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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[.]
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 ... 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, The asymptotic behavior of solutions to nonlocal parabolic... at: Seminar of Division of Mathematical Analysis at Hanoi National University of Education Seminar at Vietnam Institute for Advanced Studies in Mathematics Seminar of Division of Mathematical