We study inclusion indices relative to an interpolation scale. Applications are given to several families of functions spaces.

We consider the generalized Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equations with dissipative term. We establish the condition under which the solutions uα,β,γ converge in a strong topology to the entropy solution of a scalar conservation laws using methodology developed by Hwang and Tzavaras. First, we obtain the approximate transport equation for the given BBM–Burgers equations. Then, using the averaging lemma, we obtain the convergence.

We consider an elliptic boundary problem defined in a bounded region Ω ⊂ Rn and where the spectral parameter is multiplied by a weight function ω(x). We suppose that ω(x) ≠ 0 for x ∈ Ω, but vanishes in a specified manner on the boundary of Ω. Under limited smoothness assumptions, we derive results pertaining to existence and uniqueness of and a priori estimates for solutions of the boundary problem. If S(λ) denotes the operator pencil induced in L2(Ω) by the boundary problem with zero boundary conditions, then results are also derived pertaining to the spectral properties of S(λ).

Given a function υ ∈ BV (Ω; Rm), we introduce the notion of a minimal lifting of Dυ. We prove that every υ ∈ BV (Ω; Rm) has a unique minimal lifting, and we show that if υk → υ strictly in BV, then the minimal liftings of υk converge weakly as measures to the minimal lifting of υ. As an application, we deduce a result about weak continuity of the distributional determinant Det D2u with respect to strict convergence.

In this paper we study the convexity of the level sets of solutions of the problem where f is a suitable function with subcritical or critical growth. Under some assumptions on the Gauss curvature of ∂Ω, we prove that the level sets of the solution of (0.1) are strictly convex.

This paper considers the nonlinear stability oftravelling wavefronts of a time-delayed diffusive Nicholson blowflies equation. We prove that, under a weighted L2 norm, ifa solution is sufficiently close to a travelling wave front initially, it converges exponentially to the wavefront as t → ∞. The rate ofconvergence is also estimated.

In this paper, we show that if b(x) ≥ b∞ > 0 in Ω̄ and there exist positive constants C, δ, R0 such that where x = (y, z) ∈ RN with y ∈ Rm, z ∈ Rn, N = m + n ≥ 3, m ≥ 2, n ≥ 1, 1 < p < (N + 2)/(N − 2), ω ⊆ Rm a bounded C1,1 domain and Ω = ω × Rn, then the Dirichlet problem −Δu + u = b(x)|u|p−1u in Ω has a solution that changes sign in Ω, in addition to a positive solution.

We study the Cauchy problem for the one-dimensional equations of a viscous heat-conducting gas in the Lagrangian mass coordinates with the initial data in the Lebesgue spaces. We prove the existence, the uniqueness and the Lipschitz continuous dependence on the initial data of global weak solutions.

We introduce a new non-classical Riemann solver for scalar conservation laws with concave–convex flux-function. This solver is based on both a kinetic relation, which determines the propagation speed of (under-compressive) non-classical shock waves, and a nucleation criterion, which makes a choice between a classical Riemann solution and a non-classical one. We establish the existence of (non-classical entropy) solutions of the Cauchy problem and discuss several examples of wave interactions. We also show the existence of a class of solutions, called splitting–merging solutions, which are made of two large shocks and small bounded-variation perturbations. The nucleation solvers, as we call them, are applied to (and actually motivated by) the theory of thin-film flows; they help explain numerical results observed for such flows.

We study some properties of space-like submanifolds in Minkowski n-space, whose points are all umbilic with respect to some normal field. As a consequence of these and some results contained in a paper by Asperti and Dajczer, we obtain that being ν-umbilic with respect to a parallel light-like normal field implies conformal flatness for submanifolds of dimension n − 2 ≥ 3. In the case of surfaces, we relate the umbilicity condition to that of total semi-umbilicity (degeneracy of the curvature ellipse at every point). Moreover, if the considered normal field is parallel, we show that it is everywhere time-like, space-like or light-like if and only if the surface is included in a hyperbolic 3-space, a de Sitter 3-space or a three-dimensional light cone, respectively. We also give characterizations of total semi-umbilicity for surfaces contained in hyperbolic 4-space, de Sitter 4-space and four-dimensional light cone.

The paper considers an incompressible and irrotational flow of a fluid with constant density bounded below by a rigid horizontal bottom and above by a free surface under the influence of gravity and surface tension. It is known that the full Euler equations have travelling two-dimensional solitary-wave solutions of small amplitude for large surface tension (Bond number greater than ⅓). This paper shows that these waves are linearly unstable to three-dimensional perturbations which oscillate along the wave crest with wavenumber in a finite band. The growth rates of these unstable modes are well approximated using the linearized Kadomtsev–Petviashvili equation with positive disper

We rigorously approach the Schrödinger equation of derivative type qt + iqxx + λ |q|2qx + μq2q̄x = 0, λ ∈ R, μ ∈ C, by the cubic nonlinear Schrödinger equation AT + i AXX + i k0 (λ − μ) |A|2A = 0. We also study the case of the KdV-like equation qt + i qxx + aqxxx + i |q|2q + λ̃ (|q|2q)x + μ̃ |q|2qx = 0, λ̃ μ̃ ∈ R, arising in optical physics.

Consider the system where λ is a positive parameter and Ω is a bounded domain in RN. We prove the existence of a large positive solution for λ large when limx → ∞ (f(Mg(x))/x) = 0 for every M > 0. In particular, we do not need any monotonicity assumptions on f, g, nor any sign conditions on f(0), g(0).

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.

Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.

In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

Based on earlier results on existence, we study the asymptotic behaviour of solutions to the coalescence-breakage equations, including the volume-scattering phenomenon and high-energy collisions. The solutions are shown to converge towards one particular equilibrium, provided the kernels satisfy a kind of reversibility. We also derive stability of these equilibria in a suitable topology.

We study the existence of closed characteristics on three-dimensional energy manifolds of second-order Lagrangian systems. These manifolds are always non-compact, connected and not necessarily of contact type. Using the specific geometry of these manifolds, we prove that the number of closed characteristics on a prescribed energy manifold is bounded below by its second Betti number, which is easily computable from the Lagrangian.

In this paper we analyse a singular perturbation problem for linear wave equations with interior and boundary damping. We show how the solutions converge to the formal parabolic limit problem with dynamic boundary conditions. Conditions are given for uniform convergence in the energy space.

Given a smooth function 𝔏 of a real or complex variable and taking its values in the class of Fredholm operators of index zero in a Banach space, there are some available definitions in the literature of algebraic multiplicity of the family 𝔏 at a point x0 of the parameter at which the operator 𝔏(x0) becomes non-invertible. The purpose of this paper is to suggest an axiomatic for the multiplicity and to prove that the algebraic multiplicity is uniquely determined by a few of its properties, independently of its construction.

This paper deals with existence and non-existence of global solutions of certain fast–slow diffusion systems with nonlinear boundary conditions. Necessary and sufficient conditions for global existence of positive solutions are obtained in terms of various parameters which appear explicitly in the definition of the systems.

In this paper we present a simpler proof of a result of Lewis concerning the continuity of weak solutions to the two-dimensional thermistor problem in the case where the temperature can blow up in a region with non-empty interior. Some other regularity properties are also discussed.