We consider the hyperbolic equation uxy + c(x, y, u) =f(x, y) and the wave equation

We show that, under suitable conditions, there are bounded domains in which every solution to certain problems has a zero. Characteristic initial value problems and initial boundary value problems are considered.

We consider initial boundary value problems for the equations of linear thermoelasticity in both bounded and unbounded domains and for both nonhomogeneous and anisotropic media. For bounded domains, it is shown that the unique solution of the problem is time-asymptotically equal to the solution of a particular initial boundary value problem which is obtained from a natural decomposition of the original initial data and which represents a (in general non-vanishing) time harmonic part. For the unbounded case similar results are obtained, but now in the sense of weak convergence which lead to the result of local energy decay: the solution tends to zero in every compactum.

The purpose of this paper is to give a sharp bound for the order of instability of an unstable hyperplane of a rank 2 stable reflexive sheaf E on ℙn, in terras of the Chern classes of E; and a sharp boundfor the order of instability of an unstable hyperplane of a rank 2 nonstable reflexive sheaf E on ℙn, in terms of the Chern classes of E and the order of nonstability of E.

The paper considers the small time evolution of the interface which appears in the boundary value problem

A diffusion dominated small time outer expansion is not uniformly valid when x = O(t½ log t) and has to be supplemented by an inner expansion valid in a region where diffusion and absorption are of equal importance. For p < 1 a zero order inner solution is constructed, which is consistent with a moving interface where u is zero, and the small time development of this is given.

These results are first derived in a heuristic way using the method of matched expansions. However, an important aim of the paper is to give rigorous proofs, being one of the very few cases in nonlinear diffusion where this has been done.

Comparison principles for systems of reaction-diffusion equations coupled via both the reaction and diffusion terms are considered. Applications to the FitzHugh–Nagumo equations and models of coupled nerve fibres are included.

Let Ω be a bounded domain in ℝ2. The study, begun in Keady [13], of the boundary-value problem, for (λ/k, ψ),

is continued. Here Δ denotes the Laplacian, H is the Heaviside step function and one of λ or k is a given positive constant. The solutions considered always have ψ > 0 in Ω and λ/k > 0, and have cores

In the special case Ω = B(0, R), a disc, the explicit exact solutions of the branch τe have connected cores A and the diameter of A tends to zero when the area of A tends to zero. This result is established here for other convex domains Ω and solutions with connected cores A.

An adaptation of the maximum principles and of the domain folding arguments of Gidas, Ni and Nirenberg [9] is an important step in establishing the above result.

The Sturm–Picone comparison theorem is extended to nonself-adjoint differential equations considered on non-compact intervals.

In this paper, a counterexample is given to the H1-boundedness of solutions to a sequence of systems of linear partial differential equations uniformly satisfying a strict Legendre–Hadamard condition and whose coefficients depend on one direction only. This counterexample is relevant for the theory of homogenisation of laminated elastic materials.

The homogenisation of a linearly viscoelastic composite material is performed without any periodicity assumptions and with no restrictions on the initial data. A term of fading memory is evidenced in the expression of the homogenised stress tensor.

Suppose the linear equation x' = A(t)x has an exponential dichotomy and suppose B(t) is close to A(t) in the following sense: on any interval of length 2T, B(t) is close to some translate A(t + τ) of A(t) (actually the conditions in the paper are slightly weaker than this). Then if T is sufficiently large, the equation y' = B(t)y also has an exponential dichotomy. This generalises the usual roughness theorem for exponential dichotomies.

In this paper we study the solutions of the boundary value problem

where t ∊ℝ, x ∊ ℝN, f is a continuous function of (t,x)and locally Lipschitz in x and ω is a fixed positive number and λ ∊ ℝ. By using degree theory we prove results on the existence of solutions of (*) and the dependence of such solutions on λ. We shall prove that (*) does not have an isolated solution, and study the topological properties of the components of solutions of (*).

We generalise the Paley–Wiener closedness theorem and apply it to a class of time periodic Hamiltonians to show that all solutions to the corresponding Schrodinger equation decay.

We provide necessary and sufficient conditions for two-parameter convergence in the strong law of large numbers for U-statistics. We also obtain weak-type (1,1) inequalities for one and two-sample U-statistics of order 2 which are, in a sense, best possible.

We show that the weak convergence of a sequence of functions in a Sobolev space plus the convergence of appropriately quasiconvex “energies” imply, in fact, strong convergence. This assertion makes rigorous, for example, the heuristic principle that “quasiconvexity damps out oscillations in the gradients” of minimising sequences in the calculus of variations.

In this paper we examine the influence of capillarity on existence and uniqueness of travelling wave solutions in an isothermal system of van der Waals fluids. Existence and non-uniqueness theorems are proved using phase-space analysis and topological methods.

The nonlinear evolution problem [Bu(t)]′ = A(t, Bu)u + f(t, Bu) with B a constant linear operator and A = A(t, Bu) a time-dependent nonlinear operator from one Banach space to another, is studied. Existence and uniqueness results are obtained by making use of the theory of B-evolutions and the fractional powers of A and B. Two examples are presented in which the theory is applied to nonlinear equations with dynamic boundary conditions.

This paper is devoted to the establishment of explicit bounds on the rational function solutions of a general class of equations in several variables. The first general result on Diophantine equations over function fields was discovered in 1930 [1], as an analogy on the work of Thue on number fields: it was shown that the degrees of polynomial solutions X, Y in k[z] of f(X, Y) = c are bound for each c: here f denotes an irreducible binary form over k[z] of degree at least three, and k is an algebraically closed field of characteristic zero. The Manin-Grauert theorem [7] extended this conclusion to the rational function solutions X, Y in k(z) of any equation in two variables over k(z), provided that the curve corresponding to the equation has genus two or more: this is the analogue for function fields of the Mordell conjecture for number fields, proved by Faltings in 1983. Parsin [6] made the Manin-Grauert theorem effective by furnishing explicit bounds on the degrees of X and Y: this approach followed Grauert and Shafarevitch in relying heavily on algebraic geometry. In 1976 a different attack was made by Schmidt [8] using the theory of algebraic differential equations, first developed by Kolchin and Osgood. This method produced very good bounds for equations such as the Thue equation discussed above: for example, Schmidt provides the bound

for any rational function solution X, Y in k(z) of f(X, Y) = 1, where f denotes a binary form over k(z) of degree in X and Y at least five, and which factorises into distinct linear factors. Here deg zf is the degree of f in z. In 1983 [2] the power of Diophantine approximation provided another approach to attack Dlophantine equations.over function fields. By first establishing an inequality on solutions of the unit equation in two variables, it was shown in [3, p. 122] that the bound (1) could be improved to 36 degz f, and the condition on the degree in X and Y could be relaxed to include quartic forms. In fact the principal purpose of this new approach using Diophantine approximation had been to provide algorithms for the effective construction of all the polynomial solutions of various equations: it was a surprise to find that not only were these algorithms extremely efficient, but also that the technique extended to include rational function solutions. A full account of this approach and its consequences is given in a recent tome [3].

Let 0 < γ ≤ γ” denote the ordinates of consecutive nontrivial zeros of ζ(s) and set