The aim of the paper is to introduce and investigate a dynamical system which consists of a variational–hemivariational inequality of hyperbolic type combined with a non-linear evolution equation. Such a dynamical system arises in studies of complicated contact problems in mechanics. Existence, uniqueness and regularity of a global solution to the system are established. The approach is based on a new semi-discrete approximation with an application of a surjectivity result for a pseudomonotone perturbation of a maximal monotone operator. A new dynamic viscoelastic frictional contact model with adhesion is studied as an application, in which the contact boundary condition is described by a generalised normal damped response condition with unilateral constraint and a multivalued frictional contact law.

Adhesion between cells and other cells (cell–cell adhesion) or other tissue components (cell–matrix adhesion) is an intrinsically non-local phenomenon. Consequently, a number of recently developed mathematical models for cell adhesion have taken the form of non-local partial differential equations, where the non-local term arises inside a spatial derivative. The mathematical properties of such a non-local gradient term are not yet well understood. Here we use sophisticated estimation techniques to show local and global existence of classical solutions for such examples of adhesion-type models, and we provide a uniform upper bound for the solutions. Further, we discuss the significance of these results to applications in cell sorting and in cancer invasion and support the theoretical results through numerical simulations.

Set differential equations are usually formulated in terms of the Hukuhara differential. As a consequence, the theory of set differential equations is perceived as an independent subject, in which all results are proved within the framework of the Hukuhara calculus. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of ℝd with their support functions. Using this representation, standard existence and uniqueness theorems for ordinary differential equations can be applied to set differential equations. We provide a geometric interpretation of the main result, and demonstrate that our approach overcomes the heavy restrictions that the use of the Hukuhara differential implies for the nature of a solution.

We study a problem related to the spin-coating process in which a fluid coats a rotating surface. Our interest lies in the contact-line region for which we propose a simplified travelling wave approximation. We construct solutions to this problem by a shooting method that matches solution branches in the contact-line region and in the interior of the droplet. Furthermore, we prove uniqueness and qualitative properties of the solution connected to the fourth-order nature of the equation, such as a global maximum in the film height close to the contact line, elevated from the average height of the film.

This study deals with the existence and uniqueness of solutions to dynamical problems of finite freedom involving unilateral contact and Coulomb friction. In the frictionless case, it has been established [P. Ballard, Arch. Rational Mech. Anal. 154 (2000) 199–274] that the existence and uniqueness of a solution to the Cauchy problem can be proved under the assumption that the data are analytic, but not if they are assumed to be only of class C∞. Some years ago, this finding was extended [P. Ballard and S. Basseville, Math. Model. Numer. Anal. 39 (2005) 59–77] to the case where Coulomb friction is included in a model problem involving a single point particle. In the present paper, the existence and uniqueness of a solution to the Cauchy problem is proved in the case of a finite collection of particles in (possibly non-linear) interactions.

This work models and analyses the dynamics of a general spring-mass-damper system that is in frictional contact with its support, taking into account frictional heat generation and a reactive obstacle. Friction, heat generation and contact are modelled with subdifferentials of, possibly non-convex, potential functions. The model consists of a non-linear system of first-order differential inclusions for the position, velocity and temperature of the mass. The existence of a global solution is established and additional assumptions yield its uniqueness. Nine examples of conditions arising in applications, for which the analysis results are valid, are presented.

We consider a class of quasi-variational inequalities arising in a large number of mathematical models, which describe quasi-static processes of contact between a deformable body and an obstacle, the so-called foundation. The novelty lies in the special structure of these inequalities that involve a history-dependent term as well as in the fact that the inequalities are formulated on the unbounded interval of time [0, +∞). We prove an existence and uniqueness result of the solution, then we complete it with a regularity result. The proofs are based on arguments of monotonicity and convexity, combined with a fixed point result obtained in [22]. We also describe a number of quasi-static frictional contact problems in which we model the material's behaviour with an elastic or viscoelastic constitutive law. The contact is modelled with normal compliance, with normal damped response or with the Signorini condition, as well, associated to versions of Coulomb's law of dry friction or to the frictionless condition. We prove that all these models cast in the abstract setting of history-dependent quasi-variational inequalities, with a convenient choice of spaces and operators. Then, we apply the abstract results in order to prove the unique weak solvability of each contact problem.

In this article, we show the convergence of a class of numerical schemes for certain maximal monotone evolution systems; a by-product of this resultsis the existence of solutions in cases which had not been previouslytreated. The order of these schemes is 1/2 in general and 1 when the only non Lipschitz continuous term is the subdifferential of the indicatrix of a closed convex set. In the case of Prandtl'srheological model, our estimates in maximum norm do not dependon spatial dimension.

A method originating from statistical mechanics (low-density and high-temperature expansions) is used to prove the existence and uniqueness of a stationary regime for switching networks on finite or infinite graphs. The main assumption is that the message (customer) flows circulating through the network are ‘localized' in the sense that, for any message, the probability of having a long path is rapidly decreasing (and, moreover, a path of a ‘typical' message consists of one line). The switching rule combines message-switching and circuit-switching principles. The stationary regime for the network under consideration may be treated as a ‘small perturbation' of the ‘idealized' regime in the totally decoupled network where all the messages have single line paths.