Interior Transition Layers Research Articles

Passive and active oxidation of Al in the Al–Al2O3 composite refractory at high temperatures in air

The active and passive oxidation of Al in Al–Al2O3 composites was investigated in air at high temperatures. As fresh air impinged on the surface of the Al–Al2O3 composite sample, a thin Al2O3 layer was preferentially formed via passive oxidation of Al, which consumed most of the oxygen and lowered the local Po2 inside the sample. At low Po2, active oxidation of Al occurred and predominated in the interior transition layer, which further lowered the local Po2 to extremely low values within the non-oxide stability field. The local equilibrium PAlxOy generated by the active oxidation of Al reached a micro-positive pressure state and served as a gas barrier, hindering the inward diffusion of O2. The Po2 inside the composite was lowered by Al in the outer layers via a self-gettering process. Subsequently, Al4O4C and Al2OC-AlN solid solution were formed by both the direct reaction of Al(l) and the indirect reaction of AlxOy(g). After sintering in air, the Al–Al2O3 composite exhibited a functional gradient structure with a thin oxide layer and an interior non-oxide-reinforced corundum. A reaction mechanism model was established.

Locations of interior transition layers to inhomogeneous transition problems in higher -dimensional domains

We consider the following inhomogeneous problems \[ \begin{cases} \epsilon^{2}\mbox{div}(a(x)\nabla u(x))+f(x,u)=0 & \text{ in }\Omega,\\ \frac{\partial u}{\partial \nu}=0 & \text{ on }\partial \Omega,\\ \end{cases} \]where $\Omega$ is a smooth and bounded domain in general dimensional space $\mathbb {R}^{N}$, $\epsilon >0$ is a small parameter and function $a$ is positive. We respectively obtain the locations of interior transition layers of the solutions of the above transition problems that are $L^{1}$-local minimizer and global minimizer of the associated energy functional.

Bifurcation tearing in a forced Duffing equation

The structure of solution branches of the equation ε 2 u x x = u 3 − λ u + δ g ( x ) , for x ∈ R is studied, for real constants, ε , λ and δ , where g ( x ) is periodic in x and u satisfies Neumann boundary conditions. For α : = u ( 0 ) bifurcation structure in the α – δ plane occurs that for small ε leads to families of singular asymptotic solutions with interior transition layers between outer solutions. In contrast, bifurcation in the α – λ plane produces families of oscillatory solutions that for small ε have large numbers, n , of oscillations between envelope outer solutions. The n label of the solution branch is defined by its local bifurcation at ε 2 n 2 = λ > 0 . Further the bifurcation structure on the n branch is classified by the integer n being either even or odd. When n is even the pitchfork bifurcation in the α – δ plane leads to connected branches of solutions in the α – λ – δ space. But when n is odd the presence of g ( x ) leads to cusp catastrophes (for small ε ) in the α – δ plane and these give rise to disconnected (or isolated) branches of solution in the α – λ plane that appear to be torn from the main solution branch. The integer n acts as a count of the number of oscillations on these solution branches. As ε 2 λ − 1 → 0 we see layer behaviour occurring about the top and bottom outer solutions and as ε 2 λ − 1 → ∞ we find convergence to the middle outer solution.

Assessment of coatings for protection of cement paste against microbial induced deterioration through image analysis

In this study, a laboratory method and equipment was developed to accelerate and study the microbial induced deterioration of cementitious materials, and three types of coatings, namely, cement-based capillary crystalline waterproofing coating (CCCWC), cement-based bactericidal coating (CBC) and epoxycoaltarpitch coating (ECTPC) were applied onto the cement paste surface, and their protecting effects were investigated. After immersion in sewage, the microstructure and element distribution of cement paste were characterized by scanning electron microscopy-electron dispersive spectrometry (SEM-EDS). Besides, the distribution of dead/live cells within the biofilm attached to various specimens was examined by confocal scanning laser microscopy (CLSM). To make a quantitative study, the image analysis was adopted. Since both the images obtained belonged to RGB color system, it was found that the image processing program of MATLAB is simple but very helpful for such task. The results indicate that for uncoated specimen, there were the most content of S and the least contents of Ca and Si. However, for coated specimens, the contents of S decreased and that of Ca and Si increased accordingly. ECTPC had the best protective effect on the microbial induced deterioration of concrete, followed by CBC and CCCWC. Besides the physical barrier effect, the bactericidal function of coating has significant contribution for protecting concrete against sewage attack. Furthermore, the image analysis was used to determine the penetration depth of element S in cement paste quantitatively, and based on the change of the S concentration with penetration depth, the surface specimens can be divided into three zones: interior deterioration layer, transition layer and non-corroded layer. The combined use of micro-scale characterization and image analysis can provide a quick, accurate and labor saving method for the deterioration study of coated concrete in sewage environment.

Recent progresses in boundary layer theory

In this article,we review recent progresses in boundary layer analysis of somesingular perturbation problems.Using the techniques of differential geometry,an asymptotic expansion of reaction-diffusion or heat equations in a domain with curved boundaryis constructed and validated in some suitable functional spaces.In addition, we investigate the effect of curvatureas well as thatof an ill-prepared initial data.Concerning convection-diffusion equations, the asymptotic behavior of their solutionsis difficult and delicate to analyze because it largely depends on the characteristics of the corresponding limit problems, which are first order hyperbolic differential equations. Thus, the boundary layer analysis is performed on relatively simpler domains,typically intervals, rectangles, or circles.We consider also the interior transition layers at the turning point characteristics in an interval domain and classical (ordinary), characteristic (parabolic) and corner (elliptic) boundary layers in a rectangular domainusing the technique of correctors and the tools of functional analysis.The validity of our asymptotic expansions is also established in suitable spaces.

Qualitative analysis of a Lotka-Volterra competition system with advection

We study a diffusive Lotka-Volterra competition system with advection under Neumann boundary conditions. Our system models a competition relationship that one species escape from the region of high population density of their competitors in order to avoid competitions. We establish the global existence of bounded classical solutions for the system in one-dimensional domain. For multi-dimensional domains, globally bounded classical solutions are obtained for a parabolic-elliptic system under proper assumptions on the system parameters. These global existence results make it possible to study bounded steady states in order to model species segregation phenomenon. We then investigate the stationary problem in one-dimensional domains. Through bifurcation theory, we obtain the existence of nonconstant positive steady states, which are small perturbations from the positive equilibrium; we also study the stability of these bifurcating solutions when the diffusion coefficient of the escaper is large and the diffusion coefficient of its competitor is small. In the limit of large advection rate, we show that the reaction-advection-diffusion system converges to a shadow system involving the competitor population density and an unknown positive constant. The existence and stability of positive solutions to the shadow system have also been obtained through bifurcation theories. Finally, we construct positive solutions with an interior transition layer to the shadow system when the crowding rate of the escaper and the diffusion rate of its interspecific competitors are sufficiently small. The transition-layer solutions can be used to model the species segregation phenomenon.

Moving fronts in integro-parabolic reaction-advection-diffusion equations

We consider initial-boundary value problems for a class of singularly perturbed nonlinear integro-differential equations. In applications, they are referred to as nonlocal reactionadvection-diffusion equations, and their solutions have moving interior transition layers (fronts). We construct the asymptotics of such solutions with respect to a small parameter and estimate the accuracy of the asymptotics. To justify the asymptotics, we use the asymptotic differential inequality method.

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

A semilinear reaction–diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive parameter \({\varepsilon^2}\) , is considered. It can have multiple solutions. The numerical computation of solutions having interior transition layers is analysed. It is demonstrated that the accurate computation of such solutions is exceptionally difficult. To address this difficulty, we propose an artificial-diffusion stabilization. For both standard and stabilised finite difference methods on suitable Shishkin meshes, we prove existence and investigate the accuracy of computed solutions by constructing discrete sub- and super-solutions. Convergence results are deduced that depend on the relative sizes of \({\varepsilon}\) and N, where N is the number of mesh intervals. Numerical experiments are given in support of these theoretical results. Practical issues in using Newton’s method to compute a discrete solution are discussed.

Interior layers for an inhomogeneous Allen–Cahn equation

We consider the problem ε 2 Δ u + V ( y ) u ( 1 − u 2 ) = 0 in Ω , ∂ u ∂ n = 0 on ∂ Ω where Ω is a smooth and bounded domain in R 2 and V is a positive smooth function in Ω ¯ . Let Γ be a closed, non-degenerate geodesic with respect to the metric d s 2 = V ( y ) ( d y 1 2 + d y 2 2 ) in Ω. We prove that there exist two interior transition layer solutions u ε ( 1 ) , u ε ( 2 ) when ε is sufficiently small. One of the layer solutions u ε ( 1 ) approaches −1 in Ω 1 and +1 in Ω 2 = Ω \\ Ω ¯ 1 as ε tends to 0, while the other solution u ε ( 2 ) exhibits a transition layer in the opposite direction.

Transition layer for the heterogeneous Allen–Cahn equation

We consider the equation(1)ɛ2Δu=(u−a(x))(u2−1)in Ω,∂u∂ν=0on ∂Ω, where Ω is a smooth and bounded domain in Rn, ν the outer unit normal to ∂Ω, and a a smooth function satisfying −1<a(x)<1 in Ω¯. We set K, Ω+ and Ω− to be respectively the zero-level set of a, {a>0} and {a<0}. Assuming ∇a≠0 on K and a≠0 on ∂Ω, we show that there exists a sequence ɛj→0 such that Eq. (1) has a solution uɛj which converges uniformly to ±1 on the compact sets of Ω± as j→+∞. This result settles in general dimension a conjecture posed in [P. Fife, M.W. Greenlee, Interior transition layers of elliptic boundary value problem with a small parameter, Russian Math. Surveys 29 (4) (1974) 103–131], proved in [M. del Pino, M. Kowalczyk, J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal. 38 (5) (2007) 1542–1564] only for n=2.

Resonance and Interior Layers in an Inhomogeneous Phase Transition Model

We consider the problem $\ve^2 \Delta u + (u-a(x))(1-u^2)=0$ in $\Omega, \frac{\partial u}{\partial \nu}=0$ on $\partial \Omega$, where Ω is a smooth and bounded domain in $\R^2$, $ -1 <a(x) <1$. Assume that $ \Gamma = \{ x \in \Omega, a(x)=0 \}$ is a closed, smooth curve contained in Ω in such a way that $\Omega = \Omega_{+} \cup\Gamma\cup \Omega_{-}$ and $\frac{\partial a}{\partial n } >0$ on Γ, where n is the outer normal to $\Omega_+$. Fife and Greenlee [Russian Math. Surveys, 29 (1974), pp. 103–131] proved the existence of an interior transition layer solution $u_\ve$ which approaches ‐1 in $\Omega_{-}$ and +1 in $\Omega_{+}$, for all ε sufficiently small. A question open for many years has been whether an interior transition layer solution approaching 1 in $\Omega_{-}$ and ‐1 in $\Omega_{+}$ exists. In this paper, we answer this question affirmatively when $n=2$, provided that ε is small and away from certain critical numbers. A main difficulty is a resonance phenomenon induced by a large number of small critical eigenvalues of the linearized operator.

Equivalence of Two Sets of Transition Points Corresponding to Solutions with Interior Transition Layers

We establish the equivalence of two sets of transition points corresponding to solutions of singularly perturbed boundary-value problems with interior boundary layers. The first set appears in the formalism for constructing the asymptotics of the solution of a boundary-value problem and the second, in the direct scheme formalism for constructing the asymptotics of the solution of a variational problem.

Existence of solutions with interior transition layers touching the boundary

For the case of Dirichlet boundary conditions, the existence of a solution with interior transition layer touching the boundary is proved.

Axially Symmetric Boundary Value Problem Describing the Distribution of Charges in Semiconductors

We prove that bifurcation solutions to the stationary problem exist and can be extended with respect to parameter. The problem under consideration has a solution with the so-called interior transition layer phenomenon. In the nonstationary case, we establish the existence and uniqueness of a solution for any t > 0. Under certain assumptions, the nonstationary problem determines a dynamical system in some compact set. Bibliography: 9 titles.

On the global domain of influence of stable solutions with interior layers in the two-dimensional case

We consider the initial-boundary problem for the singularly perturbed non-stationary reaction-diffusion equation with non-linearity of cubic type. We impose conditions sufficient for the existence of a stable stationary solution with an interior transition layer in the neighbourhood of a certain curve. We investigate the set of initial functions belonging to the domain of influence of this solution.

Interior transition layers of solutions to the perturbed elliptic Sine-Gordon equation on an interval

We consider the perturbed elliptic Sine-Gordon equation on an interval-u″t+γsinu(t)=μf(u(t)),t ∈I := (-T, T),u(t) > 0,t ∈I,u(±T)=0 where λ, μ>0 are parameters andT>0 is a constant. By applying variational methods subject to the constraint depending on λ, we obtain eigenpairs (μ,u)=(μ(λ),u λ) which solve this eigenvalue problem for a given λ>0. Then we study the asymptotic behavior ofu λ and μ(λ) as λ→∞. Especially, we study the location of interior transition layers ofu λ as λ→∞.

Axis-symmetric boundary-value problem describing distribution of charges in semiconductors

Under certain assumptions, the existence of two families of solutions to the stationary problem is proved and asymptotic expansions with respect to a small parameter are established. The stability of these families of solutions is shown. The problem under consideration has a solution connected with the so-called interior transition layer. In the case of a nonstationary problem, the existence and uniqueness of a solution are established for any t>0. Under certain assumptions, the nonstationary problem defines a dynamical system in some compact set. Bibliography: 5 titles.

Asymptotic behavior of interior transition layers for nonlinear boundary-value problem with small parameter

The system of equations $$(D(|E|)(n' + nE))' = 0, E' = f - n, 0< x< 1,$$ with the boundary conditions $$E(0) = a_0 , E(1) = \alpha _1 , D(|E(0)|)(n'(0) + n(0)E(0)) = j_0 $$ is considered. The solvability of this boundary-value problem and properties of the family of solutions are studied under the condition that the diffusion coefficient is negative. Bibliography: 5 titles.

The Asymptotic Solution of a Semiconductor Device Problem Involving Reverse Bias

A standard drift diffusion model for a one-dimensional stationary semiconductor device is considered in the case of reverse bias. The resulting boundary value problem is scaled and analyzed when two small physical parameters are of comparable size. The limiting solution is obtained in both an interior depletion region and in the complementary intervals of zero space charge. The challenging question of the structure of the corresponding outer expansions and of their corrections in interior transition and endpoint boundary layers of rapid change is also studied.

Diffusive logistic equations with indefinite weights: population models in disrupted environments

SynopsisThe dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form ut = d Δu + [m(x) — cu]u in Ω × (0, ∞), where u represents the population density, c, d &gt; 0 are constants describing the limiting effects of crowding and the diffusion rate of the population, respectively, and m(x) describes the local growth rate of the population. If the environment ∞ is bounded and is surrounded by uninhabitable regions, then u = 0 on ∂∞× (0, ∞). The growth rate m(x) is positive on favourablehabitats and negative on unfavourable ones. The object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled. The models are shown to possess a unique, stable, positive steady state (implying persistence for the population) provided l/d&gt; where is the principle positive eigenvalue for the problem — Δϕ=λm(x)ϕ in Χ,ϕ=0 on ∂Ω. Analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat. In the limit as the diffusion rate d ↓ 0, the solutions tend toward the positive part of m(x)/c, and if m is discontinuous develop interior transition layers. The analysis uses bifurcation and continuation methods, the variational characterisation of eigenvalues, upper and lower solution techniques, and singular perturbation theory.