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.

We consider Hill's equation on the interval [0, a] with periodic and semi-periodic boundary condition. We derive the asymptotic form of the eigenvalues without smoothness conditions on q.

In this work we study the existence and qualitative properties of non-negative solutions of the Lotka—Volterra models with nonlinear diffusion under homogeneous Dirichlet boundary conditions. We consider the three typical interactions: prey—predator, competition and symbiosis. Unlike the linear diffusion models, non-trivial non-negative solutions can exist which are not strictly positive. Sufficient conditions in terms of the coefficients involved in the setting of the models are given, assuring that one species (or both) does not survive on a set of its habitat (called ‘dead core’) of positive measure.

We show by elementary methods that there are symplectic embeddings from standard (R2n, ω0) into (Σ x R2n−2, ω ⊕ ω0) and (T2n−2k x R2k, ω ⊕ ω0), where (Σ, ω) is a closed two-dimensional symplectic manifold, and (T2n−2k, ω) is the torus with a constant symplectic form ω. Some estimates of Gromov's symplectic capacity are given for bounded domains in these manifolds.

In a previous paper, I investigated the use (for the inverse scattering problem) of the resonant frequencies and the associated eigen far-fields. I showed that the shape of a sound soft obstacle is uniquely determined by a knowledge of one resonant frequency and one associated eigen far-field. Inverse obstacle scattering problems are ill-posed in the sense that a small error in the measurement may imply a large error in the reconstruction. This is contrary to the idea of continuity. I proved that, by adding some a priori information, the reconstruction becomes continuous. More precisely, continuity holds if we assume that the obstacle lies a fixed and known compact set.

The goal of this paper is to extend these results to the case of absorbing obstacles.

The paper is concerned with nonlinear equations of critical Sobolev growth involving the p-Laplace operator. These equations generalize the more classical scalar curvature equation.

We introduce a new variant to prove the regularity of solutions to transport equations of the Vlasov type. Our approach is mainly based on the proof of propagation of velocity moments, as in a previous paper by Lions and Perthame. We combine it with moment lemmas which assert that, locally in space, velocity moments can be gained from the kinetic equation itself. We apply our theory to two cases. First, to the Vlasov–Poisson system, and we solve a long-standing conjecture, namely the propagation of any moment larger than two. Next, to the Vlasov–Stokes system, where we prove the same result for fairly singular kernels.

The spectral theory for non-self-adjoint elliptic boundary problems involving an indefinite weight function has only been established for the case of higher-order operators under the assumption that the reciprocal of the weight function is essentially bounded. In this paper we are concerned with the spectral theory for a case where the weight function vanishes on a set of positive measure.

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.

Bifurcation of periodic and chaotic solutions is investigated for coupled perturbed autonomous ordinary differential equations (ODEs) when the first unperturbed ODE has a homoclinic solution and the second unperturbed ODE possesses a Hopf singularity.

We consider the following elliptic equation:where m > 0, f(x, u)/u tends to a positive constant as u → + ∞. Here, the nonlinear term f(x, u) does not satisfy the usual condition, that is, for some θ > 0,which is important in using the mountain pass theorem. The aim of this paper is to discuss how to use the mountain pass theorem to show the existence of a positive solution to the present problem when we lose the above condition. Furthermore, if f(x, u) ≡ f(u), we also prove that the above problem has a ground state by using the artificial constraint method.

We give a structure theorem for simplicial affine semigroups. From this result we deduce characterizations of some properties of semigroup rings of simplicial affine semigroups. We also compute an upper bound for the cardinality of a minimal presentation of a simplicial affine semigroup.

Let M be a compact manifold with smooth boundary. We study the heat content asymptotics on M defined by a time-dependent heat source and time-dependent boundary conditions.

This note develops a connection formula for spectral functions given in a recent paper by Gilbert and Harris.

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 study the diffusive Nicholson's blowflies equation. Generalizing previous works, we model the generation delay by using an integral convolution over all past times and results are obtained for general delay kernels as far as possible. The linearized stability of the non-zero uniform steady state is studied in detail, mainly by using the principle of the argument. Global stability both of this state and of the zero state are studied by using energy methods and by employing a comparison principle for delay equations. Finally, we consider what bifurcations are possible from the non-zero uniform state in the case when it is unstable.

We consider the universal phantom map out of a non-finite loop space. First we obtain a necessary and sufficient condition for the universal phantom map out of ΩG for a simply connected compact Lie group G to be essential. Next we prove that the universal phantom map out of ΩkX is essential for all k ≥ 2 if X is a simply connected non-contractible finite CW-complex. Ingredients in the proof are the Browder's ∞-implication argument and the Eilenberg–Moore spectral sequence.

Using a norm inequality for singular integral operators in pairs of weighted Lebesgue spaces we are able to prove existence and uniqueness results for solutions of nonlinear RiemannHilbert problems with non-compact Lipschitz continuous restriction curves.

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.

The Stokes problem on a domain with edge singularities is considered. The decomposition of the solution into a regular part and blocks of singular functions is established. This, together with the tangential regularity of the solution, leads to a global regularity result in suitable weighted Sobolev spaces, the properties of which are investigated. The global regularity is exploited to generate an optimally convergent semi-discrete mesh refinement mixed finite-element method. In the particular case of a prismatic domain, the Fourier finite-element method, which is a fully discrete scheme, is implemented.