We consider the Cauchy problem for the porous medium equation in one space dimension, with initial data which are locally integrable. We measure the asymptotic behaviour of the initial data near infinity in an integral sense and relate this to the pointwise rate of growth or decay of solution for large time. The emphasis is on a novel comparison method wherein the initial data are rearranged on the ×-axis to form a sequence of Dirac δ-masses. By using the explicit solution in the latter case, we derive upper and lower bounds for the solution to the original problem by comparisons.

Conditions, generalizing the usual Liouville–Green conditions, are obtained under which the equation {r(x)y′}′ + q(x)y = 0, where q and r are positive and have derivatives of a sufficiently high order, has solutions of the form y ∼ (rq)–¼e10. 0 ∼ f (q/r)½ dx as x →∞. A related result is obtained for a system of two first-order equations y′ = A(x)y, where the eigenvalues of A are pure imaginary.

This paper studies how, in a difference equation, uniform asymptotic stability for commensurable delays implies uniform asymptotic stability globally in the delays. A result is also given on renorming the space C= C([ – ∥r∥, 0]; ℂn so that a difference operator has norm less than one if it is uniformly asymptotically stable globally in the delays.

Estimates on the gradient of solutions to the Dirichlet problem for a semilinear elliptic equation are given when the nonlinearity in the equation is quadratic with respect to the gradient of the solution. These estimates extend results of F. Tomi to less smooth boundary data and results of the author to the full quadratic growth.

Decay estimates for the energy are derived for the initial boundary value problem of the wave equation with a degenerate dissipative term:

where Ω is a bounded domain in Rn, a(×) is a nonnegative function such that a1 ∊Lp(Ω) for some p > 0 and f is a function tending to 0 rapidly as t → ∞

Let F(X, Y, Z) be a non-singular quadratic form with rational coefficients. The curve EF(x2, y2, z2) = 0 is of genus 3. A procedure is described for deciding whether there is an effective divisor on E of degree 3 defined over the rationals. There is such a divisor if and only if there is a point on E defined over some algebraic number field of odd degree. An example is constructed for which there is no such divisor although (i) there are points on E defined over all p-adic fields and over the reals and (ii) there are infinitely many rational points on each of the three curves F(X, y2, z2) = 0, F(x2, Y, z2) = 0 and F(x2, y2, Z) = 0.

A sufficient condition on the angles of a bounded open subset Ω of ℝn is given for the best possible regularity of a solution to a class of parabolic problems with non-linear mixed boundary conditions.

We study the possibility of perturbing a matrix A by a diagonal matrix so that an eigenvalue problem with leading matrix A has specifiedeigenvalues when A is replaced by A+D. The particular cases presented are the one-parameter generalized eigenvalue problem (A× = λB μ×, a two-parameter eigenvalue problem (A + λB + μC)× = 0, a linked system ofsuch two-parameter problems and a quadratic eigenvalue problem (A + λB + λ2C)× = 0. The work extends results of Hadeler for the classical problem A× = λ×.

Extensions of the integral version of Hardy's Inequality were given by Kadlec and Kufner (1967) and by Copson (1976). This paper provides several levels of further generalization of their results, obtained mostly by specializing four main inequalities. Most of the inequalities have the form ∥Kf∥ ≤ C ∥f∥, where K is an integral transform and ∥.∥ is a generalized Lp-norm; some have the inequality sign reversed. Best possible constants C are obtained in several cases, under mild extra hypotheses.

In this article, we study the functional Where Ω ⊂ ĝn is a bounded open set and u: Ω ×(0, T)→ ĝm and when F: Rnm →R fails to be quasiconvex. We show that with respect to strong convergence of ∂u/∂t and weak convergence of ∇×u, the above functional behaves as where QF is the lower quasiconvex envelope of F.

The aim of the present paper is to establish some new integral inequalities of Opial type in two independent variables. Our results are the two independent variable generalizations of some of the inequalities recently established by the present author and in special cases yield the two independent variable analogue of the Opial inequality and its generalization given by G. S. Yang.

The solutions of the differential equation Lny + p(×)y = 0, where Ln is a disconjugate operator, are classified according to their behaviour as × →∞. The solution space is decomposed into disjoint sets. We study the dominance properties of the solutions which belong to different sets.

A weak (enthalpy) formulation of the problem of a free boundary moving in the thermal concentrated capacity is given. The problem is to solve the heat equation in a given domain, while on a part of the boundary of this domain the solution (or rather its trace) solves a Stefan problem with forced convection. The existence of a global weak solution is proved by the method of finite differences. Some regularity is obtained from this proof, and the continuity of the temperature is proved. The uniqueness, which is related to the existence of mushy regions, is discussed. A classical enthalpy formulation is conjectured.

Let K be the class of all operators T in a Banach space × which have the property that, for any pair of integers (n, k) with n ≧2 and l≦ k ≦ n – l, there exists a constant Cnk such that

for all fϵdom Tn. If T ϵ K, then the best possible constant for the norm inequality (*) is the smallest non-negative value of the constant Cnk in (*). Any operator T which is the adjoint of a maximal symmetric operator in a Hilbert space belongs to the class K, as was shown by Ljubič [Izv. Akad. Nauk SSSR, Ser. Mat. 24 (1960), 825–864].

This article is concerned with the computation of the best possible constant for the differentiation operator Tf=if′ on the maximal domain in L2(0, ∞). Three algorithms, proposed by Ljubič [ibid.] and Kupcov [Trudy Mat. Inst. Steklov. 138 (1975)], are discussed and related to one another, asymptotic expressions (valid for large n) and numerical values of the best possible constant are presented, and the extremals (i.e. the elements / ∈ dom Tn for which equality holds in (*) with the best possible constant) are given.

We discuss the nonlinear eigenvalue problem

with r(–×) = –r(×) and r'≧0.

For ε = h =0, the solution to Problem P is wellknown, and every solution, except u = 0 and u =1, is unstable with respect to the corresponding parabolic problem.

We give some results on existence and uniqueness for the solution of elliptic boundary value problems of type

when the βi are not necessarily smooth.

Littlewood showed that the forced Van der Pol oscillator with 0<b<⅔ and k large normally has subharmonic solutions of order 2n + l where n ≅ O([⅔−b]k). Numerical experiments suggest that n ≅ (⅔ –b)k/3 as k →∞. A refinement of Littlewood's calculation is given which leads to this result.

Let D be a compact convex planar domain containing the origin, the boundary of which is of class C∞ and has finite non-vanishing curvature throughout. For the number A(i) of lattice points in the “blown up” domain √tD, the estimate

is established. This is a generalization of Hardy's classical result for the circle problem. The proof is based on asymptotic formulae for certain exponential integrals due to E. Hlawka.

We consider a method for determining the asymptotic solution to a sufficiently wide class of ordinary linear homogeneous differential equations in a sector of a complex plane or of a Riemann surface for large values of the independent variable z. The main restriction of the method is the condition that the coefficients in the equation should be analytic and single-valued functions in the sector for | z | ≫ 1 possessing the power order of growth for |z| → ∞. In particular, the coefficients can be any powerlogarithmic functions. The equation

can be taken as a model equation. Here ai are complex numbers, aij are real numbers (i = 1,2,…, n; j = 0, 1, …, m) and ln1 Z≡ln z, lnsz= lnlnS−1z = S = 2, … It has been shown that the calculation of asymptotic representations for solution to any equation in the class considered may be reduced to the solution of some algebraic equations with constant coefficients by means of a simple and regular procedure. This method of asymptotic integration may be considered as an extension (to equations with variable coefficients) of the well known integration method for linear differential equations with constant coefficients. In this paper, we consider the main case when the set of all roots of the characteristic polynomial possesses the property of asymptotic separability.

General convergence theorems are established for the lower part of the intermediate operator spectra arising in lower bound methods of Aronszajn type.Convergence of the T*T method is considered in detail, as are constructions that allow perturbations that are nonclosable in the underlying Hilbert space. Problems with essential spectra are admitted, as is the use of infinite rank projections capable of displacing essential spectra. These last features are necessary for a theory applicable to Schrodinger operators corresponding to atomic systems.