We prove the existence of a stationary solution of any given mass to the coagulation-fragmentation equation without assuming a detailed balance condition, but assuming instead that aggregation dominates fragmentation for small particles while fragmentation predominates for large particles. We also show the existence of a self-similar solution of any given mass to the coagulation equation and to the fragmentation equation for kernels satisfying a scaling property. These results are obtained, following the theory of Poincaré-Bendixson on dynamical systems, by applying the Tykonov fixed point theorem on the semigroup generated by the equation or by the associated equation written in "self-similar variables". Moreover, we show that the solutions to the fragmentation equation with initial data of a given mass behaves, as t → +∞, as the unique self similar solution of the same mass.  2004 Elsevier SAS. All rights reserved. RésuméPour toute masse donnée, nous démontrons l'existence d'au moins une solution stationnaire pour l'équation de coagulationfragmentation. Nous ne faisons pas d'hypothèse d'équilibre en détails sur les coefficients mais nous supposons que la coagulation domine la fragmentation pour les particules de petite taille et que la fragmentation est prépondérante pour les particules de grande taille. Nous démontrons également l'existence de solutions auto-similaires pour l'équation de coagulation et pour l'équation de fragmentation sous une hypothèse d'homogéneité sur les noyaux. Ces résultats sont obtenus, s'inspirant de la preuve

We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity as well as the sticky particles model. We prove (local in time) non-concentration estimates in Orlicz spaces, from which we deduce weak stability and existence theorem. Strong stability together with uniqueness and instantaneous appearance of exponential moments are proved under additional smoothness assumption on the initial datum, for a restricted class of collision rates. Concerning the long-time behaviour, we give conditions for the cooling process to occur or not in finite time. (2000): 76P05 Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05]. Mathematics Subject Classification

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Conditions for the emergence of strong Turing–Hopf instabilities in the Lengyel–Epstein CIMA reaction–diffusion model are found. Under these conditions, time periodic spatially inhomogeneous solutions can be induced by diffusive instability of the spatially homogeneous limit cycle emerging at a supercritical Bautin–Hopf bifurcation about the unstable steady state of the reaction system. We report numerical simulations by an Alternating Directions Implicit (ADI) method that show the formation of twinkling patterns for a chosen parameter value, thus confirming our theoretical results.

Our concern is the study of degenerate Hopf bifurcation of smooth planar dynamical systems near isolated singular points. To do so, we propose to split up the definition of degeneracy into two types.Degeneracy of first kind shall means that no limit cycle surrounding the steady state can emerge after or before the critical point, with the possible emergence of limit cycles surrounding the point at infin-1 ity. Degeneracy of second kind shall means that either several limit cycles or semistable cycles as a limiting case, emerge surrounding the steady state super or subcritically. In degenerate bifurcation of second kind we also show that the radius of the emerging cycle tends to zero with an "anomalous" order as the bifurcation parameter tends to the critical value. Finally, we give a sufficient condition for degenerate bifurcations of second kind up to 6-jet-equivalence, and show some "typical" forms for degenerate bifurcations.

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We examine the appearance of Turing instabilities of spatially homogeneous periodic solutions in reaction-diffusion equations when such periodic solutions are consequence of Hopf bifurcations. First, we asymptotically develop limit cycle solutions associated to the appearance of Hopf bifurcations in reaction systems. Particularly, we will show conditions to the appearance of multiple limit cycles after Hopf bifurcation. Then, we propose expansions to normal modes associated with Turing instabilities from spatially homogeneous periodic solutions associated to limit cycles which appear as a consequence of a Hopf bifurcation. Finally, we discuss examples of reaction-diffusion systems arising in biology and chemistry in which can be observed spatial and time-periodic patterning.