Given a complete m-primary ideal J in a local regular two-dimensional ring (R,m), we describe every adjacent complete ideal above J as the integral closure of some ideal (f, g) for suitable f, g associated to J. We also provide a geometrical procedure that gives its base points, thus determining its equisingularity class. We decompose the set IJ of these adjacent ideals in terms of the Rees valuations of J. As a consequence, we obtain a geometrical characterization of the finiteness of IJ.

We apply the theory of planar dynamical systems to carry out a qualitative analysis for the planar dynamical system corresponding to the fluidized-bed modelling equation. We obtain the global phase portraits of this system under various parameter conditions and the existence conditions of bounded travelling-wave solutions of this equation. According to the discussion on relationships between the behaviours of bounded travelling-wave solutions and the dissipation coefficients ε and δ, we find a critical value λ0 for arbitrary travelling-wave velocity υ. This equation has a unique damped oscillatory solution as ∥ε + δυ∥ < λ0 and ∥ε + δυ∥ ≠ 0, while it has a unique monotone kink profile solitary-wave solution as ∥ε + δυ∥ > λ0. By means of the undetermined coefficients method, we obtain the exact bell profile solitary-wave solution and monotone kink profile solitary-wave solution. Meanwhile, we obtain the approximate damped oscillatory solution. We point out the positions of these solutions in the global phase portraits. Finally, based on integral equations that reflect the relationships between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate damped oscillatory solutions are presented.

Using variational methods we study the non-existence and multiplicity of non-negative solutions for a class of quasilinear elliptic equations of p(x)-Laplacian type with nonlinear boundary conditions of the form

where Ω; is a bounded domain with smooth boundary, n is the outer unit normal to ∂Ω and λ is a parameter. Furthermore, we want to emphasize that g : ∂Ω × [0,∞)→ ℝ is a continuous function that may or may not satisfy the Ambrosetti–Rabinowitz-type condition.

We rigorously derive a weak form of the Lifshitz–Slyozov–Wagner equation as the homogenization limit of a Stefan-type problem describing reaction-controlled coarsening of a large number of small spherical particles. Moreover, we deduce that the effective mean-field description holds true in the particular limit of vanishing surface-area density of particles.

Let −DpD be a differential operator on the compact interval [−b, b] whose leading coefficient is positive on (0, b] and negative on [−b, 0), with fixed, separated, self-adjoint boundary conditions at b and −b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded sesquilinear forms by a generalization of Kato's representation theorems. The theory of non-semibounded sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal sesquilinear form. Moreover, among all closed forms associated with the self-adjoint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Kreĭn–Feller operators with a single concentrated mass.

We study the Cauchy problem of certain Boussinesq-α equations in n dimensions with n = 2 or 3. We establish regularity for the solution under ▽u ∈ L1 (0, T; Ḃ0∞,∞(ℝn)). As a corollary, the smooth solution of the Leray-α–Boussinesq system exists globally, when n = 2. For the Lagrangian averaged Boussinesq equations, a regularity criterion ▽θ ∈ L1(0, T;L∞(ℝ2)) is established. Other Boussinesq systems with partial viscosity are also discussed in the paper.

For a family of semigroups Sε(t) : ℌε → ℌε depending on a perturbation parameter ε ∈ [0, 1], where the perturbation is allowed to become singular at ε = 0, we establish a general theorem on the existence of exponential attractors εε satisfying a suitable Hölder continuity property with respect to the symmetric Hausdorff distance at every ε ∈ [0, 1]. The result is applied to the abstract evolution equations with memory

where kε(s) = (1/ε)k(s/ε) is the rescaling of a convex summable kernel k with unit mass. Such a family can be viewed as a memory perturbation of the equation

formally obtained in the singular limit ε → 0.

Dirichlet–Neumann operators arise in many applications in the sciences, and this has inspired a number of studies on their analytical properties. In this paper we further investigate the analyticity properties of Dirichlet–Neumann operators as functions of the boundary shape. In particular, we study the size of the disc of convergence of their Taylor-series representation. For this we use a complexification technique which requires a novel reformulation of the problem, coupled with methods for systems of elliptic partial differential equations. Numerical results to illustrate our theoretical conclusions are presented.

We study the existence of a positive solution of a p-superlinear equation involving the p-Laplacian operator. The main difficulty here is that the nonlinearity considered does not necessarily verify the well-known Ambrosetti–Rabinowitz condition. As an application, by performing an adequate change of variables we obtain an existence result of a quasilinear equation depending on the gradient.

We study orthogonal projections of embedded surfaces M in H3+ (−1) along horocycles to planes. The singularities of the projections capture the extrinsic geometry of M related to the lightcone Gauss map. We give geometric characterizations of these singularities and prove a Koenderink-type theorem that relates the hyperbolic curvature of the surface to the curvature of the profile and of the normal section of the surface. We also prove duality results concerning the bifurcation set of the family of projections.

This paper investigates the existence of solutions for boundary variational–hemivariational inequalities of elliptic type at resonance as well as at non-resonance. Using the notion of the generalized gradient of Clarke and the property of the first eigenfunction, we also build a Landesman–Lazer theory in the non-smooth framework of variational–hemivariational inequalities of elliptic type.

We show the existence and non-existence of positive solutions to a system of singular elliptic equations with the Dirichlet boundary condition.