Subcritical fronts are shown to exist in a quintic version of the well-known complex Ginzburg–Landau equation, which has a subcritical pitchfork as well as a supercritical saddle-node bifurcation. The fronts connect a finite amplitude plane wave state to a stable zero solution. The unstable manifold at finite amplitude and stable manifold of vanishing amplitude solutions are shown to intersect transversely on an invariant zero-wavenumber manifold with parameters set to be real. By the persistence of transverse intersection, frontal connections exist for a continuum of nearly real fronts parametrised by appropriate variables that exhibit some interesting changes in dimension.

Consider the reaction-diffusion equation in ℝN × ℝ+: ut − h2 Δu + Φ(u) = 0, where Φ is the derivative of a bistable even potential, and h is a small parameter. If the initial data have a smooth noncritical zero set, we prove that an interface appears in time O(log (h−1)), and that the solution stays close to it for at least time O(1/√h).

There is a large number of papers in which attractors of parabolic reaction-diffusion equations in bounded domains are investigated. In this paper, these equations are considered in the whole unbounded space, and a theory of attractors of such equations is built. While investigating these equations, specific difficulties arise connected with the noncompactness of operators, with the continuity of their spectra, etc. Therefore some new conditions on nonlinear terms arise. In this paper weighted spaces are widely applied. An important feature of this problem is worth mentioning: namely, properties of semigroups corresponding to equations with solutions in spaces of growing and of decreasing functions essentially differ.

Using the theory of generalised characteristics, we study the structure of BV solutions of genuinely nonlinear systems of two conservation laws whose shock and rarefaction wave curves of the first family are straight lines. We also establish a priori estimates on the variation of the solution similar to those obtained earlier by Glimm and Lax.

We consider the scattering of time harmonic electromagnetic waves by an inhomogeneous medium of compact support, i.e. the permittivity ε = ε(x) and the conductivity σ = σ(x) are functions of x ∊ ℝ3. Existence, uniqueness and regularity results are established for the direct scattering problem. Then, based on existence and uniqueness results for the exterior and interior impedance boundary value problem, a method is presented for solving the inverse scattering problem.

A function space approach is employed to obtain bifurcation functions for which the zeros correspond to the occurrence of periodic or aperiodic solutions near heteroclinic or homoclinic cycles. The bifurcation function for the existence of homoclinic solutions is the limiting case where the period is infinite. Examples include generalisations of Silnikov's main theorems and a retreatment of a singularly perturbed delay differential equation.

In this paper we prove existence of multiple positive solutions for a Neumann problem in ℝN/(0, R), R large, with a superquadratic and odd nonlinearity. The proof is based on the fact that in such a situation the minimum of the corresponding energy functional (which is achieved) is not an even function and that there is quite a large gap (for large R) between such a minimum and the minimum of the same functional on even functions. In the set of functions whose energy lies in such a gap, we can apply index theory to prove the desired multiplicity result.

The system of differential equations ∇f = M∇g, where M is a given square matrix, arises in many contexts. A complete solution to this problem in the case when M is a constant matrix is presented here. Applications to continuum mechanics and biHamiltonian systems are indicated.

An element α of Pn, the semigroup of all partial transformations of {1,2,…, n}, is said to have projection characteristic (r, s), or to belong to the set [r, s], if dom α= r, im α = s. Let E be the set of all idempotents in Pn\[n, n] and E1, the set of those idempotents with projection characteristic (n, n − 1) or (n − 1, n − 1). For α in Pn\[n, n], we define a number g(α), called the gravity of α and closely related to the number denned in Howie [5] for full transformations, and we obtain the result that

Let d(α) be the defect of α, and for any real number x let [x] be the least integer m such that m ≧ x. Then by analogy with the results of Saito [9] we have that

α ϵ Ek(α) and α ∉ Ek(α)

where k(α) = [g(α)/d(α)] or [g(α)/d(α)+ 1. Following Howie, Lusk and McFadden [6] we then explore connections between the defect and the gravity of α. Letting K(n, r) be the subsemigroup of Pn consisting of all elements of rank r or less, we prove a result, corresponding to that of Howie and McFadden [7] for total transformations, that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n + 1, r + 1), the Stirling number of the second kind.

Given a decreasing sequence of domains Ωn converging in measure to some domain Ω0, a sequence of subspaces V of a Hilbert space V is constructed in such a way that the convergence of the solutions of u −Δu = f on Ωn with Neumann Boundary Condition is given in terms of the convergence of the orthogonal projections Pn on Vn. Under dissipative assumptions, we can obtain continuation results for equations like u −Δu = f(x,u ∇u).