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The semigroup methods are known as a powerful tool for analyzing nonlinear diffusion equations and systems. The author has studied abstract parabolic evolution equations and their applications to nonlinear diffusion equations and systems for more than 30 years. He gives first, after reviewing the theory of analytic semigroups, an overview of the theories of linear, semilinear and quasilinear abstract parabolic evolution equations as well as general strategies for constructing dynamical systems, attractors and stable-unstable manifolds associated with those nonlinear evolution equations. In the second half of the book, he shows how to apply the abstract results to various models in the real world focusing on various self-organization models: semiconductor model, activator-inhibitor model, B-Z reaction model, forest kinematic model, chemotaxis model, termite mound building model, phase transition model, and Lotka-Volterra competition model. The process and techniques are explained concretely in order to analyze nonlinear diffusion models by using the methods of abstract evolution equations. Thus the present book fills the gaps of related titles that either treat only very theoretical examples of equations or introduce many interesting models from Biology and Ecology, but do not base analytical arguments upon rigorous mathematical theories.

Evolution equations --- Differential equations, Parabolic --- Mathematics --- Calculus --- Physical Sciences & Mathematics --- Evolution equations. --- Differential equations, Parabolic. --- Parabolic differential equations --- Parabolic partial differential equations --- Evolutionary equations --- Equations, Evolution --- Equations of evolution --- Mathematics. --- Dynamics. --- Ergodic theory. --- Partial differential equations. --- Biomathematics. --- Partial Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Mathematical and Computational Biology. --- Differential equations, Partial --- Differential equations --- Differential equations, partial. --- Differentiable dynamical systems. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics --- Partial differential equations --- Biology --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics

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This second volume continues the study on asymptotic convergence of global solutions of parabolic equations to stationary solutions by utilizing the theory of abstract parabolic evolution equations and the Łojasiewicz–Simon gradient inequality. In the first volume of the same title, after setting the abstract frameworks of arguments, a general convergence theorem was proved under the four structural assumptions of critical condition, Lyapunov function, angle condition, and gradient inequality. In this volume, with those abstract results reviewed briefly, their applications to concrete parabolic equations are described. Chapter 3 presents a discussion of semilinear parabolic equations of second order in general n-dimensional spaces, and Chapter 4 is devoted to treating epitaxial growth equations of fourth order, which incorporate general roughening functions. In Chapter 5 consideration is given to the Keller–Segel equations in one-, two-, and three-dimensional spaces. Some of these results had already been obtained and published by the author in collaboration with his colleagues. However, by means of the abstract theory described in the first volume, those results can be extended much more. Readers of this monograph should have a standard-level knowledge of functional analysis and of function spaces. Familiarity with functional analytic methods for partial differential equations is also assumed.

Mathematical analysis. --- Analysis (Mathematics). --- Functional analysis. --- Measure theory. --- Analysis. --- Functional Analysis. --- Measure and Integration. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- 517.1 Mathematical analysis --- Mathematical analysis --- Anàlisi matemàtica --- Anàlisi funcional --- Teoria de la mesura --- Anells (Àlgebra) --- Integrals generalitzades --- Integral de Lebesgue --- Teoria ergòdica --- Teoria de la mesura geomètrica --- Àlgebres de mesura --- Càlcul funcional --- Càlcul de variacions --- Àlgebres de Hilbert --- Àlgebres topològiques --- Anàlisi funcional no lineal --- Anàlisi microlocal --- Espais analítics --- Espais de Hardy --- Espais d'Orlicz --- Espais funcionals --- Espais vectorials normats --- Espais vectorials --- Filtres digitals (Matemàtica) --- Funcionals --- Funcions vectorials --- Multiplicadors (Anàlisi matemàtica) --- Pertorbació (Matemàtica) --- Teoria d'operadors --- Teoria de distribucions (Anàlisi funcional) --- Teoria de functors --- Teoria de l'aproximació --- Teoria del funcional de densitat --- Teoria espectral (Matemàtica) --- Equacions funcionals --- Equacions integrals --- Matemàtica --- Àlgebra lineal --- Anàlisi combinatòria --- Anàlisi de Fourier --- Anàlisi estocàstica --- Anàlisi matemàtica no-estàndard --- Anàlisi numèrica --- Funcions --- Matemàtica per a enginyers --- Sèries infinites --- Teoria del potencial (Matemàtica) --- Teories no lineals --- Rutes aleatòries (Matemàtica) --- Àlgebra --- Càlcul --- Evolution equations. --- Evolutionary equations --- Equations, Evolution --- Equations of evolution --- Differential equations

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This second volume continues the study on asymptotic convergence of global solutions of parabolic equations to stationary solutions by utilizing the theory of abstract parabolic evolution equations and the Łojasiewicz-Simon gradient inequality. In the first volume of the same title, after setting the abstract frameworks of arguments, a general convergence theorem was proved under the four structural assumptions of critical condition, Lyapunov function, angle condition, and gradient inequality. In this volume, with those abstract results reviewed briefly, their applications to concrete parabolic equations are described. Chapter 3 presents a discussion of semilinear parabolic equations of second order in general n-dimensional spaces, and Chapter 4 is devoted to treating epitaxial growth equations of fourth order, which incorporate general roughening functions. In Chapter 5 consideration is given to the Keller-Segel equations in one-, two-, and three-dimensional spaces. Some of these results had already been obtained and published by the author in collaboration with his colleagues. However, by means of the abstract theory described in the first volume, those results can be extended much more. Readers of this monograph should have a standard-level knowledge of functional analysis and of function spaces. Familiarity with functional analytic methods for partial differential equations is also assumed.

Functional analysis --- Mathematical analysis --- Mathematics --- Measuring methods in physics --- Mathematical physics --- differentiaalvergelijkingen --- analyse (wiskunde) --- functies (wiskunde) --- meettechniek --- wiskunde

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Functional analysis --- Mathematical analysis --- Mathematics --- Measuring methods in physics --- Mathematical physics --- differentiaalvergelijkingen --- analyse (wiskunde) --- functies (wiskunde) --- meettechniek --- wiskunde