The equations for three-wave interaction describe the resonant, quadratic, nonlinear interaction of three waves. They are obtained as amplitude equations in an asymptotic reduction of the basic equations of nonlinear optics, fluid mechanics, and plasma physics. These equations are completely integrable and have been the subject of intensive research in the last years. It is the purpose of this paper to prove exact estimates between the approximations obtained via this system and solutions of the original physical system. Although the three-wave interaction model is believed to describe a number of different physical models we restrict attention to its application as a model of the resonant interaction of water waves subject to weak surface tension.

We study a transport model for populations whose individuals move according to a velocity jump process and stop moving in areas which provide shelter or food. This model has direct applications in ecology (e.g. homeranges, territoriality, stream ecosystems, travelling waves) or cellular biology (e.g. movement of bacteria or movement of proteins in the cell nucleus). In this paper we consider a general model from a mathematical point of view. This provides general insight into the features of these models, which in turn is useful in the modelling process. We consider a singular perturbation expansion and show that the leading order term of the outer solution satisfies a reaction-advection-diffusion equation. The advective term describes taxis toward homeranges or toward regions of shelter. The reaction terms are given by “effective” birth and death rates. Within this framework, the parameters of the reaction-advection-diffusion model (like mobility, drift, birth or death rates) are directly related to the individual movement behaviour of the species at hand (like velocity, frequency of directional changes, response to spatial in-homogeneities, death, or reproduction). We prove that in a homogeneous environment the diffusion limit approximates the solution of the resting-phase transport model to second order in the perturbation parameter.

We present an $n$-population generalization of the Lighthill–Whitham and Richards traffic flow model. This model is analytically interesting because of several non-standard features. For instance, it leads to non-classical shocks and enjoys an unexpected stability in spite of the presence of umbilic points. Furthermore, while satisfying all the minimal ‘common sense’ requirements, it also allows for a description of phenomena often neglected by other models, such as overtaking.

In this paper, we study a class of linear functional differential equations of which the pantograph equation is a prominent member. Specifically, we study the existence of solutions holomorphic at a fixed point of the functional argument. The local theory for equations with attracting fixed points is known [17, 13], but little is known about the case where the fixed point is repelling. We formulate an eigenvalue problem for the repelling fixed point case and show that the corresponding spectrum is discrete. Hence, that holomorphic solutions occur only as special cases. The second order pantograph equation is used to illustrate this result. A key step in this process is to reformulate the problem in terms of a compact operator. Aside from exploiting well known results for the spectra of such operators, we use results such as the Fredholm Alternative to derive existence results for the non-homogeneous problem.

In the limit of small activator-diffusivity $\varepsilon$, a formal asymptotic analysis is used to derive a differential equation for the motion of a one-spike solution to a simplified form of the Gierer–Meinhardt activator-inhibitor model in a two-dimensional domain. The analysis, which is valid for any finite value of the inhibitor diffusivity $D$ with $D\,{\gg}\,\varepsilon^2$, is delicate in that two disparate scales $\varepsilon$ and ${-1/\ln\varepsilon}$ must be treated. This spike motion is found to depend on the regular part of a reduced-wave Green's function and its gradient. Limiting cases of the dynamics are analyzed. For $D$ small with $\varepsilon^2 \,{\ll}\, D \,{\ll}\, 1$, the spike motion is metastable. For $D\,{\gg}\, 1$, the motion now depends on the gradient of a modified Green's function for the Laplacian. The effect of the shape of the domain and of the value of $D$ on the possible equilibrium positions of a one-spike solution is also analyzed. For $D\,{\ll}\,1$, stable spike-layer locations correspond asymptotically to the centres of the largest radii disks that can be inserted into the domain. Thus, for a dumbbell-shaped domain when $D\,{\ll}\,1$, there are two stable equilibrium positions near the centres of the lobes of the dumbbell. In contrast, for the range $D\,{\gg}\,1$, a complex function method is used to derive an explicit formula for the gradient of the modified Green's function. For a specific dumbbell-shaped domain, this formula is used to show that there is only one equilibrium spike-layer location when $D\,{\gg}\,1$, and it is located in the neck of the dumbbell. Numerical results for other non-convex domains computed from a boundary integral method lead to a similar conclusion regarding the uniqueness of the equilibrium spike location when $D\,{\gg}\,1$. This leads to the conjecture that, when $D\,{\gg}\, 1$, there is only one equilibrium spike-layer location for any convex or non-convex simply connected domain. Finally, the asymptotic results for the spike dynamics are compared with corresponding full numerical results computed using a moving finite element method.

Suppose f = (f1, …, fm) is a solution of a non-hyperbolic quasi-linear system of the formwith fk = φk on ∂Ω, where each fi is a bounded function in C2(Ω) ∩ C0(Ω̄). For a system of equations that can have a slightly more general form than above, when Ω is an unbounded open subset of a slab SM andas |X| → ∞ in a specified manner and when certain other conditions are satisfied, a Phragmèn–Lindelöf theorem that yields the limits at infinity of the functions fk(X), k = 1, …, m, is proven.

In this paper we study the problem of the axial symmetry of solutions of some semilinear elliptic equations in unbounded domains. Assuming that the solutions have Morse index one and that the nonlinearity is strictly convex in the second variable, we are able to prove several symmetry results in Rn and in the exterior of a ball. The case of some bounded domains is also discussed.

We describe products of arbitrary L-, R- and H-classes with the set of idempotents of full transformation semigroups.

We consider two types of equations on a cylindrical domain Ω × (0, ∞), where Ω is a bounded domain in RN, N ≥ 2. The first type is a semilinear damped wave equation, in which the unbounded direction of Ω × (0, ∞) is reserved for time t. The second type is an elliptic equation with a singled-out unbounded variable t. In both cases, we consider solutions that are defined and bounded on Ω × (0, ∞) and satisfy a Dirichlet boundary condition on ∂Ω × (0, ∞). We show that, for some nonlinearities, the equations have bounded solutions that do not stabilize to any single function φ: Ω → R, as t → ∞; rather, they approach a continuum of such functions. This happens despite the presence of damping in the equation that forces the t derivative of bounded solutions to converge to 0 as t → ∞. Our results contrast with known stabilization properties of solutions of such equations in the case N = 1.

We propose a notion of weak solution for certain nonlinear contractive equations: ‘dissipative solution’. We show that this notion coincides with the viscosity solutions for Hamilton–Jacobi equations, as introduced by Crandall and Lions, and with the entropy solutions for conservation laws introduced by Kružkov. The proposed notion is based on the properties of accretive operators.

We consider the coupled Schrödinger–Korteweg–de Vries systemwhich arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Ground states of the system are, by definition, minimizers of the energy functional subject to constraints on conserved functionals associated with symmetries of the system. In particular, ground states have a simple time dependence because they propagate via those symmetries. For a range of values of the parameters α, β, γ, δi, ci, we prove the existence and stability of a two-parameter family of ground states associated with a two-parameter family of symmetries.

The purpose of this paper is to study the stability of some unilateral free-discontinuity problems in two-dimensional domains, with the density of the volume part having p-growth, with 1 < p < ∞, under perturbations of the discontinuity sets in the Hausdorff metric.

For a scalar Lotka–Volterra-type delay equation ẋ(t) = b(t)x(t)[1 − L(xt)], where L: C([−r, 0];R) → R is a bounded linear operator and b a positive continuous function, sufficient conditions are established for the boundedness of positive solutions and for the global stability of the positive equilibrium, when it exists. Special attention is given to the global behaviour of solutions for the case of L a positive linear operator. The approach used for this situation is applied to address the global asymptotic stability of delayed logistic models in the more general form ẋ(t) = b(t)x(t)[a(t) − L(t, xt)], with L(t, ·) being linear and positive.

This article represents another step in our programme of obtaining a Galois theory and a coGalois theory when we have a category C and a given enveloping (for Galois) or covering (for coGalois) class. More precisely, in this paper, we study what should be understood by a conormal morphism between two objects of a given category and we characterize conormal morphisms between finite abelian groups when the covering class under consideration is that of torsion-free abelian groups.

We prove localization estimates for general 2mth-order quasilinear parabolic equations with boundary data blowing up in finite time, as t → T−. The analysis is based on energy estimates obtained from a system of functional inequalities expressing a version of Saint-Venant's principle from the theory of elasticity. We consider a special class of parabolic operators including those having fixed orders of algebraic homogenuity p > 0. This class includes the second-order heat equation and linear 2mth-order parabolic equations (p = 1), as well as many other higher-order quasilinear ones with p ≠ 1. Such homogeneous equations can be invariant under a group of scaling transformations, but the corresponding least-localized regional blow-up regimes are not group invariant and exhibit typical exponential singularities ~ e(T−t)−γ → ∞ as t → T−, with the optimal constant γ = 1/[m(p + 1) − 1] > 0. For some particular equations, we study the asymptotic blow-up behaviour described by perturbed first-order Hamilton–Jacobi equations, which shows that general estimates of exponential type are sharp.