We prove a lower bound for the Dirichlet heat kernel pD(x,y;t), where x and y are a visible pair of points in an open set D in ℝm.

This paper considers the problem of asymptotic decay as t → ∞ of solutions of the wave equation utt − Δu = −a(x)β(ut, ∇u), (t,x) ∊ ℝ+ × Ω (a bounded, open, connected set in ℝN, N≧ 1, with smooth boundary), u =0 on ℝ+ × ∂Ω. The nonlinear function β is not assumed to be globally Lipschitz continuous, β(0, y2, …, yN+1) = 0, y1 β(y1…,yN+1) ≧ 0 for all y ∊ ℝN+1; β is not assumed to be monotone in y1. Under additional restrictions on the kernel of β, conditions are given which imply that [u, ut,] converges to [0,0] weakly in H = H10(Ω) × L2(Ω) as t → ∞. The work generalises earlier results of Dafermos and Haraux where strong decay in H as t → ∞ was obtained in the case β(y1 …, yN+1) = q(y1), q monotone on ℝ.

In some cases, a reaction–diffusion system can be transformed into an abstract equation where the linear part is given by a polynomial of a linear operator, say Multiparameter bifurcation for this equation is considered as the coefficients of the operator polynomial in are varied.

We consider the following eigenvalue problem:

We prove the existence of H1(Rn)∩L∞(Rn) bifurcation at λ=0 but only require aij(x, t) (i,j= 1, 2, …,n) and f(x, t) to satisfy certain conditions in theneighbourhood of Rn × {0}.

We consider the Dirichlet problem for harmonic maps from the disc D2 into the sphere S2, with prescribed boundary values γ:∂D2→S2, and we prove that if γ is not a rational function, one can find infinitely many nonhomotopic harmonic maps which agree with γon ∂D2.

We study a special class of linear differential operators well-behaved with respect to weakconvergence. Questions related to weak lower semicontinuity, associated Young measures, weak continuity and quasi-convexity are addressed. Specifically, it is shown that the well-known necessary conditions for weak lower semicontinuity are also sufficient in this case. Some examples are given, including a discussion on how well the operator curl fits inthis context.

In this note, we consider the Hardy-Littlewood maximal function on R for arbitrary measures, as was done by Peter Sjögren in a previous paper. We determine the best constant for the weak type inequality.

Theorems of Gersgorin-type are established for a diagonally dominant, unbounded, infinite matrix operator A acting on lp for some l ≦p≦∞. The results are established using an approximating sequence of infinite matrices An that converges to A in the generalised sense as n → ∞. This constructive approach admits approximation of the spectral properties of A by those of An.

We prove that for every dense Gδ set H, there exists a continuous function f, such that f intersects every analytic function in finitely many points and f is infinitely differentiable exactly at the points of H. This answers a problem of S. Agronsky, A. M. Bruckner, M. Laczkovich and D. Preiss. They proved a result which implies that every continuous function with finite intersections with analytic functions is infinitely differentiable at the points of a dense Gδ set.

In this paper we study the equilibrium equations for axisymmetric deformations of isotropic circular plates in tension. We give results on the global multiplicity of solutions and study the stability of the trivial homogeneous solution for large displacements.

We consider the nonlinear elliptic problem at resonance, Δu + λ1u + f(x, u) = h(x) in Ω, u = 0 on ∂Ω, where Ω is a bounded smooth domain in ℝN, λl is the first eigenvalue of –Δ in Ω and h(x) is orthogonal to the first eigenfunction. We give some conditions of solvability in terms of the primitive of f with respect to u.

We study the behaviour of solutions u = um of ut, + (um)x = 0 for t > 0, x ∊ R, u(x, 0) = u0(x), u0 ≧0, u0 ∊ L1(R) as m → ∞. This is a singular perturbation problem about m = ∞ if u0 > 1 on a set of positive measure. It is shown that the limit exists and satisfies the stationary equation

We study the asymptotic behaviour as x →∞ of the solutions of the ordinary differential equation problem

This equation generalises the ordinary differential equation obtained by studying the blow-up of the similarity solutions of the semilinear parabolic partial differential equation vt=vxx = ev. We show that if λ≦1, all solutions of (*) tend to —∞ as rapidly as the function —exp (x2/4) (E- solutions). However, if λ>1, then there also exists a solution which tends to –∞, like 2λlog(x) (L-solutions). Thus, the case λ = 1, for which (*) reduces tothe Kassoy equation, is the borderline between two quite different forms of asymptotic behaviour of the function u(x).

In this note we demonstrate that a conjecture about a necessary and sufficient condition for the existence of a centre of cubic equations is not true.

Let N:ℝ→ℝ be a locally Lipschitzian map such that (y + l)N(y)>0 for all y ≠ –1 and such that N(y)=1 + y for – 1 ≦ y ≦ 3. For any positive number α the equation y'(t) αy(t–1)N(y(t)) has, aside from the constantsolutions y(t) ≡ –1, and y(t) ≡–1 solution y(t) such that y(t + 4) = y(t) for all real t If N(y) = 1 + y for all y, one obtains Wright's equation, which isknown to have periodic solutions of minimal period p (depending on α) arbitrarily close to 4. Some results concerning nonexistence of periodic solutions of period 4 of other differential-delay equations are also proved. In all cases the method of proof consists in analysing an associated fourth-order system of ordinary differential equationsand showing that this system has no nonconstant periodic solutions.

We consider a weakly coupled set of two partial differential equations where the coupling matrix has variable elements and the principal part of each equation is the same uniformly elliptic operator. Weobtain necessary conditions that the system of equations can be decoupled. By decoupling the system and using a positivity lemma due to Hess and Kato, we determine the algebraic sign of the solution components. This work extends recent results of de Figueiredo and Mitidieri. Further, one can use these results to determine the sign of the solution to certain fourth order elliptic boundary value problems.

In this paper we prove that the domain of hyperbolicity of the polynomial xn + λ2nn−2+λ3xn−3+ … + λn,λiϵR intersected by the half-space λ2 ≧ – 1, has the property of Whitney, i.e., every two points of this set can be connected by a piecewise-smooth curve belonging to it, whose length is ≦C times greater than the euclidian distance between the points, where the constant C does not depend on the choice of the points. Parallel with this, we show that the values x1≦x2≦…≦xn of a random variable are uniquely determined by the corresponding probabilities and by thefirst n moments.

Depending upon the initial data associated with the fundamental matrix, the function M(λ), used to generate L2-solutions of homogeneous linear differential systems, may vary. We show that there is a matrix bilinear transformation between such functions M(λ) with different initial data and illustrate how the result can be used to simplify the calculation of a specific M(λ)-function for a scalar second-order problem.