dlotko, Global Attractor in Abstract Parabolic Problems, Cambridge University Press, 2000 [28] I.D Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, ACTA Scientific Publishing House, 2002 [29] F.J.S.A Corrˆea, S.D.B Menezes and J Ferreira (2004), On a class of problems involving a nonlocal operator, Appl Math Comp 147, 475-489 [30] M Cuesta (2001), Eigenvalue problems for the p-Laplacian with indefinite weights, Electron J Differential Equations, No 33, pp [31] E.B Davies, Spectral Theory and Differential Operators, Cambridge University Press,1995 [32] E DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, 1993 [33] S Duo, H Wang and Y Zhang (2019), A comparative study on nonlocal diffusion operators related to the fractional Laplacian, Discrete Contin Dyn Syst Ser B 24, no 1, 231-256 [34] Y Fu and M Xiang (2016), Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent, Appl Anal 95, no 3, 524-544 [35] C.G Gal and M Warma (2016), Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Contin Dyn Syst 36, 1279-1319 [36] P.G Geredeli (2015), On the existence of regular global attractor for p-Laplacian evolution equation, Appl Math Optim 71, 517-532 [37] P.G Geredeli and A Khanmamedov (2013), Long-time dynamics of the parabolic p-Laplacian equation, Commun Pure Appl Anal 12, 735-754 [38] P Grisvard, Elliptic problems in nonsmooth domains, volume 69 of Classics in Applied Mathematics Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA: Reprint of the 1985 original With a foreword by Susanne C, Brenner (2011) [39] Q.Y Guan (2006), Integration by parts formula for regional fractional Laplacian, Comm Math Phys 266, 289-329 [40] Q.Y Guan and Z.M Ma (2005), Boundary problems for fractional Laplacians, Stoch Dyn 5, 385-424 [41] J.K Hale, Asymptotic behavior of dissipative systems, AMS, Providence, RJ, 1988 103 [42] Y Han and Q Li (2018), Threshold results for the existence of global and blowup solutions to Kirchhoff equations with arbitrary initial energy, Comput Math Appl 75, no 9, 3283-3297 [43] O Ladyzhenskaya, Attractors for semigroups and evolution equations Lezioni Lincee, [Lincei Lectures] Cambridge University Press, Cambridge, 1991 [44] N Laskin (2000), Fractional quantum mechanics and L´evy path integrals, Phys Lett A 268, no 4-6, 298-305 [45] J Li and Y Han (2019), Global existence and finite time blow-up of solutions to a nonlocal p-Laplace equation, Math Model Anal 24, no 2, 195-217 [46] J.L Lions, Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non Lin´eaires, Dunod, Paris, 1969 [47] J L Lions and E Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol 1, Springer-Verlag Berlin Heidelberg 1972 [48] Q.F Ma, S.H Wang and C.K Zhong(2002), Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Math J 51 (6) [49] S.B de Menezes (2006), Remarks on weak solutions for a nonlocal parabolic problem, Int J Math Math Sci., Art ID 82654, 10 pp [50] C Mu, D Liu and S Zhou (2010), Properties of positive solutions for a nonlocal reaction-diffusion equation with nonlocal nonlinear boundary condition J Korean Math Soc 47, no 6, 1317-1328 [51] E Di Nezza, G Palatucci and E Valdinoci (2012), Hitchhiker’s guide to the fractional Sobolev spaces, Bull Sci Math 136, 521-573 [52] A.A Ovono (2010), Asymptotic behaviour for a diffusion equation governed by nonlocal interactions, Electron J Differential Equations, No 134, 16 pp [53] N Pan, B Zhang and J Cao (2017), Degenerate Kirchhoff-type diffusion problems involving the fractional p-Laplacian, Nonlinear Anal Real World Appl 37, 56-70 [54] P Pucci, M Xiang and B Zhang (2017), A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian, Discrete Contin Dyn Syst 37, no 7, 4035-4051 [55] C Raffaele (2018), What Do You Mean by “Nonlinear Eigenvalue Problems”?, Axioms, volume: Issues: 2, 39-69 104 [56] C.A Raposo, M Sep´ ulveda, O.V Villagr´an, D.C Pereira and M.L Santos (2008), Solution and asymptotic behaviour for a nonlocal coupled system of reactiondiffusion, Acta Appl Math 102, 37-56 [57] G Raugel, Global attractors in partial differential equations Handbook of dynamical systems, Vol 2, 885–982, North-Holland, Amsterdam, 2002 [58] R.J Robalo, R.M P Almeida, M.C Coimbra and J Ferreira (2014), A reactiondiffusion model for a class of nonlinear parabolic equations with moving boundaries: existence, uniqueness, exponential decay and simulation, Appl Math Model.38, no 23, 5609-5622 [59] J.C Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001 [60] J.C Robinson, J.L Rodrigo and Witold Sadowski, The Three-Dimensional Navier-Stokes Equations: Classical Theory, Cambridge University Press 2016 [61] X Ros-Oton and J Serra (2014), The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J Math Pures Appl 101, 275-302 [62] J Simsen and J Ferreira (2014), A global attractor for a nonlocal parabolic problem, Nonlinear Stud 21, 405-416 [63] P Souplet (1998), Blow-up in Nonlocal Reaction-Diffusion Equations, SIAM J Math Anal., 29(6), 1301–1334 [64] W.A Strauss, Partial Differential Equations: An Introduction, Wiley, 1992 [65] C Sun and C Zhong (2005), Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal 63 , no 1, 49-65 [66] C Sun, S Wang and C Zhong (2007), Global attractors for a nonclassical diffusion equation,Acta Math Sin (Engl Ser.) 23 (2007), no 7, 1271-1280 [67] R Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York, 1997 [68] R Temam, Navier-Stokes Equations: Theory and Numerical Analysis, 2nd edition, North-Holland, Amsterdam, 1979 [69] J.L V´azquez (2014), Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin Dyn Syst Ser S 7, no 4, 857885 105 [70] Y Wang, C Mu and Z Xiang (2007), Properties of positive solution for nonlocal reaction-diffusion equation with nonlocal boundary, Bound Value Probl, Art ID 64579, 12 pp [71] M Warma (2015), The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal 42, 499-547 [72] M Warma (2015), A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains, Commun Pure Appl Anal 14, 2043-2067 [73] M Xiang, V.D R˘adulescu and B Zhang (2018), Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions, Nonlinearity 31, no 7, 32283250 [74] M Yang, C Sun and C Zhong (2007), Global attractors for p-Laplacian equation, J Math Anal Appl 327, 1130-1142 [75] H.M Yin (2004), On a class of parabolic equations with nonlocal boundary conditions, Journal of Mathematical Analysis and Applications, vol 294, no 2, pp 712-728 [76] C Zhang, F Li and J Duan (2018), Long-time behavior of a class of nonlocal partial differential equations, Discrete Contin Dyn Syst Ser B 23, 749-763 [77] S Zheng and M Chipot (2005), Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot Anal 45, 301-312 [78] C Zhong, M Yang and C Sun (2006), The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reactiondiffusion equations, J 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