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Consider the Cauchy problem u t − u x x − F ( u ) = 0 ; x ∈ ℝ , t > 0 u ( x , 0 ) = u 0 ( x ) ; x ∈ ℝ where u 0 ( x ) is continuous, nonnegative and bounded, and F( u ) = u p with p > 1, or F( u ) = e u . Assume that u blows up at x = 0 and t = T > 0. In this paper we shall describe the various possible asymptotic behaviours of u ( x , t ) as ( x , t ) → (0, T). Moreover, we shall show that if u 0 ( x ) has a single maximum at x = 0 and is symmetric, u 0 ( x ) = u 0 (− x ) for x > 0, there holds 1) If F( u ) = u p with p > 1, then lim t ↑ T u ( ξ ( ( T − t ) | log ( T − t ) | ) 1 / 2 , t ) × ( T − t ) 1 / ( p − 1 ) = ( p − 1 ) − ( 1 / ( p − 1 ) ) [ 1 + ( p − 1 ) ξ 2 4 p ] − ( 1 / ( p − 1 ) ) uniformly on compact sets |ξ| ≦ R with R > 0, 2) If F( u ) = e u , then lim t ↑ T ( u ( ξ ( ( T − t ) | log ( T − t ) | ) 1 / 2 , t ) + log ( T − t ) ) = − log [ 1 + ξ 2 4 ] uniformly on compact sets |ξ| ≦ R with R > 0. [ABSTRACT FROM AUTHOR]

A system of particles, in general d-dimensional space, that interact by means of pair potentials and adjust their positions according to the gradient flow dynamics induced by the total energy of the system is studied. The case when the range of the interaction is of the same order as the mean interparticle distance is considered. It is also assumed that particles, locally, are located close to some crystallographic lattice. An appropriate system of equations that describes the evolution of macroscopic deformation of the crystallographic lattice, as well as the system that describes the evolution of the main crystallographic directions is derived. Well-posedness of the derived system is studied as well as the stability of the particle system. Same examples of potentials that yield stable and unstable systems are given. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this note we derive a model describing the two-dimensional viscous flow driven by surface tension and gravity of a thin liquid film near a stagnation point. In the thin-film approximation of such a flow, accumulation takes place where the combined effects of gravity and surface tension stop the flow. These stagnation points are characterised to leading order by the geometry of the substrate. We first derive the thin-film approximation that describes the flow away from such accumulation regions. Then, assuming the existence of isolated stagnation points, we derive the boundary layer equation describing the inner structure of solutions describing accumulation. The existence of these solutions has been proved by the authors elsewhere. Finally, in order to justify the model we prove the existence of curves that give a substrate with an isolated stagnation point. [Copyright &y& Elsevier]

Abstract: In this paper we construct classical solutions of a family of coagulation equations with homogeneous kernels that exhibit the behaviour known as gelation. This behaviour consists in the loss of mass due to the fact that some of the particles can become infinitely large in finite time. [Copyright &y& Elsevier]

The purpose of this paper is to prove well-posedness for a problem that describes the dynamics of a set of points by means of a system of parabolic equations. It has been seen in Velázquez (Point dynamics in a singular limit of the Keller–Segel model. (1) motion of the concentration regions, SIAM J. Appl. Math., to appear) that the considered model is the limit of a singular perturbation problem for a system of the Keller–Segel type. [Copyright &y& Elsevier]

In this paper, the stability of a blow-up mechanism for a particular Keller-Segel model that yields chemotactic aggregation is considered. It is shown using matched asymptotic methods that the considered blow-up mechanism is stable. More precisely, small perturbations of the initial data produce a small shifting on the blow-up time and the blow-up point but not modification in the blow-up mechanism. [ABSTRACT FROM AUTHOR]

We show the existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation for homogeneous kernels satisfying C 1 ( x − a y b + x b y − a ) ≤ K ( x , y ) ≤ C 2 ( x − a y b + x b y − a ) with a > 0 and b < 1 . This covers especially the case of Smoluchowski's classical kernel K ( x , y ) = ( x 1 / 3 + y 1 / 3 ) ( x − 1 / 3 + y − 1 / 3 ) . For the proof of existence we take a self-similar solution h ε for a regularized kernel K ε and pass to the limit ε → 0 to obtain a solution for the original kernel K . The main difficulty is to establish a uniform lower bound on h ε . The basic idea for this is to consider the time-dependent problem and to choose a special test function that solves the dual problem. [ABSTRACT FROM AUTHOR]

Abstract: In this article, an existence theorem of global solutions with small initial data belonging to , for a chemotaxis system is given on the whole space , . In the case , our global solution is integrable with respect to the space variable on some time interval, and then conserves the mass for a short time, at least. The system consists of a chemotaxis equation with a logarithmic term and an ordinary equation without diffusion term. [Copyright &y& Elsevier]

Abstract: In this paper we propose a mechanism for the formation of paths of minimal length between two points (trails) by a collection of individuals undergoing reinforced random walks. This is the case, for instance, of ant colonies in search for food and the development of ant trails connecting nest and food source. Our mechanism involves two main ingredients: (1) the reinforcement due to the gradients in the concentration of some substance (pheromones in the case of ants) and (2) the persistence understood as the tendency to preferably follow straight directions in the absence of any external effect. Our study involves the formulation and analysis of suitable Markov chains for the motion in simple labyrinths, that will be understood as graphs, and numerical computations in more complex graphs reproducing experiments performed in the past with ants. [Copyright &y& Elsevier]

Abstract: The LSW model with encounters has been suggested by Lifshitz and Slyozov as a regularization of their classical mean-field model for domain coarsening to obtain universal self-similar long-time behavior. We rigorously establish that an exponentially decaying self-similar solution to this model exist, and show that this solutions is isolated in a certain function space. Our proof relies on setting up a suitable fixed-point problem in an appropriate function space and careful asymptotic estimates of the solution to a corresponding homogeneous problem. [Copyright &y& Elsevier]

Abstract: We study an integro-differential equation modeling angular alignment of interacting bundles of cells or filaments. A bifurcation analysis of the related stationary problem was done by Geigant and Stoll in [E. Geigant, M. Stoll, Bifurcation analysis of an orientational aggregation model, J. Math. Biol. 46 (6) (2003) 537–563]. Here we analyze the time-dependent problem and prove that the type of alignment (one- or multi-directional) depends on the initial distribution, the interaction potential, and the preferred optimal orientation of the bundles of cells or filaments. Our main technical tool is the analysis of the evolution of suitable functionals for the cell density, which allows to also specify the direction(s) where the final alignment takes place. [Copyright &y& Elsevier]