Travelling wavefronts for a system of two reaction–diffusion equations are studied. The existence of a family of wavefronts (one for each wave speed) as well as the existence of the minimal speed (in the case that the Lewis number is greater than $1$) are proved. Asymptotic formulae for the wavefronts are established. New results are obtained when applying the main theorems to an isothermal autocatalytic chemical reaction system.

Riesz transforms $R_\mu$ related to the Bessel operators

$$ \Delta_\mu=x^{-\mu-1/2}Dx^{2\mu+1}Dx^{-\mu-1/2} $$

are studied in this work. We develop for $R_\mu$ a theory that runs parallel to that for the Euclidean Hilbert transform. It is proved that $R_\mu$ is actually a Calderón–Zygmund singular integral operator. Also, $R_\mu$ is seen to be the boundary value of the appropriate harmonic extension for this context. Finally, we analyse weighted inequalities involving $R_\mu$.

We study a one-dimensional non-local variant of Fisher's equation describing the spatial spread of a mutant in a given population, and its generalization to the so-called monostable nonlinearity. The dispersion of the genetic characters is assumed to follow a non-local diffusion law modelled by a convolution operator. We prove that, as in the classical (local) problem, there exist travelling-wave solutions of arbitrary speed beyond a critical value and also characterize the asymptotic behaviour of such solutions at infinity. Our proofs rely on an appropriate version of the maximum principle, qualitative properties of solutions and approximation schemes leading to singular limits.

We prove that the integral of the product of two functions over a symmetric set in $\mathbb{S}^1\times\mathbb{S}^1$, defined as $E=\{(x,y)\in\mathbb{S}^1\times\mathbb{S}^1:d(\sigma_1(x),\sigma_2(y))\leq\alpha\}$ (where $\sigma_1$, $\sigma_2$ are diffeomorphisms of $\mathbb{S}^1$ with certain properties and $d$ is the geodesic distance on $\mathbb{S}^1$), increases when we pass to their symmetric decreasing rearrangement. We also give a characterization of the diffeomorphisms $\sigma_1$, $\sigma_2$ for which the rearrangement inequality holds. As a consequence, we obtain the result for the integral of the function $\varPsi(f(x),g(y))$ (where $\varPsi$ is a supermodular function) with a kernel given as $k[d(\sigma_1(x),\sigma_2(y))]$, with $k$ decreasing.

A new method for asymptotic summation of linear systems of difference equations is proposed and studied. It is based on the introduction of a certain summation equation that pinpoints sufficient conditions for asymptotic summation. These conditions serve as a framework from which new and old theorems follow. In particular the analogues of the fundamental theorems of Levinson and Hartman and Wintner are shown to follow from one and the same framework. Examples are given that are not amenable to other techniques.

Let $D$ be a bounded, finitely connected domain in $\mathbb{C}$ without isolated points in the boundary and let $f$ be a continuous function on $bD$. Let $\tilde{f}$ be a continuous extension of $f$ to $\bar{D}$. We prove that $f$ extends holomorphically through $D$ if and only if the degree of $\tilde{f}+h$ is non-negative for every holomorphic function $h$ on $D$ such that $\tilde{f}+h$ is bounded away from $0$ near $bD$.

We investigate the existence and properties of the Jost solution associated with the differential equation $-y''+q(x)y=\lambda y$, $x\geq0$, for a class of real- or complex-valued slowly decaying potentials $q$. In particular, it is shown how the traditional condition $q\in L(\mathbb{R}^{+})$ for the existence of the Jost solution can be replaced by $q'\in L(\mathbb{R}^{+})$ for a class of potentials considered here. We also examine the asymptotics of the Titchmarsh–Weyl function for a class of real- or complex-valued slowly decaying potentials and the form of the spectral density for a class of real-valued slowly decaying potentials.

A real polynomial $P$ of degree $n$ in one real variable is hyperbolic if its roots are all real. A real-valued function $P$ is called a hyperbolic polynomial-like function of degree $n$ if it has $n$ real zeros and $P^{(n)}$ vanishes nowhere. Denote by $x_k^{(i)}$ the roots of $P^{(i)}$, $k=1,\dots,n-i$, $i=0,\dots,n-1$. Then, in the absence of any equality of the form

\begin{equation} x_i^{(j)}=x_k^{(l)} \tag{*} \label{*} \end{equation}

for all $i<j$ we have

\begin{equation} x_k^{(i)}<x_k^{(j)}<x_{k+j-i}^{(i)} \tag{**} \label{**} \end{equation}

(the Rolle theorem). For $n\geq4$ (respectively, for $n\geq5$) not all arrangements without equalities \eqref{*} of $\tfrac12n(n+1)$ real numbers $x_k^{(i)}$ and compatible with \eqref{**} are realizable by the roots of hyperbolic polynomials (respectively, of hyperbolic polynomial-like functions) of degree $n$ and of their derivatives. For $n=5$ and when

$$ x_1^{(1)}<x_1^{(3)}<x_2^{(1)}<x_3^{(1)}< x_2^{(3)}<x_4^{(1)} $$

we show that all such 102 arrangements are realizable by hyperbolic polynomial-like functions (of which 66 are obtained by hyperbolic polynomials and another 8 by perturbations of such).

We study the solutions of topological type for a class of self-dual vortex theories in two dimensions. We consider the regime corresponding to the limit of small vortex core size with respect to the separation distance between vortices, namely as the scaling parameter $\delta>0$ tends to zero. Using a gluing technique for the corresponding nonlinear elliptic equation on the plane, with any number (finite or countable) of prescribed singular sources, we prove the existence of multi-vortex solutions which behave as a single vortex solution near each vortex point, up to an error exponentially small, as $\delta\to0$. Moreover, in the physically relevant cases, namely when the vortex points are either finite or periodically arranged in the plane, we prove that the multi-vortex solution satisfying a ‘topological condition' is unique, for $\delta>0$ sufficiently small.

In this paper, we study the Cauchy problem of a cubic autocatalytic chemical reaction system

$$ u_{1,t}=u_{1,xx}-u_1u^{2}_2,\qquad u_{2,t}=du_{2,xx}+u_1u^{2}_2 $$

with non-negative initial data, where the constant $d>0$ is the Lewis number. Our purpose is to study the global dynamics of solutions under mild decay of initial data as $|x|\rightarrow\infty$. In particular, we show that, for a substantial class of $L^1$ initial data, the exact large-time behaviour of solutions is characterized by a universal, non-Gaussian spatio-temporal profile, subject to the apparent conservation of total mass.

The main purpose of this paper is to study the mean value properties of certain Hardy sums over a short interval by using the mean-value theorems of the Dirichlet $L$-functions. Our main result is a mean-value formula for these sums.