We consider the coupled Schrödinger–Korteweg–de Vries system which 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.

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.

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.

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.

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.

Suppose f = (f1, …, fm) is a solution of a non-hyperbolic quasi-linear system of the form with 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 and as |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 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.