This book is intended for graduate students and research mathematicians working in partial differential equations.

This book is intended for graduate students and research mathematicians working in partial differential equations.

This volume is a collection of articles discussing the most recent advances on various topics in partial differential equations. Many important issues regarding evolution problems, their asymptotic behavior and their qualitative properties are addressed. The quality and completeness of the articles will make this book a source of inspiration and references in the future. Contents: Steady Free Convection in a Bounded and Saturated Porous Medium (S Akesbi et al.); Quasilinear Parabolic Functional Evolution Equations (H Amann); A Linear Parabolic Problem with Non-Dissipative Dynamical Boundary Conditions (C Bandle & W Reichel); Remarks on Some Class of Nonlocal Elliptic Problems (M Chipot); On Some Definitions and Properties of Generalized Convex Sets Arising in the Calculus of Variations (B Dacorogna et al.); Note on the Asymptotic Behavior of Solutions to an Anisotropic Crystalline Curvature Flow (C Hirota et al.); A Reaction-Diffusion Approximation to a Cross-Diffusion System (M Iida et al.); Bifurcation Diagrams to an Elliptic Equation Involving the Critical Sobolev Exponent with the Robin Condition (Y Kabeya); Ginzburg-Landau Functional in a Thin Loop and Local Minimizers (S Kosugi & Y Morita); Singular Limit for Some Reaction Diffusion System (K Nakashima); Rayleigh-Benard Convection in a Rectangular Domain (T Ogawa & T Okuda); Some Convergence Results for Elliptic Problems with Periodic Data (Y Xie); On Global Unbounded Solutions for a Semilinear Parabolic Equation (E Yanagida). Key Features An accessible presentation of the latest, cutting-edge topics in partial differential equations Written by leading scholars in related fields Readership: Graduate students and researchers in partial differential equations and nonlinear science.

This volume is a collection of articles discussing the most recent advances on various topics in partial differential equations. Many important issues regarding evolution problems, their asymptotic behavior and their qualitative properties are addressed. The quality and completeness of the articles will make this book a source of inspiration and references in the future.

This handbook is volume II in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems. Much of this field has developed from interactions with other areas of science, and this volume shows how concepts of dynamical systems further the understanding of mathematical issues that arise in applications. Although modeling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behavior. However, the authors and editors have made an effort to ensure readability on a non-technical level for mathematicians from other fields and for other scientists and engineers. The eighteen surveys collected here do not aspire to encyclopedic completeness, but present selected paradigms. The surveys are grouped into those emphasizing finite-dimensional methods, numerics, topological methods, and partial differential equations. Application areas include the dynamics of neural networks, fluid flows, nonlinear optics, and many others.While the survey articles can be read independently, they deeply share recurrent themes from dynamical systems. Attractors, bifurcations, center manifolds, dimension reduction, ergodicity, homoclinicity, hyperbolicity, invariant and inertial manifolds, normal forms, recurrence, shift dynamics, stability, to namejust a few, are ubiquitous dynamical concepts throughout the articles.

Let $V = {\mathbb R}^{p,q}$ be the pseudo-Euclidean vector space of signature $(p,q)$, $p\ge 3$ and $W$ a module over the even Clifford algebra $C\! \ell^0 (V)$. A homogeneous quaternionic manifold $(M,Q)$ is constructed for any $\mathfrak{spin}(V)$-equivariant linear map $\Pi : \wedge^2 W \rightarrow V$. If the skew symmetric vector valued bilinear form $\Pi$ is nondegenerate then $(M,Q)$ is endowed with a canonical pseudo-Riemannian metric $g$ such that $(M,Q,g)$ is a homogeneous quaternionic pseudo-Kahler manifold. If the metric $g$ is positive definite, i.e. a Riemannian metric, then the quaternionic Kahler manifold $(M,Q,g)$ is shown to admit a simply transitive solvable group of automorphisms. In this special case ($p=3$) we recover all the known homogeneous quaternionic Kahler manifolds of negative scalar curvature (Alekseevsky spaces) in a unified and direct way. If $p>3$ then $M$ does not admit any transitive action of a solvable Lie group and we obtain new families of quaternionic pseudo-Kahler manifolds. Then it is shown that for $q = 0$ the noncompact quaternionic manifold $(M,Q)$ can be endowed with a Riemannian metric $h$ such that $(M,Q,h)$ is a homogeneous quaternionic Hermitian manifold, which does not admit any transitive solvable group of isometries if $p>3$. The twistor bundle $Z \rightarrow M$ and the canonical ${\mathrm SO}(3)$-principal bundle $S \rightarrow M$ associated to the quaternionic manifold $(M,Q)$ are shown to be homogeneous under the automorphism group of the base. More specifically, the twistor space is a homogeneous complex manifold carrying an invariant holomorphic distribution $\mathcal D$ of complex codimension one, which is a complex contact structure if and only if $\Pi$ is nondegenerate. Moreover, an equivariant open holomorphic immersion $Z \rightarrow \bar{Z}$ into a homogeneous complex manifold $\bar{Z}$ of complex algebraic group is constructed. Finally, the construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any $\mathfrak{spin}(V)$-equivariant linear map $\Pi : \vee^2 W \rightarrow V$ a homogeneous quaternionic supermanifold $(M,Q)$ is constructed and, moreover, a homogeneous quaternionic pseudo-Kahler supermanifold $(M,Q,g)$ if the symmetric vector valued bilinear form $\Pi$ is nondegenerate.