In a paper read to the Society on 15th March last (ante, p. 501), I gave a series of proofs of the theorem that four colours suffice for a map. All of these were long, and I felt that, while more than sufficient to prove the truth of the theorem, they gave little insight into its real nature and bearings. A somewhat similar remark may, I think, be made about Mr Kempe's proof.

But a remark incidentally made in the abstract of my former paper has led me to a totally different mode of attacking the question, which puts its nature in a clearer light. I have therefore withdrawn my former paper, as in great part superseded by the present one.

The remark referred to is to the effect that, if an even number of points be joined, so that three (and only three) lines meet in each, these lines may be coloured with three colours only, so that no two conterminous lines shall have the same colour. (When an odd number of the points forms a group, connected by one line only with the rest, the theorem is not true.)

Elastic effects can have a pronounced effect on the phase-separation process in solids. The classical Ginzburg—Landau energy can be modified to account for such elastic interactions. The evolution of the system is then governed by diffusion equations for the concentrations of the alloy components and by a quasi-static equilibrium for the mechanical part. The resulting system of equations is elliptic-parabolic and can be understood as a generalization of the Cahn—Hilliard equation. In this paper we give a derivation of the system and prove an existence and uniqueness result for it.

We investigate the Cauchy problem for a hydrodynamic model for semiconductors. An existence theorem of global weak solutions with large initial data is obtained by using the fractional step Lax—Friedrichs scheme and Godounov scheme.

In this paper we apply the Painlevé tests to the damped, driven nonlinear Schrödinger equation

where a(x, t) and b(x, t) are analytic functions of x and t, todetermine under what conditions the equation might be completely integrable. It is shown that (0.1) can pass the Painlevé tests only if

where α0(t),α1(t) and β(t) are arbitrary, real analytic functions of time. Furthermore, it is shown that in this special case, (0.1) may be transformed into the original nonlinear Schrödinger equation, which is known to be completely integrable.

We consider the above equation on the interval 0 ≦ x ≦ 1 subject to Neumann boundary conditions with f(u) = F′(u) where F is a double well energy density function with equal minima. Our previous work [3] proved the existence and persistence of very slowly evolving patterns (metastable states) in solutions with two-phase initial data. Here we characterise these metastable states in terms of the global unstable manifolds of equilibria, as conjectured by Fusco and Hale [6].

We consider the existence of normalized solutions in H1(ℝN) × H1(ℝN) for systems of nonlinear Schr¨odinger equations, which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz, one is led to coupled systems of elliptic equations of the form

and we are looking for solutions satisfying

where a1> 0 and a2> 0 are prescribed. In the system, λ1 and λ2 are unknown and will appear as Lagrange multipliers. We treat the case of homogeneous nonlinearities, i.e. , with positive constants β, μi, pi, ri. The exponents are Sobolev subcritical but may be L2-supercritical. Our main result deals with the case in which in dimensions 2 ≤ N ≤ 4. We also consider the cases in which all of these numbers are less than 2 + 4/N or all are bigger than 2 + 4/N.

The classically well-known relation between the number of linearly independent solutions of the electro- and magnetostatic boundary value problems (harmonic Dirichlet and Neumann vector fields) and topological characteristics (genus and number of boundaries) of the underlying domain in 3-dimensional euclidean space is investigated in the framework of Hilbert space theory. It can be shown that this connection is still valid for a large class of domains with not necessarily smooth boundaries (segment property). As an application the inhomogeneous boundary value problems of electro- and magnetostatics are discussed.

A differential inequality technique is used to determine a lower bound on the blow-up time for solutions to the heat equation subject to a nonlinear boundary condition when blow-up of the solution does occur. In addition, a sufficient condition which implies that blow-up does occur is determined.

Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows, where they often emerge on timescales much shorter than the viscous timescale, and then dominate the dynamics for very long time intervals. In this paper we propose a dynamical systems explanation of the metastability of an explicit family of solutions, referred to as bar states, of the two-dimensional incompressible Navier–Stokes equation on the torus. These states are physically relevant because they are associated with certain maximum entropy solutions of the Euler equations, and they have been observed as one type of metastable state in numerical studies of two-dimensional turbulence. For small viscosity (high Reynolds number), these states are quasi-stationary in the sense that they decay on the slow, viscous timescale. Linearization about these states leads to a time-dependent operator. We show that if we approximate this operator by dropping a higher-order, non-local term, it produces a decay rate much faster than the viscous decay rate. We also provide numerical evidence that the same result holds for the full linear operator, and that our theoretical results give the optimal decay rate in this setting.

This paper is devoted to the problem of finite-dimensional reduction for parabolic partial differential equations. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mañé projection theorems. The recent counter-examples showing that the underlying dynamics may in a sense be infinite dimensional if the spectral gap condition is violated, as well as a discussion of the most important open problems, are also included.

This paper deals with the coupled chemotaxis-haptotaxis model of cancer invasion given by

where χ, ξ and μ are positive parameters and Ω ⊂ ℝn (n ≥ 1) is a bounded domain with smooth boundary. Under zero-flux boundary conditions, it is shown that, for any μ > χ and any sufficiently smooth initial data (u0, w0) satisfying u0 ≥ 0 and w0 > 0, the associated initial–boundary-value problem possesses a unique global smooth solution that is uniformly bounded. Moreover, we analyse the stability and attractivity properties of the non-trivial homogeneous equilibrium (u, v, w) ≡ (1,1, 0) and establish a quantitative result relating the domain of attraction of this steady state to the size of μ. In particular, this will imply that whenever u0 > 0 and 0 < w0 < 1 in there exists a positive constant μ* depending only on χ, ξ, Ω, u0 and w0 such that for any μ < μ* the above global solution (u, v, w) approaches the spatially uniform state (1, 1, 0) as time goes to infinity.

Using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the type

The first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u. The second one deals with a class of nonlinearities with explicit x dependence.

In this work, we study a mesh termination scheme in acoustic scattering, known as the perfectly matched layer (PML) method. The main result of the paper is the following. Assume that the scatterer is contained in a bounded and strictly convex artificial domain. We surround this domain by a PML of constant thickness. On the peripheral boundary of this layer, a homogenous Dirichlet condition is imposed. We show in this paper that the resulting boundary-value problem for the scattered field is uniquely solvable for all wavenumbers and the solution within the artificial domain converges exponentially fast toward the full-space scattering solution when the layer thickness is increased. The proof is based on the idea of interpreting the PML medium as a complex stretching of the coordinates in Rn and on the use of complexified layer potential techniques.

We establish some bifurcation results for the boundary-value problem −Δu = g (u) + λ|∇u|p + μf (x, u) in Ω, u > 0 in Ω, u = 0 on δΩ, where Ω is a smooth bounded domain in RN, λ, μ ≥ 0, 0 < p ≤ 2, f is non-decreasing with respect to the second variable and g is unbounded around the origin. The asymptotic behaviour of the solution around the bifurcation point is also established, provided g(u) behaves like u−α around the origin, for some 0 < α < 1. Our approach relies on finding explicit sub- and supersolutions combined with various techniques related to the maximum principle for elliptic equations. The analysis we develop in this paper shows the key role played by the convection term |∇u|p.

We investigate the singular problem

where Ω is a bounded smooth domain, k a bounded, nonnegative measurable function and v Ω 0. For the solution u to this problem, which is shown to exist if k(x) > 0 on some subset of Ω with positive measure, a uniform bound for |∇u| in Ω is derived when k(x) ≧ ψ (dist (x, ∂Ω)) with ψ (s)/sv ∈ Lp(0, a) for some a > 0, p > 1.

The following results are obtained for symmetric differential expressions of arbitrary order r ≧ 1 with matrix coefficients on the half-line, with a non-negative (and possibly identically degenerate) weight matrix W(t), and with a spectral parameter λ: upper and lower bounds for the deficiency indices N(λ) are found; it is proved that N(λ) is independent of λ for Im λ< 0and for Im λ>0; under very general conditions it is proved that the maximum possible values of the deficiency indices in the half-planesIm λ≷0 can only be attained simultaneously; sufficient conditions for first-order expressions to be quasi-regular are derived; and it isshown that a symmetric system of any order reduces to a canonical, first-order system.

Examples are constructed, and the case of the whole line is also touched on.

We show that every orthogonality-preserving linear map between normed spaces is a scalar multiple of an isometry. Using this result, we generalize Uhlhorn's version of Wigner's theorem on symmetry transformations to a wide class of Banach spaces.

Solutions with peaks near the critical points of Q(x) are constructed for the problem

We establish the existence of 2k −1 positive solutions when Q(x) has k non-degenerate critical points in ℝN

In this paper, we study all the stationary solutions of the form u(r)einθ to the complex-valued Ginzburg–Landau equation on the complex plane: here (r, θ) are the polar coordinates, and n is any real number. In particular, we show that there exists a unique solution which approaches to a nonzero constant as r → ∞.

This paper deals with entire solutions of a bistable reaction—diffusion equation for which the speed of the travelling wave connecting two constant stable equilibria is zero. Entire solutions which behave as two travelling fronts approaching, with super-slow speeds, from opposite directions and annihilating in a finite time are constructed by using a quasi-invariant manifold approach. Such solutions are shown to be unique up to space and time translations.