The paper is concerned with a non-local time-delayed reaction–diffusion equation. We prove the (time) asymptotic stability of a travelling wavefront without a smallness assumption on its wavelength, i.e. the so-called strong wavefront, by means of the (technical) weighted energy method, when the initial perturbation around the wave is small. The exponential convergent rate is also given. Selection of a suitable weight plays a key role in the proof.

Motivated by the aim of modelling the mechanical behaviour of biological gels (such as collagen gels) which have a fibrous microstructure, we consider the extensional flow of a thin two-dimensional film of incompressible, transversely isotropic viscous fluid. Neglecting inertia, and the effects of gravity and surface tension, leading-order equations are derived from a perturbation expansion of the full flow problem in powers of the (small) inverse aspect ratio. The existence and uniqueness of the solution of the reduced system of equations for small times is then proven. Special cases, in which the solution may be determined explicitly, are considered and we discuss the physical interpretation of the results.

We study the evolution of solutions to the initial-boundary-value problem

\begin{alignat*}{3} u_t&=(u^m)_{xx}+\lambda(u^q)_x, & \quad x&>0,&\quad t&\in(0,T), \\[2pt] -(u^m)_x(0,t)&=u^p(0,t), & &&t&\in(0,T), \\[2pt] u(x,0)&=u_0(x), & \quad x&>0,\end{alignat*}

and give a rather complete characterization, in terms of the parameters $m\ge1$, $p,q>0$ and $\lambda>0$, of whether all solutions are global in time or, on the contrary, there exist blow-up solutions. We show that the presence of the convective term has a preventive effect on the blow-up (with respect to the case $\lambda=0$) and gives rise to a collapse of the region where all solutions blow up in this case. On the other hand, a new Fujita-type phenomenon takes place at the level $p=q$ and $0<\lambda<1$.

A system of differential equations describing the joint motion of thermo-elastic porous body and slightly compressible viscous thermofluid occupying pore space is considered. Although the problem is correct in an appropriate functional space, it is very hard to tackle due to the fact that its main differential equations involve non-smooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification under various conditions imposed on physical parameters is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As a result, we derive Biot's system of equations of thermo-poroelasticity, a similar system, consisting of anisotropic Lamé equations for a thermoelastic solid coupled with acoustic equations for a thermofluid, Darcy's system of filtration, or acoustic equations for a thermofluid, according to ratios between physical parameters. The proofs are based on Nguetseng's two-scale convergence method of homogenization in periodic structures.

In the analysis of stability in bifurcation problems it is often assumed that the (appropriate reduced) equations are in normal form. In the presence of symmetry, the truncated normal form is an equivariant polynomial map. Therefore, the determination of invariants and equivariants of the group of symmetries of the problem is an important step. In general, these are hard problems of invariant theory and, in most cases, they are tractable only through symbolic computer programs. Nevertheless, it is desirable to obtain some of the information about invariants and equivariants without actually computing them, for example, the number of linearly independent homogeneous invariants or equivariants of a certain degree. Generating functions for these dimensions are generally known as ‘Molien functions'.

We obtain formulae for the number of linearly independent homogeneous invariants or equivariants for Hopf bifurcation in terms of characters. We also show how to construct Molien functions for invariants and equivariants for Hopf bifurcation. Our results are then applied to the computation of the number of invariants and equivariants for Hopf bifurcation for several finite groups and the continuous group $\mathbb{O}(3)$.

We study the coupled surface and grain boundary motion in a bi-crystal in the context of the ‘quarter loop’ geometry. Two types of normal curve velocities are involved in this model: motion by mean curvature and motion by surface diffusion. Three curves meet at points where junction conditions are given. A formulation that describes the coupled normal motion of the curves and preserves arc length parametrisation up to scaling is proposed. The formulation is shown to be well-posed in a simple, linear setting. Equations and junction conditions are approximated by finite difference methods. Numerical convergence to exact travelling wave solutions is shown. The method is applied to other problems of physical interest.

Linear stability theory is developed for an activator–inhibitor model where fractional derivative operators of generally different exponents act both on diffusion and reaction terms. It is shown that in the short wave limit the growth rate is a power law of the wave number with decoupled time scales for distinct anomaly exponents of the different species. With equal anomaly exponents an exact formula for the anomalous critical value of reactants diffusion coefficients' ratio is obtained.

We consider the following problem:

\begin{alignat*}{2}-\text{div}(A(x,u)\nabla u)&=u^s+f(x) & \quad &\text{in }\varOmega, \\ u(x)&\ge0 & & \text{in }\varOmega, \\ u(x)&=0 & & \text{on }\p\varOmega,\end{alignat*}

where $\varOmega$ is an open bounded subset of $\mathbb{R}^N$, $N\ge3$, and

$$A:\varOmega\times\mathbb{R}\rightarrow M_{N\times N}$$

is an elliptic matrix such that when $u\to\infty$ is non-coercive.

In this paper, we study the multiplicity of positive solutions for the following semilinear elliptic equation:

\begin{alignat*}{2} -\Delta u+\lambda u&=f(x)u^{p-1}+h(x)u^{q-1} &\quad&\text{in }\mathbb{R}^{N}, \\ u&>0 &&\text{in }\mathbb{R}^{N}, \\ u&\in H^{1}(\mathbb{R}^{N}),\end{alignat*}

where $1\leq q<2<p<2^{\ast}$ ($2^{\ast}=2N/(N-2)$ if $N\geq3$ and $2^{\ast}=\infty$ if $N=1,2$), $\lambda>0,h\in L^{2/(2-q)}(\mathbb{R}^{N})\setminus\{0\}$ is non-negative and $f\in C(\mathbb{R}^{N})$. We will show how the shape of the graph of $f(x)$ affects the number of positive solutions.

We consider a single-species structured population with distributed maturity and spatial diffusion in a cylindrical domain subject to Neumann and Robin boundary conditions. We first establish the threshold property of the reaction–diffusion system with distributed delay and non-local interaction in a corresponding lower-dimensional domain, so that the system approaches either an extinction state or a stable spatially varying pattern. We then investigate the transition from the extinction state to the stable pattern of the system in the cylindrical domain.

In this article, we show that the magnetohydrodynamic system in $\mathbb{R}^N$ with variable density, variable viscosity and variable conductivity has a local weak solution in the Besov space $\dot{B}^{N/p_1}_{p_1,1}(\mathbb{R}^N)\times\dot{B}^{(N/p_2)-1}_{p_2,1}(\mathbb{R}^N) \times\dot{B}^{(N/p_2)-1}_{p_2,1}(\mathbb{R}^N)$ for all $1<p_2<+\infty$ and some $1<p_1\leq2N/3$ if the initial density approaches a positive constant. Moreover, this solution is unique if we impose the restrictive condition $1<p_2\leq2N$. We also prove that the constructed solution exists globally in time if the initial data are small. In particular, this allows us to work in the framework of Besov space with negative regularity indices and this fact is particularly important when the initial data are strongly oscillating.

In this paper, assume that $q$ is a positive continuous function in $\mathbb{R}^{N}$ satisfying suitable conditions. We prove that the Dirichlet problem $-\Delta u+u=q(z)|u|^{p-2}u$ in an exterior domain admits at least two positive solutions and a solution which changes sign.

The study of the ring of all formal series $a_{0}+a_{1}\binom{x}{1}+a_{2} \binom{x}{2}+\cdots$ with integer coefficients, denoted by $\mathbb{Z}[\hspace{-1.6pt}[\binom{x}{1},\binom{x}{2},\dots]\hspace{-1.6pt}]$, or $\mathbb{Z}[\hspace{-1.6pt}[\binom{x}{n}]\hspace{-1.6pt}]_{n\geq0}$ for short, is motivated by the elementary number theoretical properties of the binomial coefficients. The binomial polynomials as well as the binomial coefficients and their generalizations can be found in different branches of mathematics, e.g. in algebra, analysis, combinatorics and in topology. The question of finding the remainder when dividing $\binom{n}{k}$ by a prime (Lucas's 1878 theorem) leads to base-$p$ expansions in the binomial coefficients and the consideration of integer-valued polynomials with rational coefficients. And although the study of these polynomials dates back to the seventeenth century, the study of this set as a ring began in 1936 with Skolem. More generally, the bijective correspondence between the set of functions defined on the set of non-negative integers and the series $a_{0}+a_{1}\binom{x}{1}+a_{2}\binom{x}{2}+\cdots$ is used by Mahler in the field of $p$-adic analysis and naturally leads to the expansion of Skolem's approach and the definition of $\mathbb{Z}[\hspace{-1.6pt}[\binom{x}{1},\binom{x}{2},\dots]\hspace{-1.6pt}]$ or in fact of $R[\hspace{-1.6pt}[\binom{x}{1},\binom{x}{2},\dots]\hspace{-1.6pt}]$ (with $R$ any ring). Iwasawa used such series in connection with the $p$-adic $L$-functions but without considering the ring.

We consider a least-energy solution to a nonlinear elliptic equation arising in a nonlinear optics model. The equation is linear in a given bounded domain $D$ and nonlinear in $\mathbb{R}^N\setminus D$. We consider a singular limiting problem to obtain a relation between a location of a spiky solution and a geometry of $D$.