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.

Associated to every plane curve there is the locus of centres of circles bitangent to that curve, the so-called symmetry set of the curve. We can view this set as the spine of our curve, which can be recovered by taking the envelope of circles of varying radii along this spine. Varying the symmetry set in some isotopy while keeping the radius function fixed may be viewed as crudely modelling motion of the original curve viewed as a biological object. Fixing the symmetry set and varying the radius function can be considered to model growth crudely. In this paper we shall describe the generic changes in the curves which take place in the process of growth and motion, and outline the corresponding results for spheres centred on a space curve. We also use the idea of a stratified Morse function to describe the generic changes which occur in one parameter families of (full) bifurcation sets in the plane. Applying this to the bifurcation set of distance squared functions we find all the transitions of a symmetry set (and evolute) which occur in a generic isotopy of a plane curve.

This paper describes the set GmU of effective conductivity tensors of mixtures generated by two isotropic materials taken in prescribed proportions m1 and m2 We describe microstructures which realise any point of GmU for n-dimensional space.

We consider a cylinder with arbitrary cross section moving in a viscous incompressible fluid parallel to a plane wall. Formal asymptotic expansions of the solution for small Reynolds numbers are constructed by using boundary integral equations of the first kind. In contrast to the problem without a wall, we show that there exists a unique solution to the zeroth order problem. However, the problem considered here is still singular in the sense that we find the Stokes paradox in the next higher order problem. A justification of the formal asymptotic expansion for the first two terms is established rigorously.

A Ljapunov equation XL − BX = C appears in stabilisation studies of linear systems. Here, the operators L, B, and C are given linear operators working in infinite-dimensional Hilbert spaces, which are derived from a specific control system. We have so far considered the case where L is a general elliptic operator of order 2 in a bounded domain of an Euclidean space. When L is instead a self-adjoint elliptic operator working in an interval of ℝ1, we derive here a stronger geometrical character of the solution X to the Ljapunov equation. The result is applied to stabilisation of one-dimensional diffusion equations.

Some second order ordinary differential equations of the form ξ2ϕ″ + ξ2(N − 1)″′/r + ½(ϕ − ϕ3) + ½k = 0 are studied. Properties such as existence and monotonicity of solutions are considered for N ≧ 1, ξ > 0 and two sets of boundary conditions. For N = 1, some explicit results are obtained for small ξ. These ODE's arise from a phase field approach to free boundary problems involving a phase transition.

The exact number of positive solutions of Δu + f(u) = 0 on finite balls in ℝN is determined. The assumptions about f(u) are similar to those imposed by Serrin and the second author in a previous study of uniqueness of the positive solution when the spatial domain is all of ℝN (see [7, 8]). For finite balls of sufficiently large radius it is shown here that there are exactly two positive and, hence, radial solutions. To this end, we first prove the linear nondegeneracy of the positive solution of ℝN. This is obtained by applying the technique of monotone separation of graphs [7] to the linearised equations. Somewhat sharper estimates are required here (see Part I, Section 2).

Upper bounds are obtained for the Hausdorff dimension of compact invariant sets of ordinary differential equations which are periodic in the independent variable. From these are derived sufficient conditions for dissipative analytic n-dimensional ω-periodic differential equations to have only a finite number of ω-periodic solutions. For autonomous equations the same conditions ensure that each bounded semi-orbit converges to a critical point. These results yield some information about the Lorenz equation and the forced Duffing equation.

Various physical problems in diffraction theory lead us to study modifications of the Sommerfeld half-plane problem governed by two proper elliptic partial differential equations in complementary ℝ3 half-spaces Ω± and we allow different boundary or transmission conditions on two half-planes, which together form the common boundary of Ω±.

By using asymptotic estimates for the eigenvalues and eigenfunctions or irregular boundary value problems, we state necessary conditions for the pointwise convergence and for the divergence of the corresponding eigenfunction expansions.

In this paper it is proved that, for scalar-type operators a and b on an infinite dimensional separable complex Hilbert space H, the generalised derivation Δa,b, defined for bounded linear operators x onℋ by the equation Δa,bx = ax − xb, is a (scalar-type) prespectral operator of the class (the trace class operators on ℋ) if and only if at least one of the spectra σ(a)or σ(b)is finite. It is shown also that the same condition is necessary and sufficient for Δa,b restricted to any one of the von Neumann-Schatten classes(p≠2) to be a spectral operator (of scalar type). Our results may be compared with those of J. Anderson and C. Foiaş, who established in [1] that, for scalar-type a, b, Δa,b is a (scalar-type) spectral operator if and only if both spectra, σ(a) and σ(b), are finite. However, we use different and more direct methods to show the existence or nonexistence of the spectral resolution of identity for Δa,b.

In this paper we consider Hopf bifurcation in the presence of O(2) symmetry. The system of reaction diffusion equations ut, = D(µ)uxx + f(µ, u) provided with periodic boundary conditions may serve as a model problem. We prove the bifurcation of a torus of standing waves and two circles of travelling waves and we compute the stability (with asymptotic phase) of these periodic solutions, giving explicit formulae. Finally we demonstrate how a small perturbation which breaks part of the symmetry leads to secondary bifurcation.

This paper considers existence of periodic solutions for vector Liénard differential equations

In our main result we write

where Q(t, x) is a symmetric matrix and h(t, x) is sublinear. The key assumption relates the asymptotic behaviour as x →+ ∞ of the eigenvalues of Q(t, x) to the spectrum of the linear operator −d2/dt2 Several choices for Q(t, x) are considered which lead to known theorems and extend others. In the case of the Duffing equation

the assumptions are weakened.

Our approach is based on Leray-Schauder's degree theory and a priori estimates.

We study the initial value problem for the nonlinear Schrödinger equation

Under suitable regularity assumptions on f and ø and growth and sign conditions on f, it is shown that the maximum norms of solutions to (*) decay as t→² ∞ at the same rate as that of solutions to the free Schrödinger equation.

We consider a class of non-linear evolution equations subject to a periodic forcing term. Using bifurcation theory we obtain results on the existence and number of periodic solutions. The theory applies to semi-linear diffusion equations defined on bounded or unbounded domains.

Dominance properties of solutions to Lny + p(x)y = 0, where Ln is a disconjugate operator, are compared to dominance properties of solutions to its adjoint.

This paper discusses a few problems on the size of the set of invariant means of an amenable semigroup posed by Maria M. Klawe, Alan L. T. Paterson and M. Rajagopalan and P. V. Ramakrishnan ([4], [5], [8] and [9]).

The semi-linear equation Δu − λu + h(x)uσ = 0 is studied on all of d-dimensional Euclidean space. In the bifurcation problem a non-trivial solution is sought for small λ which tends to zero with λ. The asymptotic dependence of the solution on λ is examined. For fixed λ = 1 the existence of non-degenerate non-trivial solutions is proved for generic measurable h(x) sufficiently near to a constant, provided d = 1 or 3. The two problems are seen to be interdependent. The bifurcation problem at λ = 0 is particularly interesting as the linearised equation is of non-Fredholm type.

Professor R. D. Brown has brought to our attention an error in our Theorem 4.2. This would, in turn, invalidate Theorem 3.6 which depends on Theorem 4.2. The basic flaw is that the alignment hypothesis of Proposition 4.1 is not satisfied by the product space projections, Qn. Interestingly, such product space constructions are a common computational device in applications of the T*T method, but apparently seldom satisfy the alignment hypothesis of Proposition 4.1. We wish here to correct these theorems, and comment on the resulting implications for Example 6.1. In independent and as yet unpublished work, Professor Brown has obtained further new results on convergence of the method of intermediate problems.

Solutions to semilinear parabolic equations of the form ut = Δu + f(u), x in Ω, which blow up at some finite time t* are investigated for “slowly growing” functions f. For nonlinearities such as f(s) = (2 +s)(ln(2 +s))1+b with 0 < b < l,u becomes infinite throughout Ω as t→t* −. It is alsofound that for marginally more quickly growing functions, e.g. f(s) = (2 + s)(ln(2 +s))2, u is unbounded on some subset of Ω which has positive measure, and is unbounded throughout Ω if Ω is a small enough region.