We consider a dissipative reaction–diffusion equation on a thin L-shaped domain (with the thinness measured by a parameter ε); we determine the limit equation for ε = 0 and prove the upper semicontinuity of the global attractors at ε = 0. We also state a lower semicontinuity result. When the limit equation is one-dimensional, we prove convergence of any orbit to a singleton.

In this paper we show that each quasiperiodic standing wave solution of the real Ginzburg–Landau equation which is on the global branch emanating from the Eckhaus unstable periodic orbit is itself unstable. A rigorous proof of the instability is given by showing that the linearised operator about such a solution has spectrum which contains an interval along the unstable axis of the spectral plane. The proof employs some geometric and topological methods arising from a dynamical systems approach to the analysis of the eigenvalue problem for the linearised operator.

In [3], uniform asymptotic expansions were derived for solutions of the oblate spheroidal wave equation (z2 − 1)d2p/dz2 + 2zdp/dz − (λ + μ2/(z2 − 1)) p = 0, for the case where the parameter μ is real and non-negative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u → ∞, uniform asymptotic expansions were derived involving elementary, Airy and Bessel functions, these being valid in certain subdomains of the complex z plane. In this paper the complementary case, where λ is real and negative, is considered. Asymptotic expansions are derived which are valid in certain subdomains of the half-plane |arg (z)| ≦ π/2, uniformly valid for u → ∞ with λ /u2 fixed and negative, and 0 ≦ μ/u ≦ − ½λ /u2 − δ, where δ is an arbitrary positive constant. Explicit error bounds are available for all the approximations.

In a recent work, G. Anzellotti and the present authors introduced a notion of variation for functions defined over a rectifiable current. In this paper, we give a definition of a curvature varifold slightly different from that of Hutchinson (equivalent in the nonoriented case) and we study the variation properties, in the sense of [2], of the normal to a rectifiable current when the associated varifold is a curvature varifold.

Oscillation, comparison and asymptotic theory for the Sturm-Liouville problem

with 1/p, q, r ε L1 ([0, 1]), p, r > 0, are studied subject to eigenvalue-dependent boundary conditions

This continues previous work on cases with (− 1)j δj ≦ 0 where δj = ajdj − bjcj. We now consider the remaining sign conditions for δj, exploiting the interplay between the graph of cot θ− (λ, 1), for a modified Prüfer angle θ−, and the eigencurves of a related two-parameter problem.

We show that for every finite set A and for every natural number n, there exists a natural number N such that every word of length N over the alphabet A has, for every permutation π of the numbers 1,…,n, a representation of the form Xw1 … wnzwπ(1) … wπ(n) Y, where X, Y are words and w1,…,wn, z are nonempty words over A.

The paper provides an example of an integral functional in more than two dimensions, with a symmetric and positively defined quadratic integrand q, exhibiting the Lavrentieff phenomenon on a ball B and on a linear boundary datum u0, i.e. for which

The example is also utilised to discuss nonidentity between some relaxation procedures for a quadratic integral functional and to provide a weighted Sobolev space in the Hilbert case in which smooth functions are not dense.

We would like to obtain the transmutation operator V, associated with the self-adjoint operators −d2/(w(x) dx2) and (−d2/(w(x) dx2)) + h(x), where w(x) ≎ xa as x → 0. We shall show that V = 1 + K, where K is a lower triangular Volterra operator.

The problem of approximation of attractors for semidynamical systems (SDSs) in a metric space is studied. Let some (exact) SDS possessing an attractor M be inaccurately defined, i.e. let another (approximate) SDS, which is close in some sense to the exact one, be given. The problem is to construct a set , which is close to M in the Hausdorff metric.

The suggested procedure for constructing is finite, which makes it possible to use it in computations. The results obtained are suitable for numerical approximation of attractors for a rather large class of semidynamical systems, including ones generated by the Lorenz equations and the Navier–Stokes equations.

In this paper we consider mathematical models inspired by the mechanisms of biological evolution. We take populations which are subject to interaction and mutation. In the cases we consider, the interaction is through competition or through a prey-predator relationship. The models consider the specific characteristics as taking values in real intervals and the equations are of the integro—differential type. In the case of competition, thanks to the fact that some of the equations have solutions which are quite explicit, we succeed in proving the existence of attracting stationary solutions. In the case of prey and predator, using techniques of dynamical systems in infinite-dimensional spaces, we succeed in showing the existence of a global attractor, which in some instances reduces to a point. Our analysis takes into account having δ distributions, corresponding to all individuals having the same characteristics, as possible populations.

In this paper, we consider an n-dimensional semilinear equation of parabolic type with a discontinuous source term arising from combustion theory. We prove local existence for a classical solution having a ‘regular’ free boundary. In this regard, the free boundary is a surface through which the discontinuous source term exhibits a switch-like behaviour. We specify conditions under which this solution and its free boundary are global in time; moreover, we exhibit a special domain for which, for t tending to infinity, such a global-in-time solution converges, together with its free boundary, to the solution of the stationary problem and to its regular free boundary (which is proved to exist), respectively. We also prove uniqueness and continuous dependence theorems.

We study the problem when a ring which is an extension of a commutative idempotent ring by a commutative idempotent ring is commutative. In particular, we answer Sands' question showing that the class of commutative idempotent rings whose every homomorphic image has zero annihilator is a maximal but not the largest radical class consisting of commutative idempotent rings.

This paper addresses the realisation of ordinary differential equations (ODEs) by retarded functional differential equations (FDEs) in finite-dimensional invariant manifolds, locally around equilibrium points. A necessary and sufficient condition for realisability of C1 vector fields is established in terms of their linearisations at the equilibrium.

It is also shown that any arbitrary finite jet of vector fields of ODEs can be realised without any further restrictions than those imposed by the realisability of its linear term, a fact of relevance for discussing the flows defined by FDEs around singularities, and their bifurcations. Besides, it is proved that such a realisation can always be achieved with FDEs whose nonlinearities are defined in terms of a finite number of delayed values of the solutions.

Given an elliptic operator L on a bounded domain Ω ⊆ Rn, and a positive Radon measure μ on Ω, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains Ωh ⊇ Ω with the following property: for every f ∈ H−1(Ω) the sequence uh of the solutions of the Dirichlet problems Luh = f in Ωh, uh = 0 on ∂Ωh, extended to 0 in Ω\Ωh, converges to the solution of the “relaxed Dirichlet problem” Lu + μu = f in Ω, u = 0 on ∂Ω.

Complex and chaotic structures in certain dynamical systems in biology arise as a consequence of noncomplete integrability of two-degree-of-freedom Hamiltonian systems. A study of this problem is made using Ziglin theory and implemented with the aid of the Kovacic algorithm.

We establish lower and upper bounds which are valid for the overall conductivity of twodimensional composites. They are based on a method which modifies the so-called translation method in a way which makes it effectively much more flexible. When specialised to composites of n > 2 isotropic phases, the new bounds are often strictly better than all the previously known ones. From the mathematical point of view, the improvement is due mainly to a new regularity result in p.d.e.s [2]. From the physical point of view the latter can be interpreted as a result bounding in a suitable sense the fluctuations of the ‘electric field’.

We investigate the Cauchy problem for a hydrodynamic model for semiconductors. An existence theorem of global weak solutions with large initial data is obtained by using the fractional step Lax—Friedrichs scheme and Godounov scheme.

In the first part of this paper, Yau's estimates for positive L-harmonic functions and Li and Yau's gradient estimates for the positive solutions of a general parabolic heat equation on a complete Riemannian manifold are obtained by the use of Bakry and Emery's theory. In the second part we establish a heat kernel bound for a second-order differential operator which has a bounded and measurable drift, using Girsanov's formula.

In this paper we use arguments based on Picard-Fuchs equations and transversality of intersections of level curves to obtain an exact count of the number of stationary solutions of the one-dimensional Cahn-Hilliard equation with a cubic nonlinearity.

Let f be an odd, C2 function on [− 1, 1], which vanishes at ± 1, and such that f′(O) < 0, f′ (±1) > 0 and u ↦ f(u)/u is increasing. Dang, Fife and Peletier [5] showed that there is a unique solution u with values in [−1, 1] of

which has the same sign as xy. The linearised operator around u is B defined by

It is proved here that the spectrum of B contains at least one negative eigenvalue, that all eigenfunctions corresponding to negative eigenvalues have the symmetries of the square, and that for Allen–Cahn's nonlinearity (f(u) = 2u3 − 2u), there is exactly one negative eigenvalue.