Given a set valued mapping ∑: ℝ → ∑n, we prove a closure property with respect to -convergence for the differential inclusion

under very mild assumptions on ∑.

In this paper, we prove that the Hele–Shaw problem with kinetic condition and surface tension is the limit case of the supercooled Stefan problem in the classical sense when specific heat ε goes to zero. The method is the use of a fixed-point theorem; the key is to construct a workable function space. The main feature is to obtain the existence and the uniform estimates with respect to ε > 0 at the same time for the solutions of the supercooled Stefan problem. For the sake of simplicity, we only consider the case of one phase, although the method used here is also applicable in the case of two phases.

We consider the question: When do two ordinary linear differential expressions commute? It turns out that the set of all expressions which commute with a given one form a commutative ring. Here we study the algebraic structure of these rings. As an application a complete characterization of normal differential expressions is obtained.

Dominance properties of solutions to Lny + p(x)y = 0, where Ln is a disconjugate operator, are compared to dominance properties of solutions to its adjoint.

Comparison principles for systems of reaction-diffusion equations coupled via both the reaction and diffusion terms are considered. Applications to the FitzHugh–Nagumo equations and models of coupled nerve fibres are included.

We prove certain L2(ℝn) bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the k-plane transform. As the estimates are L2-based, they follow from bilinear identities: in particular, these are the analogues of a known identity for paraboloids, and may be seen as higher-dimensional versions of the classical L2(ℝ2)-bilinear identity for Fourier extension operators associated to curves in ℝ2.

The generalized Bendixson's criterion for non-existence of non-constant periodic solutions of an autonomous system dx/dt = f(x), x ∈ D, is further extended to systems when the condition f ∈ C1(D, RN) is violated. In particular, the criterion can be applied to region-wise linear systems in RN.

Complex linear differential equations with entire coefficients are studied in the situation where one of the coefficients is an exponential polynomial and dominates the growth of all the other coefficients. If such an equation has an exponential polynomial solution $f$, then the order of $f$ and of the dominant coefficient are equal, and the two functions possess a certain duality property. The results presented in this paper improve earlier results by some of the present authors, and the paper adjoins with two open problems.

1. A reagent (p-cresol) has been used to determine the activity of tyrosinase in the potato tuber.

2. The course of the reaction tyrosinase-p-cresol has been shown to be mono-molecular, with a short linear period at the commencement. From observations of this period, constants for the enzyme activities have been obtained and a temperature coefficient deduced.

3. The enzyme activity has been shown to depend upon stage of maturity of tuber, variety of tuber, and disease infection, but to be independent of tuber weight, environment, season, and storage conditions.

4. The probable significance of the results has been discussed.

The writer wishes to thank the Department of Agriculture for Scotland for a Grant which enabled the work to be carried out, and also Dr A. Lauder and Dr A. M. Smith of the Edinburgh and East of Scotland College of Agriculture for their interest and assistance throughout the course of the work.

Infection mechanism plays a significant role in epidemic models. To investigate the influence of saturation effect, a nonlocal (convolution) dispersal susceptible-infected-susceptible epidemic model with saturated incidence is considered. We first study the impact of dispersal rates and total population size on the basic reproduction number. Yang, Li and Ruan (J. Differ. Equ. 267 (2019) 2011–2051) obtained the limit of basic reproduction number as the dispersal rate tends to zero or infinity under the condition that a corresponding weighted eigenvalue problem has a unique positive principal eigenvalue. We remove this additional condition by a different method, which enables us to reduce the problem on the limiting profile of the basic reproduction number into that of the spectral bound of the corresponding operator. Then we establish the existence and uniqueness of endemic steady states by a equivalent equation and finally investigate the asymptotic profiles of the endemic steady states for small and large diffusion rates to provide reference for disease prevention and control, in which the lack of regularity of the endemic steady state and Harnack inequality makes the limit function of the sequence of the endemic steady state hard to get. Finally, we find whether lowing the movements of susceptible individuals can eradicate the disease or not depends on not only the sign of the difference between the transmission rate and the recovery rate but also the total population size, which is different from that of the model with standard or bilinear incidence.

It is shown that classical solutions backward in time to the alternatives to the Navier-Stokes equations proposed by Ladyzhenskaya cannot exist for all time. Estimates are obtained for the maximum interval of existence.

A generic bifurcation theory is developed which is somewhat different from the approach in [4]. We put special emphasis on equations satisfying additional symmetry properties and on the non-generic bifurcation sets arising in this context. We apply our results on the von Kármán equations for the buckling of a rectangular plate under a compressive thrust and a normal load.

A new class of irregular boundary value problems—non-regular in the sense of Birkhoff—is studied. This class of strongly irregular problems includes the class of boundary value problems with irregular decomposing boundary conditions. For each strongly irregular problem we can find a problem with irregular decomposing boundary conditions so that we have equiconvergence with respect to Riesz typical means of the eigenfunction expansions arising from these two problems of an arbitrary summable function.

A closed summation operator, whose spectrum lies within a certain region, generates a derivation and antiderivation, and an Euler–Maclaurin sum formula among these three operators.

For subsets in the standard symplectic space $(\mathbb {R}^{2n},\omega _0)$ whose closures are intersecting with coisotropic subspace $\mathbb {R}^{n,k}$ we construct relative versions of the Ekeland–Hofer capacities of the subsets with respect to $\mathbb {R}^{n,k}$, establish representation formulas for such capacities of bounded convex domains intersecting with $\mathbb {R}^{n,k}$. We also prove a product formula and a fact that the value of this capacity on a hypersurface $\mathcal {S}$ of restricted contact type containing the origin is equal to the action of a generalized leafwise chord on $\mathcal {S}$.

We consider a singularly perturbed convection–diffusion equation, $-\varepsilon\Delta u+\bm{v}\cdot\bm{\nabla}u=0$, defined on a rectangular domain $\varOmega\equiv\{(x,y)\mid 0\leq x\leq\pi a,0\leq y\leq\pi\}$, $a>0$, with Dirichlet-type boundary conditions discontinuous at the points $(0,0)$ and $(\pi a,0)$: $u(x,0)=1$, $u(x,\pi)=u(0,y)=u(\pi a,y)=0$. An asymptotic expansion of the solution is obtained from a series representation in two limits, namely when the singular parameter $\varepsilon\to0^+$ (with fixed distance to the points $(0,0)$ and $(\pi a,0)$), and when $(x,y)\to(0,0)$ or $(x,y)\to(\pi a,0)$ (with fixed $\varepsilon$). It is shown that the first term of the expansion at $\varepsilon=0$ contains a linear combination of error functions. This term characterizes the effect of the discontinuities on the $\varepsilon$-behaviour of the solution $u(x,y)$ in the boundary or the internal layers. On the other hand, near the points of discontinuity $(0,0)$ and $(\pi a,0)$, the solution $u(x,y)$ is approximated by a linear function of the polar angle.

In this note we study the asymptotic behaviour of a product of matrices where Pj is a matrix of transition probabilities in a non-homogeneous finite Markov chain. We give conditions that (i) the rows of P(n) tend to identity and that (ii) P(n) tends to a limit matrix with identical rows.

In this paper we prove the existence and analyticity of solitary waves of generalized Kadomtsev–Petviashvili equations satisfying a set of conditions on linear and nonlinear terms which determine their behaviour at infinity and around 0.

We prove that, contrary to the L1-Nash inequality, there exists a second best constant for the L2-Nash inequality on any smooth compact Riemannian manif

We compute critical groups in semilinear elliptic boundary-value problems in which the nonlinear term may fail to have asymptotic limits at zero and at infinity. As applications, we prove several new existence results.