In the first part, we study several exterior boundary-value problems covering three types of semilinear equations: elliptic, parabolic and hyperbolic. By a unified approach, we show that these problems share a common critical behaviour. In the second part we prove a blow-up result for an inhomogeneous porous medium equation with the critical exponent, which was left open in a previous paper.

By a new method it is proved that a non-linear elliptic boundary value problem of rather general type admits a weak solution lying between a given weak lower solution ϕ and a given weak upper solution ψ≧ϕ

This paper is devoted to the study of Nicholson's blowflies equation with diffusion: a kind of time-delayed reaction diffusion. For any travelling wavefront with speed c > c* (c* is the minimum wave speed), we prove that the wavefront is time-asymptotically stable when the delay-time is sufficiently small, and the initial perturbation around the wavefront decays to zero exponentially in space as x → −∞, but it can be large in other locations. The result develops and improves the previous wave stability obtained by Mei et al. in 2004. The new approach developed in this paper is the comparison principle combined with the technical weighted-energy method. Numerical simulations are also carried out to confirm our theoretical results.

We give an explicit expression for the quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function in terms of the singular values. This envelope is also the convex, polyconvex and rank 1 convex envelope of the Saint Venant–Kirchhoff stored energy function. Moreover, it coincides with the Saint Venant–Kirchhoff stored energy function itself on, and only on, the set of matrices whose singular values arranged in increasing order are located outside an ellipsoid. It vanishes on, and only on, the set of matrices whose singular values are less than 1. Consequently, a Saint Venant–Kirchhoff material can be compressed under zero external loading.

This paper is concerned with the asymptotic behaviour of solutions to a class of non-autonomous stochastic nonlinear wave equations with dispersive and viscosity dissipative terms driven by operator-type noise defined on the entire space $\mathbb {R}^n$. The existence, uniqueness, time-semi-uniform compactness and asymptotically autonomous robustness of pullback random attractors are proved in $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$ when the growth rate of the nonlinearity has a subcritical range, the density of the noise is suitably controllable, and the time-dependent force converges to a time-independent function in some sense. The main difficulty to establish the time-semi-uniform pullback asymptotic compactness of the solutions in $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$ is caused by the lack of compact Sobolev embeddings on $\mathbb {R}^n$, as well as the weak dissipativeness of the equations is surmounted at light of the idea of uniform tail-estimates and a spectral decomposition approach. The measurability of random attractors is proved by using an argument which considers two attracting universes developed by Wang and Li (Phys. D 382: 46–57, 2018).

In this paper we study travelling front solutions of a certain food-limited population model incorporating time-delays and diffusion. Special attention is paid to the modelling of the time delays to incorporate associated non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. For a particular class of delay kernels, existence of travelling front solutions connecting the two spatially uniform steady states is established for sufficiently small delays. The approach is to reformulate the problem as an existence question for a heteroclinic connection in R4. The problem is then tackled using dynamical systems techniques, in particular, Fenichel's invariant manifold theory. For larger delays, numerical simulations reveal changes in the front's profile which develops a prominent hump.

We consider the inverse-scattering problem of determining the shape of a partly coated obstacle in R3 from a knowledge of the incident time-harmonic electromagnetic plane wave and the electric far-field pattern of the scattered wave. A justification is given of the linear sampling method in this case and numerical examples are provided showing the practicality of our method.

The investigation of general quasi-adequate semigroups is initiated. These are semigroups which are abundant and in which the idempotents form a subsemigroup. For such a semigroup S we study the minimum good congruence γ such that S/γ is adequate. Results on γ together with results from a previous paper of the authors are used to obtain a structure theorem for a class of quasi-adequate semigroups.

The discrete coagulation-fragmentation equations are a model for the time-evolution of cluster growth. The processes described by the model are the coagulation of clusters via binary interactions and the fragmentation of clusters. The assumptions made on the fragmentation coefficients in this paper have the physical interpretation that surface effects are not important, i.e. it is unlikely that a large cluster will fragment into two large pieces. Since solutions of the initial-value problem are not unique, we have to restrict the class of solutions. With this restriction, we prove that the fragmentation acts as a strong damping mechanism and we obtain results on the asymptotic behaviour of solutions. The main tool used is an estimate on the moments of admissible solutions.

Consider the reaction-diffusion equation in ℝN × ℝ+: ut − h2 Δu + Φ(u) = 0, where Φ is the derivative of a bistable even potential, and h is a small parameter. If the initial data have a smooth noncritical zero set, we prove that an interface appears in time O(log (h−1)), and that the solution stays close to it for at least time O(1/√h).

The main theorem states that a bounded linear operator $h$ from a unital $C^{\ast}$-algebra $A$ into a unital Banach algebra $B$ must be a homomorphism provided that $h(\bm{1})=\bm{1}$ and the following condition holds: if $x,y,z\in A$ are such that $xy=yz=0$, then $h(x)h(y)h(z)=0$. This theorem covers various known results; in particular it yields Johnson's theorem on local derivations.

We consider an infinite planar straight strip perforated by small holes along a curve. In such a domain, we consider a general second-order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm-resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm-resolvent convergence, we prove the convergence of the spectrum.

Semilinear parabolic equations of the form u1 = ∇2u + δf(u), where f is positive and is finite, are known to exhibit the phenomenon of blow-up, i.e. for sufficiently large S, u becomes infinite after a finite time t*. We consider one-dimensional problems in the semi-infinite region x>0 and find the time to blow-up (t*). Also, the limiting behaviour of u as t→t*- and x→∞ is determined; in particular, it is seen that u blows up at infinity, i.e. for any given finite x, u is bounded as t→t*. The results are extended to problems with convection.

The modified equation xu, = uxx +f(u) is discussed. This shows the possibility of blow-up at x =0 even if u(0, f) = 0. The manner of blow-up is estimated.

Finally, bounds on the time to blow-up for problems in finite regions are obtained by comparing u with upper and lower solutions.

The dynamical behaviour of a slender rod is analyzed here in terms of a generalization of Euler's elastica theory. The model includes a linear stress-strain relation but nonlinear geometric terms. Properties of the rod may vary along its length and various boundary conditions are considered. A rotational inertia term that is neglected in many theories is retained, and is essential to the analysis. By use of the equivalence of an energy and a Sobolev norm, and by reformulation of the equations as a semilinear system, global existence of solutions is proved for any smooth initial data. Equilibrium solutions that are stable in the static sense of minimizing the potential energy are then proved to be stable in the dynamic sense due to Liapounov.

We consider a common abstraction of de Morgan algebras and Stone algebras which we call an MS-algebra. The variety of MS-algebras is easily described by adjoining only three simple equations to the axioms for a bounded distributive lattice. We first investigate the elementary properties of these algebras, then we characterise the least congruence which collapses all the elements of an ideal, and those ideals which are congruence kernels. We introduce a congruence which is similar to the Glivenko congruence in a p-algebra and show that the location of this congruence in the lattice of congruences is closely related to the subdirect irreducibility of the algebra. Finally, we give a complete description of the subdirectly irreducible MS-algebras.

The paper presents a list of unsolved problems about operators on Hilbert space, accompanied by just enough definitions and general discussion to set the problems in a reasonable context. The subjects are: quasitriangular matrices, the resemblances between normal and Toeplitz operators, dilation theory, the algebra of shifts, some special invariant subspaces, the category (in the sense of Baire) of the set of non-cyclicoperators, non-commutative(i.e. operator) approximation theory, infinitary operators, and the possibility of attacking invariance problems by compactness or convexity arguments.

We consider a class of measure-valued Markov processes constructed by taking a superprocess over some underlying Markov process and conditioning it to stay alive forever. We obtain two representations of such a process. The first representation is in terms of an “immortal particle” that moves around according to the underlying Markov process and throws off pieces of mass, which then proceed to evolve in the same way that mass evolves for the unconditioned superprocess. As a consequence of this representation, we show that the tail σ-field of the conditioned superprocess is trivial if the tail σ-field of the underlying process is trivial. The second representation is analogous to one obtained by LeGall in the unconditioned case. It represents the conditioned superprocess in terms of a certain process taking values in the path space of the underlying process. This representation is useful for studying the “transience” and “recurrence” properties of the closed support process.

For scalar variational problems

subject to linear boundary values, we determine completely those integrands W: ℝn → ℝ for which the minimum is not attained, thereby completing previous efforts such as a recent nonexistence theorem of Chipot [9] and unifying a large number of examples and counterexamples in the literature.

As a corollary, we show that in case of nonattainment (and provided W grows superlinearly at infinity), every minimising sequence converges weakly but not strongly in W1,1(Ω) to a unique limit, namely the linear deformation prescribed at the boundary, and develops fine structure everywhere in Ω, that is to say every Young measure associated with the sequence of its gradients is almost-nowhere a Dirac mass.

Connections with solid–solid phase transformations are indicated.

Let Ω ⊂ Rn be open. Given a homeomorphism of finite distortion with |Df| in the Lorentz space Ln−1, 1 (Ω), we show that and f−1 has finite distortion. A class of counterexamples demonstrating sharpness of the results is constructed.

A nonlocal eigenvalue problem of the form u″ + a(x)u + Bu = λu with homogeneous Dirichlet boundary conditions is considered, where B is a rank-one bounded linear operator and x belongs to some bounded interval on the real line. The behaviour of the eigenvalues is studied using methods of linear perturbation theory. In particular, some results are given which ensure that the spectrum remains real. A Sturm-type comparison result is obtained. Finally, these results are applied to the study of some nonlocal reaction–diffusion equations.