Focal point and conjugacy criteria for the half-linear second-order differential equationare obtained using the generalized Riccati transformation. An oscillation criterion is given in case when the function c(t) is periodic.

We consider the system of reaction-diffusion equations known as the Sel'kov model. This model has been applied to various problems in chemistry and biology. We obtain a priori bounds on the size of the positive steady-state solutions of the system defined on bounded domains in Rn, 1 ≤ n ≤ 3 (this is the physically relevant case). Previously, such bounds had been obtained in the case n = 1 under more restrictive hypotheses. We also obtain regularity results on the smoothness of such solutions and show that non-trivial solutions exist for a wide range of parameter values.

The main result of this paper is the determination of all plane polynomial vector fields that admit a prescribed collection of algebraic curves as invariant sets. As an application, the polynomial vector fields admitting certain types of algebraic integrating factors are characterized.

We consider some geometric aspects of regular Sturm—Liouville problems. First, we clarify a natural geometric structure on the space of boundary conditions. This structure is the base for studying the dependence of Sturm—Liouville eigenvalues on the boundary condition, and reveals many new properties of these eigenvalues. In particular, the eigenvalues for separated boundary conditions and those for coupled boundary conditions, or the eigenvalues for self-adjoint boundary conditions and those for non-self-adjoint boundary conditions, are closely related under this structure. Then we give complete characterizations of several subsets of boundary conditions such as the set of self-adjoint boundary conditions that have a given real number as an eigenvalue, and determine their shapes. The shapes are shown to be independent of the differential equation in question. Moreover, we investigate the differentiability of continuous eigenvalue branches under this structure, and discuss the relationships between the algebraic and geometric multiplicities of an eigenvalue.

Let D be a domain in Rn with bounded complement and let n ≠ 2. For the initial-boundary value problemwe prove that there are no non-trivial global (non-negative) solutions if 0 < n (p − 1) ≤ 2 and there exist both global non-trivial and non-global solutions if n (p − 1) > 2.

Consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent region bounded by free surfaces, the motion being driven solely by a constant surface tension acting at the free boundaries. When the fluid region is simply-connected, it is known that this Stokes flow problem is closely related to a Hele-Shaw free boundary problem when the zero-surface-tension model is employed. Specifically, if the initial configuration for the Stokes flow problem can be produced by injection at N points into an empty Hele-Shaw cell, then so can all later configurations. Moreover, there are N invariants; while the N points at which injection must take place move, the amount to be injected at each of these points remains the same. In this paper, we consider the situation when the fluid region is doubly-connected and show that, provided the geometry has an appropriate rotational symmetry, the same results continue to hold and can be exploited to determine the solution of the Stokes flow problem.

Sufficient conditions are obtained for a ring R, faithfully graded by a bisimple inverse semigroup S, to be (a) prime and (b) right primitive, these conditions being on the subring RG consisting of all elements of R with support contained in G, a maximal subgroup of S. Earlier results on semigroup rings arise as special cases.

The entire positive solutions of a conformally invariant biharmonic equation in Rn will be classified using the method of moving spheres. As a byproduct, one also shows that any entire non-negative solution of the equation Δ2u = up with 1 ≤ p < (n + 4)/(n−4) with n ≥ 2 is zero.

We consider nonnegative steady-state solutions of the evolution equation

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Our class of coefficients f, g allows degeneracies at h = 0, such as f(0) = 0, as well as divergences like g(0) = ±∞. We first construct steady states and study their regularity. For f, g > 0 we construct positive periodic steady states, and non-negative steady states with either zero or nonzero contact angles. For f > 0 and g < 0, we prove there are no non-constant positive periodic steady states or steady states with zero contact angle, but we do construct non-negative steady states with nonzero contact angle. In considering the volume, length (or period) and contact angle of the steady states, we find a rescaling identity that enables us to answer questions such as whether a steady state is uniquely determined by its volume and contact angle. Our tools include an improved monotonicity result for the period function of the nonlinear oscillator. We also relate the steady states and their scaling properties to a recent blow-up conjecture of Bertozzi and Pugh.

It is known that the condition ‘either ∂L (F) ≠ Ø or there exist υ1,…,υq ∈ Rnsuch thatF ∈ int co {υ1,…,υq} characterizes solvability of the problemwith f(·) = 〈F,·〉.

We extend this result to the case of lower semicontinuous integrands L : Rn → R.

We also show that validity of this condition for all F ∈ Rn is both a necessary and sufficient requirement for solvability of all minimization problems with sufficiently regular Ω and f. Moreover, the assumptions on Ω and f can be completely dropped if L has sufficiently fast growth at infinity.

We study the existence of a boundary trace for minorized solutions of the equation Δu + K (x) e2u = 0 in the unit open ball B2 of R2. We prove that this trace is an outer regular Borel measure on ∂B2, not necessarily a Radon measure. We give sufficient conditions on Borel measures on ∂B2 so that they are the boundary trace of a solution of the above equation. We also give boundary removability results in terms of generalized Bessel capacities.

In this paper we give a local classification of the integral curves of implicit differential equations where F is a smooth function and p = dy/dx, at points where Fp = 0, Fpp ≠ 0 and where the discriminant {(x, y) : F = Fp = 0} has a Morse singularity. We also produce models for generic bifurcations of such equations and apply the results to the differential geometry of smooth surfaces. This completes the local classification of generic one-parameter families of binary differential equations (BDEs).

Some new global results are given about solutions to the boundary value problem for the Euler–Lagrange equations for the Ginzburg–Landau model of a one-dimensional superconductor. The main advance is a proof that in some parameter range there is a branch of asymmetric solutions connecting the branch of symmetric solutions to the normal state. Also, simplified proofs are presented for some local bifurcation results of Bolley and Helffer. These proofs require no detailed asymptotics for solution of the linear equations. Finally, an error in Odeh's work on this problem is discussed.

In 1929, Zermelo proposed a probabilistic model for ranking by paired comparisons and showed that this model produces a unique ranking of the objects under consideration when the outcome matrix is irreducible. When the matrix is reducible, the model may yield only a partial ordering of the objects. In this paper, we analyse a natural extension of Zermelo's model resulting from a singular perturbation. We show that this extension produces a ranking for arbitrary (nonnegative) outcome matrices and retains several of the desirable properties of the original model. In addition, we discuss computational techniques and provide examples of their use.

We provide an approximation of some free discontinuity problems by local functionals with a singular perturbation of higher order. More precisely, we study the limiting behaviour of energies of the formwhere Hu denotes the Hessian matrix of u.

Let u be a bounded slowly oscillating mild solution of an inhomogeneous Cauchy problem, u̇ (t) = Au (t) + f (t), on R or R+, where A is a closed operator such that σap (A) ∩iR is countable, and the Carleman or Laplace transform of f has a continuous extension to an open subset of the imaginary axis with countable complement. It is shown that u is (asymptotically) almost periodic if u is totally ergodic (or if X does not contain c0 in the case of a problem on R). Similar results hold for second-order Cauchy problems and Volterra equations.