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Progress in Partial Differential Equations is devoted to modern topics in the theory of partial differential equations. It consists of both original articles and survey papers covering a wide scope of research topics in partial differential equations and their applications. The contributors were participants of the 8th ISAAC congress in Moscow in 2011 or are members of the PDE interest group of the ISAAC society. This volume is addressed to graduate students at various levels as well as researchers in partial differential equations and related fields. The reader will find this an excellent resource of both introductory and advanced material. The key topics are: • Linear hyperbolic equations and systems (scattering, symmetrisers) • Non-linear wave models (global existence, decay estimates, blow-up) • Evolution equations (control theory, well-posedness, smoothing) • Elliptic equations (uniqueness, non-uniqueness, positive solutions) • Special models from applications (Kirchhoff equation, Zakharov-Kuznetsov equation, thermoelasticity).

Mathematics --- Physical Sciences & Mathematics --- Calculus --- Differential equations. --- Differential equations, Partial. --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Mathematics. --- Dynamics. --- Ergodic theory. --- Partial differential equations. --- Mathematical physics. --- Partial Differential Equations. --- Ordinary Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Mathematical Applications in the Physical Sciences. --- Mathematical Physics. --- Physical mathematics --- Physics --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Math --- Science --- Differential equations, partial. --- Differential Equations. --- Differentiable dynamical systems. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics

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Progress in Partial Differential Equations is devoted to modern topics in the theory of partial differential equations. It consists of both original articles and survey papers covering a wide scope of research topics in partial differential equations and their applications. The contributors were participants of the 8th ISAAC congress in Moscow in 2011 or are members of the PDE interest group of the ISAAC society. This volume is addressed to graduate students at various levels as well as researchers in partial differential equations and related fields. The reader will find this an excellent resource of both introductory and advanced material. The key topics are: • Linear hyperbolic equations and systems (scattering, symmetrisers) • Non-linear wave models (global existence, decay estimates, blow-up) • Evolution equations (control theory, well-posedness, smoothing) • Elliptic equations (uniqueness, non-uniqueness, positive solutions) • Special models from applications (Kirchhoff equation, Zakharov-Kuznetsov equation, thermoelasticity).

Differential geometry. Global analysis --- Ergodic theory. Information theory --- Partial differential equations --- Differential equations --- Mathematics --- Mathematical physics --- differentiaalvergelijkingen --- differentiaal geometrie --- wiskunde --- fysica --- informatietheorie

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The present volume is dedicated to modern topics of the theory of hyperbolic equations such as evolution equations, multiple characteristics, propagation phenomena, global existence, influence of nonlinearities. It is addressed to beginners as well as specialists in these fields. The contributions are to a large extent self-contained. Key topics include: - low regularity solutions to the local Cauchy problem associated with wave maps; local well-posedness, non-uniqueness and ill-posedness results are proved - coupled systems of wave equations with different speeds of propagation; here pointwise decay estimates for solutions in spaces with hyperbolic weights come in - damped wave equations in exterior domains; the energy method is combined with the geometry of the exterior domain; for the critical part of the boundary a restricted localized effective dissipation is employed - the phenomenon of parametric resonance for wave map type equations; the influence of time-dependent oscillations on the existence of global small data solutions is studied - a unified approach to attack degenerate hyperbolic problems as weakly hyperbolic ones and Cauchy problems for strictly hyperbolic equations with non-Lipschitz coefficients - weakly hyperbolic Cauchy problems with finite time degeneracy; the precise loss of regularity depending on the spatial variables is determined; the main step is to find the correct class of pseudodifferential symbols and to establish a calculus which contains a symmetrizer.

Differential equations, Hyperbolic. --- Differential equations --- Schrödinger operator. --- Scattering (Mathematics) --- Pseudodifferential operators. --- Qualitative theory. --- Operators, Pseudodifferential --- Pseudo-differential operators --- Operator theory --- Scattering theory (Mathematics) --- Boundary value problems --- Differential equations, Partial --- Scattering operator --- Operator, Schrödinger --- Differential operators --- Quantum theory --- Schrödinger equation --- 517.91 Differential equations --- Hyperbolic differential equations --- Global analysis (Mathematics). --- Differential equations, partial. --- Operator theory. --- Functional analysis. --- Analysis. --- Partial Differential Equations. --- Operator Theory. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Functional analysis --- Partial differential equations --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis. --- Analysis (Mathematics). --- Partial differential equations. --- 517.1 Mathematical analysis --- Mathematical analysis

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Analytic functions. --- Polynomials.

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This book provides an overview of different topics related to the theory of partial differential equations. Selected exercises are included at the end of each chapter to prepare readers for the “research project for beginners” proposed at the end of the book. It is a valuable resource for advanced graduates and undergraduate students who are interested in specializing in this area. The book is organized in five parts: In Part 1 the authors review the basics and the mathematical prerequisites, presenting two of the most fundamental results in the theory of partial differential equations: the Cauchy-Kovalevskaja theorem and Holmgren's uniqueness theorem in its classical and abstract form. It also introduces the method of characteristics in detail and applies this method to the study of Burger's equation. Part 2 focuses on qualitative properties of solutions to basic partial differential equations, explaining the usual properties of solutions to elliptic, parabolic and hyperbolic equations for the archetypes Laplace equation, heat equation and wave equation as well as the different features of each theory. It also discusses the notion of energy of solutions, a highly effective tool for the treatment of non-stationary or evolution models and shows how to define energies for different models. Part 3 demonstrates how phase space analysis and interpolation techniques are used to prove decay estimates for solutions on and away from the conjugate line. It also examines how terms of lower order (mass or dissipation) or additional regularity of the data may influence expected results. Part 4 addresses semilinear models with power type non-linearity of source and absorbing type in order to determine critical exponents: two well-known critical exponents, the Fujita exponent and the Strauss exponent come into play. Depending on concrete models these critical exponents divide the range of admissible powers in classes which make it possible to prove quite different qualitative properties of solutions, for example, the stability of the zero solution or blow-up behavior of local (in time) solutions. The last part features selected research projects and general background material.

Partial differential equations --- Differential equations --- differentiaalvergelijkingen