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 1992, Fountain and Lewin showed that any proper ideal of an endomorphism monoid of a finite independence algebra is generated by idempotents. Here the ranks and idempotent ranks of these ideals are determined. In particular, it is shown that when the algebra has dimension greater than or equal to three the idempotent rank equals the rank.

In this paper, we show that if $K(x)$ satisfies suitable conditions, then the quasilinear Neumann problem $-\Delta_pu+|u|^{p-2}u=K(x)|u|^{q-2}u$ in an exterior domain $\varOmega$ has at least two solutions, of which one is a positive ground-state solution and one is a nodal solution.

In this paper we find an explicit moving frame along curves of Lagrangian planes invariant under the action of the symplectic group. We use the moving frame to find a family of independent and generating differential invariants. We then construct geometric Hamiltonian structures in the space of differential invariants and prove that, if we restrict them to a certain Poisson submanifold, they become a set of decoupled Korteweg–de Vries (KdV) first and second Hamiltonian structures. We find an evolution of curves of Lagrangian planes that induces a system of decoupled KdV equations on their differential invariants (we call it the Lagrangian Schwarzian KdV equation). We also show that a generalized Miura transformation takes this system to a modified matrix KdV equation. In the four-dimensional case we show that there are no unrestricted compatible geometric pairs.

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).

Following pioneering work by Fan and Slemrod, who studied the effect of artificial viscosity terms, we consider the system of conservation laws arising in liquid–vapour phase dynamics with physical viscosity and capillarity effects taken into account. Following Dafermos, we consider self-similar solutions to the Riemann problem and establish uniform total variation bounds, allowing us to deduce new existence results. Our analysis covers both the hyperbolic and the hyperbolic–elliptic regimes and apply to arbitrarily large Riemann data.

The proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system. Strong $L^1$ convergence to a weak solution of bounded variation is established in the hyperbolic regime, while in the hyperbolic–elliptic regime a stationary singularity near the axis separating the two wave fans, or more generally an almost-stationary oscillating wave pattern (of thickness depending upon the capillarity–viscosity ratio), is observed and the solution may not have globally bounded variation.

We consider the flow of a gas into a bounded tank Ω with smooth boundary ∂Q. Initially Ω is empty, and at all times the density of the gas is kept constant on ∂Ω. Choose a number R > 0 sufficiently small that, for any point x in Q having distance R from ∂Ω, the closed ball B with radius R centred at x intersects ∂Ω at only one point. We show that if the gas content of such balls B is constant at each given time, then the tank Ω must be a ball. In order to prove this, we derive an asymptotic estimate for gas content for short times. Similar estimates are also derived in the case of the evolution p-Laplace equation for p ⩾ 2.

This paper studies blowing-up properties of a unique positive principal eigenvalue for a linear elliptic eigenvalue problem with an indefinite weight function and Neumann boundary condition. Necessary and sufficient conditions for the blowing-up property are discussed, based on the variational characterization of the unique positive principal eigenvalue. A counterexample is constructed, which shows that a known necessary and sufficient condition for the blowing-up property in the Dirichlet boundary condition case no longer remains true in the Neumann case.

The Kuramoto–Sivashinsky-perturbed Korteweg–de Vries (KS–KdV) equation

$$ \partial_tu=-\partial_x^3u-\tfrac{1}{2}\partial_x(u^2)-\varepsilon(\partial_x^2+\partial_x^4)u, $$

with $0<\varepsilon\ll1$ a small parameter, arises as an amplitude equation for small amplitude long waves on the surface of a viscous liquid running down an inclined plane in certain regimes when the trivial solution, the so-called Nusselt solution, is sideband unstable. Although individual pulses are unstable due to the long-wave instability of the flat surface, the dynamics of KS–KdV is dominated by travelling pulse trains of $O(1)$ amplitude. As a step toward explaining the persistence of pulses and understanding their interactions, we prove that for $n=1$ and $2$ the KdV manifolds of $n$-solitons are stable in KS–KdV on an $O(1/\varepsilon)$ time-scale with respect to $O(1)$ perturbations in $H^n(\mathbb{R})$.

In this paper, we study the self-similar solutions and the time-asymptotic behaviour of solutions for a class of degenerate and singular diffusion equations in the form

$$ u_t=(|(p(u))_x|^{\lambda-2}(p(u))_x)_x,\quad -\infty<x<+\infty,\quad t>0, $$

where $\lambda>2$ is a constant. The existence, uniqueness and regularity for the self-similar solutions are obtained. In particular, the behaviour at two end points is discussed. Based on the monotonicity property of the self-similar solutions and the comparison principle, we also investigate the time convergence of the solution for the Cauchy problem to the corresponding self-similar solution when the initial data have some decay in space variable.

We describe a new general method for the computation of the group Aut(X) of self-homotopy equivalences of a space. It is based on the decomposition of Aut(X) induced by a factorization of X into a product of simpler spaces. Normally, such decompositions require assumptions (‘induced equivalence property’, ‘diagonalizability’), which are strongly restrictive and difficult to check. We derive computable homological criteria for an analogous assumption, called reducibility, and then show that these criteria are satisfied when the so-called atomic decomposition of the space is used. This essentially reduces the computation of Aut(X) to the computation of the group of self-equivalences of its atomic factors, and the computation of certain homotopy sets between those factors.

We derive geometrical information on smooth surfaces in $\mathbb{R}^4$ from the geometry of their images under linear projections to 3-spaces.

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, I prove that on every fractal structure with three vertices, for suitable weights, the renormalization operator has an eigenform.

This paper is devoted to the study of time-periodic solutions to the nonlinear one-dimensional wave equation with x-dependent coefficients u(x)ytt – (u(x)yx)x + g(x,t,y) = f(x,t) on (0,π) × ℝ under the periodic boundary conditions y(0,t) = y(π,t), yx(0,t) = yx(π,t) or anti-periodic boundary conditions y(0, t) = –y(π,t), yx[0,t) = – yx(π,t). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. Our main concept is that of the ‘weak solution’. For T, the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.

In this paper, we consider the following non-homogeneous elliptic problem

\begin{gather*} -\Delta u+u=\lambda(u^{p-1}+h(x))\text{ in }\varOmega, \\ u>0\text{ in }\varOmega,\quad u\in H_0^{1}(\varOmega), \end{gather*}

where $2<p<2N/(N-2)$, $N\geq3$, $\lambda>0$, $\varOmega=\bar\omega^{c}$ is an exterior domain in $\mathbb{R}^N$, where $\omega$ is a bounded set with smooth boundary, $h\in L^{2}(\varOmega)\cap L^{\beta}(\varOmega)$ ($\beta>\tfrac{1}{2}N$ if $N\geq4$ and $\beta=\tfrac{1}{2}N$ if $N=3$) is non-negative and $h(x)\not\equiv0$. We use variational methods to show that there exists a positive number $\lambda_0$ such that the equation above has at least two positive solutions if $\lambda\in(0,\lambda_0)$, no positive solution if $\lambda>\lambda_0$ and at least one positive solution if $\lambda=\lambda_0$. Furthermore, we use the Lyusternik–Schnirelman category to show that there exists a positive number $\lambda_{\ast}<\lambda_0$ such that the above equation has at least three positive solutions if $\lambda\in(0,\lambda_{\ast})$.

We consider the nonlinear elliptic eigenvalue problem $-\Delta u+u^p=\lambda u$ in $B_R$, $u>0$ in $B_R$, $u=0$ on $\partial B_R$, where $B_R:=\{|x|<R\}\subset\mathbb{R}^N$ ($N\ge2$) and $p>1$ is a constant. This equation is well known as a model equation of population density for some species when $p=2$. Here, $\lambda>0$ represents the reciprocal number of its diffusion rate and $\Vert u\Vert_1$ stands for the mass of the species. We establish the precise asymptotic formula for $\Vert u_\lambda\Vert_q$ as $\lambda\to\infty$, where $1\le q<\infty$. We also obtain the difference between $\Vert u_\lambda\Vert_\infty$ and $\Vert u_\lambda\Vert_q$ when $\lambda\gg1$.

We study homogenization processes for the heat equation in multilayers with interlayer conduction, modelled by Neumann transmission conditions. We establish the homogenized equations for three kinds of dependency of the interlayer conduction magnitude upon the interlayer distance.

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.

We characterize a topological convex space $C$ in terms of the family $\mathcal{A}(C)$ of real continuous affine functions on $C$. Our main result states that two topological convex spaces $C_1$ and $C_2$ are affine-homeomorphic if and only if $\mathcal{A}(C_1)$ and $\mathcal{A}(C_2)$ are isomorphic as ordered unital vector spaces.