We study the class of nonlinear Klein–Gordon–Maxwell systems describing a standing wave (charged matter field) in equilibrium with a purely electrostatic field. We improve some previous existence results in the case of an homogeneous nonlinearity. Moreover, we deal with a limit case, namely when the frequency of the standing wave is equal to the mass of the charged field; this case shows analogous features of the well-known ‘zero-mass case’ for scalar field equations.

Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood (see the works of Emel'yanov, Wolff and Zaharopol). For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it is shown here that (amongst other things) every power-bounded linear operator on E is mean ergodic if and only if E is reflexive if and only if E is Dedekind σ-complete and every positive power-bounded operator on E is mean ergodic if and only if every positive power-bounded operator in the strong dual E′β (no longer a Fréchet lattice) is mean ergodic. An important technique is to develop criteria that detect when E admits a (positively) complemented lattice copy of c0, l1 or l∞.

We consider a purely quasi-incompressible elasticity model. We rigorously establish asymptotic expansions of near- and far-field measurements of the transient elastic wave induced by a small elastic anomaly. Our proof uses layer potential techniques for the modified Stokes system. Based on these formulae, we design asymptotic imaging methods leading to a quantitative estimation of elastic and geometrical parameters of the anomaly.

The relation between quasi-convexity and k-quasi-convexity, k ≥ 2, is investigated. It is shown that every smooth strictly k-quasi-convex integrand with p-growth at infinity, p > 1, is the restriction to kth-order symmetric tensors of a quasi-convex function with the same growth. When the smoothness condition is dropped, it is possible to prove an approximation result. As a consequence, lower semicontinuity results for kth-order variational problems are deduced as corollaries of well-known first-order theorems. This generalizes a previous work by Dal Maso et al., in which the case where k = 2 was treated.

We give sufficient conditions on two dyadic systems to obtain the equivalence of corresponding Haar systems on dyadic weighted Lebesgue spaces on spaces of homogeneous type. In order to obtain these results, we prove a Fefferman–Stein weighted inequality for vector-valued dyadic Hardy–Littlewood maximal operators with dyadic weights in this general setting.

We give an elementary proof of the unique, global-in-time solvability of the coagulation–(multiple) fragmentation equation with polynomially bounded fragmentation and particle production rates and a bounded coagulation rate. The proof relies on a new result concerning domain invariance for the fragmentation semigroup which is based on a simple monotonicity argument.

We consider the stability of periodic travelling-wave solutions to a generalized Korteweg–de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L2-critical KdV-4 equation that arises in the study of blow-up and the KdV-½ equation, which is an idealized model for plasmas.

We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce an augmented formulation that allows for the modelling of the dynamics of interfaces between fluid flows. The main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is observed for the initial-value problem, and these solutions need to be supplemented with further admissibility conditions. This paper is devoted to investigating these issues in the setting of self-similar vanishing viscosity approximations to the Riemann problem for general hyperbolic systems. Following earlier works by Joseph, LeFloch and Tzavaras, we establish an existence theorem for the Riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization. Our main contribution consists of nonlinear wave interaction estimates for solutions that apply to resonant wave patterns.

Famous results by Rademacher, Kolmogorov and Paley and Zygmund state that random series on the torus enjoy better Lp bounds that the deterministic bounds. We present a natural extension of these harmonic analysis results to a partial-differential-equations setting. Specifically, we consider the one-dimensional nonlinear harmonic oscillator i∂tu + Δu − |x|2u = |u|r−1u, and exhibit examples for which the solutions are better behaved for randomly chosen initial data than would be predicted by the deterministic theory. In particular, on a deterministic point of view, the nonlinear harmonic oscillator equation is well posed in L2(ℝ) if and only if r ≤ 5. However, we shall prove that, for all nonlinearities |u|r−1u, r > 1, not only is the equation well posed for a large set of initial data whose Sobolev regularity is below L2, but also the flows enjoy very nice large-time probabilistic behaviour.

These results are joint work with Laurent Thomann and Nikolay Tzvetkov.

A logistic equation with infinite delay is considered under conditions that force its solution to approach a positive steady state at large times. It is shown that this rate of convergence depends on the initial history in some cases, and is independent of the history in others.

We consider the Lp norms of sums of characteristic functions of affine subspaces of a vector space V over a finite field under certain restrictions on p, dim V and the dimensions of the subspaces involved. We investigate the conditions under which these norms are increased when the affine subspaces are replaced by their parallel translates passing through 0. Applications to extremal configurations for Kakeya maximal-type inequalities are given and open questions are raised.

After a brief introduction and physical motivation, we show how the nonlinear Schrödinger (NLS) equation can be derived from a general class of nonlinear hyperbolic systems. Its purpose is to describe the behaviour of high-frequency oscillating wave packets over a large time-scale that requires us to take into account diffractive effects. We then show that the NLS approximation fails for short pulses and propose some alternative models, including a modified Schrödinger equation with improved frequency dispersion. It turns out that these models have better properties and are quite accurate for short pulses. For ultrashort pulses, however, they must also be abandoned for more complex approaches. We give the main steps for such an analysis and explain one striking fact about ultrashort pulses: their dynamics in dispersive media is linear.

The existence, stability and uniqueness of positive solutions to a semilinear elliptic system with sublinear nonlinearities are proved. It is shown that the precise global bifurcation diagram of the positive solutions is a monotone curve with different asymptotical behaviour according to the form of the nonlinearities. Equations with Hölder continuous nonlinearities are also considered.

The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

A method is proposed for reducing autonomous nonlinear boundary-value problems of partial differential equations with a particular structure and constant boundary conditions to the simple problem

This requires the solution of a non-standard two-point system of ordinary differential equations. Some applications are also presented for classical problems of mathematical physics.

This paper deals with the non-negative boundary blow-up solutions of the equation ∆u = b(x)up + c(x)uσ|∇u|q in Ω ⊂ ℝ,N, where b(x), c(x) ∈ Cγ (Ω,ℝ+) for some 0 < γ < 1 and can be vanishing or singular on the boundary, and p, σ and q are non-negative constants. The existence and asymptotic behaviour of such a solution near the boundary are investigated, and we show how the nonlinear gradient term affects the results. As a consequence of the asymptotic behaviour, we also show the uniqueness result.

We consider the equation

where

Under these conditions, (1) is correctly solvable in L1(ℝ), i.e.

(i) for any function f ∈ L1(ℝ), there exists a unique solution of (1), y ∈ L1(ℝ);

(ii) there is an absolute constant c1 ∈ (0, ∞) such that the solution of (1),

In this work we strengthen the a priori inequality (1). We find minimal requirements for a given weight function θ ∈ Lloc1(ℝ) under which the solution of (1), y ∈ L1(ℝ), satisfies the estimate

where c2 is some absolutely positive constant.

We consider a time-varying inclusion in a thermal conductor specimen. In particular, the thermal conductivity is a variable function depending on space and time with a jump of discontinuity along the interface of the unknown anomalous region. Provided with some a priori information on the conductivity and its support, we study the continuous dependence of the inclusion from infinitely many thermal measurements taken on an open portion of the boundary of our specimen. We prove a rate of continuity of logarithmic type showing, in addition, its optimality.

We review recent nonlinear partial differential equation techniques developed to address questions concerning solitons for the quartic generalized Korteweg—de Vries equation (gKdV) and other generalizations of the KdV equation. We draw a comparison between results obtained in this way and some elements of the classical integrability theory for the original KdV equation, which serve as a reference for soliton and multi-soliton problems. First, known results on stability and asymptotic stability of solitons for gKdV equations are reviewed from several different sources. Second, we consider the problem of the interaction of two solitons for the quartic gKdV equation. We focus on recent results and techniques from a previous paper by the present authors concerning the interaction of two almost-equal solitons.