Spatial and Dynamical Chaos Generated by Reaction–Diffusion Systems in Unbounded Domains

We consider in this article a nonlinear reaction–diffusion system with a transport term (L,∇ _x )u, where L is a given vector field, in an unbounded domain Ω. We prove that, under natural assumptions, this system possesses a locally compact attractor in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov’s ɛ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584–637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on . As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of .     

Spectral Properties of Schrödinger Operators on Domains with Varying Orders of Thinness

We consider the Schrödinger operator on two types of domains depending on a small parameter : dumbbell domains and thin domains with varying orders of thinness. In both situations we compare the eigenvalues and eigenvectors of the Schrödinger operator with the corresponding eigenvalues and eigenvectors of a limit operator defined on the limit domain.     

On Burenkov's extension operator preserving Sobolev–Morrey spaces on Lipschitz domains

We prove that Burenkov's extension operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n‐dimensional Euclidean space.     

Well‐posedness and regularity results for a 2m‐th order parabolic equation in symmetric conical domains of ℝN+1

In this paper, we use the domain decomposition method to prove well‐posedness and smoothness results in anisotropic weighted Sobolev spaces for a multidimensional high‐order parabolic equation set in conical time‐dependent domains of . Copyright © 2017 John Wiley & Sons, Ltd.     

Series solutions for steady unsaturated flow in irregular porous domains

For the most part, analytical solutions for steady unsaturated infiltration have been restricted to infinite and semi-infinite seepage geometries, using the quasi-linear approximation for the hydraulic conductivity. We provide analytical series methods to solve the steady quasi-linear flow equations, in finite irregular seepage geometries. Unlike the classical approach, the series method has been modified, to cater for arbitrary boundary geometry and surface recharge distributions. The matrix flux potential and the stream function both satisfy the same governing partial differential equation, and the stream function formulation is used to estimate the series coefficients. For a finite vadose zone, the stream function solution does not uniquely determine the matrix flux potential, when flux boundary conditions are used. Consequently, the stream function solution applies to a range of moisture distributions, for given infiltration and evapotranspiration rates through the surface.     

Formation and stability of a charged jet in a strong electric field

The shape of a charged jet is determined in the approximation of a strong electric field. The stability of the jet with respect to both axisymmetric and nonaxisymmetric perturbations of the sinusoidal type is investigated in the linear approximation. The domains of predominance of the axisymmetric and bending modes and the longitudinal partition mode are determined. Experimental data on the longitudinal partition of a polymeric jet into several daughter jets are given.     

Real analytic families of harmonic functions in a planar domain with a small hole

We consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter ?   and we denote by _u?$u_{?}$ the corresponding solution. The behavior of _u?$u_{?}$ for ?   small and positive can be described in terms of real analytic functions of two variables evaluated at (?,1/log??)$(?,1/\log ??)$. We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by _u?$u_{?}$ for ?   small and describe _u?$u_{?}$ by real analytic functions of ?. Then it is natural to ask what happens when ? is negative. The case of boundary data depending on ? is also considered. The aim is to study real analytic families of harmonic functions which are not necessarily solutions of a particular boundary value problem.     

The Mal'cev Rank for Modules over Pseudo-Valuation Domains

Let R be a commutative ring and A be an R-module. The Mal'cev rank μ(A) of A is the sup of genN, where N ranges over the finitely generated submodules of A, and genN is the minimum number of generators of N. We prove that μ is both sub-additive and pre-additive as an invariant of Mod(R). Our main goal is to investigate μ for modules over pseudo-valuation domains. Specifically, we establish which pseudo-valuation domains R satisfy the property that an R-module of finite Mal'cev rank must be finitely generated. We split the class 𝒞 of pseudo-valuation domains as a union 𝒞 = 𝒞₁ ∪ 𝒞₂ ∪ 𝒞₃ ∪ 𝒞₄ of suitably defined subclasses, and prove that the property holds if and only if R ∈ 𝒞₃ ∪ 𝒞₄. In that case we can describe the R-modules A where μ(A) < ∞. We also show that, for R ∈ 𝒞₄, there exist indecomposable R-modules of arbitrarily large finite Mal'cev rank.     

A Factorization Theorem for Topological Abelian Groups

Using the nice properties of the w-divisible weight and the w-divisible groups, we prove a factorization theorem for compact abelian groups K; namely, K = K _tor  × K _d , where K _tor is a bounded torsion compact abelian group and K _d is a w-divisible compact abelian group. By Pontryagin duality this result is equivalent to the same factorization for discrete abelian groups proved in [9 Galindo , J. , Macario , S. ( 2011 ). Pseudocompact group topologies with no infinite compact subsets . J. Pure and Appl. Algebra 215 : 655 – 663 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Energy decay rates of elastic waves in unbounded domain with potential type of damping

In this paper we first show that the total energy of solutions for a semilinear system of elastic waves in Rn with a potential type of damping decays in an algebraic rate to zero. We study the critical potential case and we assume that the initial data have a compact support. An application for the Euler-Poisson-Darboux type dissipation V(t,x) is obtained and in this case the compactness of the support on the initial data is not necessary. Finally, we shall discuss the energy concentration region for the linear system of elastic waves in an exterior domain.