The general theory of epidemic disease I have already considered in a communication to this Society. In that communication I showed that the course of epidemics of all forms of infectious disease obeyed certain very definite laws. In the same paper it was also shown that the distribution of epidemic disease in a uniformly populated area obeyed a law essentially similar. Certain reasons were given why the normal curve of error might be expected to give an approximate solution in both the cases considered, but why the distribution actually found (type iv.) should be the common form was not at all clear. I think, however, I have now arrived at the solution.

Positive radial solutions of elliptic equation involving supercritical growth are analysed as their supremum norm tends to infinity. It is shown that they converge, uniformly away from the origin, as well as in H1, to the unique singular solution.

We study the equations of viscoelasticity in a multidimensional setting for the ‘no-traction’ boundary data. For the sake of modelling phase transitions we do not assume elliptieity of the stored energy function W. We construct dynamics in W1,2(Ωℝn) globally in time. Next, we study the question of stability for a class of equilibria. Moreover, we show a certain kind of decay in time of solutions for arbitrary initial conditions.

This is the first of two papers in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in ℝ3. When the opening angle at the edge is variable, there appears the phenomenon of “crossing” of the exponents of singularities. For this case, we introduce the appropriate combinations of the simple tensor product singularities that allow us to give estimates in ordinary and weighted Sobolev spaces for the regular part of the solution and for the coefficients of the singularities. These combinations appear in a natural way as sections of an analytic bundle above the edge. Their behaviour is described with the help of divided differences of powers of the distance to the edge. The class of operators considered includes second-order elliptic operators with analytic complex-valued coefficients with mixed Dirichlet, Neumann or oblique derivative conditions. With our description of the singularities we are able to remove some restrictive hypotheses that were previously made in other works. In this first part, we prove the basic facts in a simplified framework. Nevertheless the tools we use are essentially the same in the general situation.

The Dirichlet, Neumann and mixed boundary-value problems for the two-dimensional Helmholtz equation in the interior or exterior of a quadrant are considered in a Sobolev space setting. It is shown that the potential operators arising in the interior problems can be used to derive systems of boundary integral equations to the exterior problems, which can be solved explicitly.

The object of the present research was to follow, by the degeneration method, the course of the fibres proceeding from definite and limited areas of the motor cortex, and to determine to what extent there is a grouping or localisation of the fibres of the pyramidal tract at different levels in the brain and spinal cord.

Some evidence indicates that spherically symmetric solutions of the compressible Euler equations blow up near the origin at some time under certain circumstances (cf. [4,19]). In this paper, we observe a criterion for L∞ Cauchy data of arbitrarily large amplitude to ensure the existence of L∞ spherically symmetric solutions in the large, which model outgoing blast waves and large-time asymptotic solutions. The equilibrium states of the solutions and their asymptotic decay to such states are analysed. Some remarks on global spherically symmetric solutions are discussed.

We analyse the dynamics of a prototype model for competing species with diffusion coefficients (d1d2) in a heterogeneous environment Ω. When diffusion is switched off, at each point x ∊ Ω we have a pair of ODE's: the kinetic. If for some x ∊ Ω kinetic has a unique stable coexistence state, we show that there exist such that for every the RD-model is persistent, in the sense that it has a compact global attractor within the interior of the positive cone and has a stable coexistence state. The same result is true if there exist xu, xv ∊ Ω such that the semitrivial coexistence states (u, 0) and (0, v) of the kinetic are globally asymptotically stable at x = xu and x = xv, respectively. More generally, our main result shows that, for most kinetic patterns, stable coexistence of xspopulations can be found for some range of the diffusion coefficients.

Singular perturbation techniques, monotone schemes, fixed point index, global analysis of persistence curves, global continuation and singularity theory are some of the technical tools employed to get the previous results, among others. These techniques give us necessary and/or sufficient conditions for the existence and uniqueness of coexistence states, conditions which can be explicitly evaluated by estimating some principal eigenvalues of certain elliptic operators whose coefficients are solutions of semilinear boundary value problems.

We also discuss counterexamples to the necessity of the sufficient conditions through the analysis of the local bifurcations from the semitrivial coexistence states at the principal eigenvalues. An easy consequence of our analysis is the existence of models having exactly two coexistence states, one of them stable and the other one unstable. We find that there are also cases for which the model has three or more coexistence states.

The following theorem is proved: Let S(t), t≧0 be a dynamical system in an infinite dimensional Banach space X such that S(t) = S1(t)+S2(t) for t≧0, where (1) uniformly in bounded sets of x in X, and (2) S2(t) is compact for t sufficiently large. Then, if the orbit {S(t)x: t ≧0} of x ∈ X is bounded in X, it is precompact in X. Applications are made to an age dependent population model, a non-linear functional differential equation on an infinite interval, and a non-linear Volterra integrodifferential equation.

For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation,

$$-\Delta u+V_{\mu, \nu} u=(I_\alpha\ast \vert u \vert ^{({N+\alpha})/{N}}){ \vert u \vert }^{{\alpha}/{N}-1}u,\quad {\rm in} \ {\open R}^N, $$

$N\ges 3$

$I_\alpha : {\open R}^N \to 0$

$N \ges 4$

$\ker (-\Delta + V_{\mu ,\nu }) = \{ 0\} $

An existence theorem is obtained for a fourth-order semilinear elliptic problem in RN involving the critical Sobolev exponent (N + 4)/(N − 4), N>4. A preliminary result is that the best constant in the Sobolev embedding L2N/(N–4) (RN) is attained by all translations and dilations of (1 + ∣x∣2)(4-N)/2. The best constant is found to be

The authors continue their study of Titchmarch-Weyl matrix M(λ) functions for linear Hamiltonian systems. A representation for the M(λ) function is obtained in the case where the system is limit circle, or maximum deficiency index, type. The representation reduces, in a special case, to a parameterization for scalar m-coefficients due to C. T. Fulton. A proof that matrix M(λ) functions are meromorphic in the limit circle case is given.

Sufficient conditions are derived for every solution of a nonlinear Schrödinger equation (or inequality) to be oscillatory in an exterior domain of En. Such results apply in particular to the n-dimensional Emden-Fowler equation. The method involves oscillatory behaviour of solutions of a nonlinear ordinary differential inequality satisfied by the spherical mean of a positive solution of the Schrödinger equation.

The “energy solutions” to the equation

have finite speed of propagation if l < q < 2 or l < r < 2. If 1 <r <2 (Vq <1) support u(· t) is uniformly bounded for t >0 (localisation property) and if q<2 ≦ r, sharp upper bounds of the interface (or free boundary) are obtained. We use a weighted energy method, the weights being powers of the distance to a variable half-space. We also study decay rates as t→∞ and extinction in finite time for bounded and unbounded domains (with null Dirichlet boundary conditions). Our equation includes the porous media equation with absorption. Analogous results hold if (−Δ)m is replaced by an appropriate quasilinear elliptic operator.

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.

In a previous paper, B.-Y. Chen defined a Riemannian invariant δ by subtracting from the scalar curvature at every point of a Riemannian manifold the smallest sectional curvature at that point, and proved, for a submanifold of a real space form, a sharp inequality between δ and the mean curvature function. In this paper, we extend this inequality to totally real submanifolds of a complex space form. As a consequence, we obtain a metric obstruction for a Riemannian manifold Mn to admit a minimal totally real (i.e. Lagrangian) immersion into a complex space form of complex dimension n. Next we investigate three-dimensional submanifolds of the complex projective space ℂP3 which realise the equality in the inequality mentioned above. In particular, we construct and characterise a totally real minimal immersion of S3 in ℂP3.

We consider the problem

where H stands for the maximal monotone graph associated with the Heaviside step function. It is shown that the problem possesses at least one (strong) solution belonging to an appropriate function space. Moreover, we prove:

(i) There is a smooth initial function u0, u0≧1, where the equality holds at exactly one point such that there are at least two different solutions corresponding to the initial data u0.

(ii) The comparison principle: The relation for any x ≠ 0, implies u1(t)>u2(t), t≧0 for any u1, u2 solving the problem with the initial data , respectively.

(iii) For a “reasonable” set of initial data the solution is uniquely determined. Moreover, the free boundary {(x, t)| u(x, t) = 1} is regular and on its complement the equation holds in a classical sense.

We study the stability of positive radially symmetric solitary waves for a three dimensional generalisation of the Korteweg de Vries equation, which describes nonlinear ion-acoustic waves in a magnetised plasma, and for a generalisation in dimension two of the Benjamin–Bona–Mahony equation.

The characterisation of the bulk energy density of the relaxation in W1, P(Ω; ℝd) of a functional

is obtained for p > q – q/N, where u ∈ W1, P(Ω; ℝd), and f is a continuous function on the set of d × N matrices verifying

for some constant C > 0 and 1 ≦ q < + ∞. Typical examples may be found in cavitation and related theories. Standard techniques cannot be used due to the gap between the exponent q of the growth condition and the exponent p of the integrability of the macroscopic strain ∇u. A recently introduced global method for relaxation and fine Sobolev trace and extension theorems are applied.

The Hardy-Rellich inequality given here generalizes a Hardy inequality of Davies, from the case of the Dirichlet Laplacian of a region Ω ⊆ ℝN to that of the higher-order polyharmonic operators with Dirichlet boundary conditions. The inequality yields some immediate spectral information for the polyharmonic operators and also bounds on the trace of the associated semigroups and resolvents.