Asymptotic Analysis

We introduce a simplified model for a multi-material made up of two elastic bodies connected by a strong thin material layer whose stiffness grows as 1/ε. The model is obtained by identifying the Γ-limit of the stored strain energy functional of the physical problem when the thickness ε of the intermediate layer tends to zero. The intermediate layer behaves as a stiffening elastic membrane. Furthermore, in the linear anisotropic case, we establish the strong convergence of the exact solution toward the solution of the limit problem.

A multi-type branching diffusion process with fast transmutations between particles are considered in this paper. The first moment equations for the process are reaction–diffusion equations. As the inverse transmutation rate approaches 0, we derive an averaging principle for these reaction–diffusion equations.

The paper contains results concerning the stability properties of solutions to a simplified Keller–Segel problem modelling chemotaxis in the whole space R². First, the existence of solutions to considered system is proved, as well as for this system with the elliptic equation replaced by its parabolic version with time derivative multiplied by a positive parameter ε. Secondly, the stability of the model is shown, namely the convergence of solutions of parabolic–parabolic system to a solution of parabolic–elliptic one, when a suitable parameter in the system under consideration converges towards zero.