Asymptotic Analysis

We study the behavior of a piezoelectric plate when its thickness tends to 0. Our analysis regorously justifies the classical a priori assumption that the electric potential is a second order polynomial with respect to the thickness variable. We prove that the electric potential appears in the limit bidimensional equation only. Moreover, we obtain the very contribution of the piezoelectric constants in the bending operator.

We study the large time behavior of the solutions to the Cauchy problem for the system with relaxation source u_t+v_x=0, v_t+a²u_x=f(u)−v, with $(x,t)\in{\mathbb{R}} \times{\mathbb{R}} ^{+}$ , for the initial data (u,v)=(u₀,v₀) at t=0, with $u_{0},v_{0}\in L^{1}({\mathbb{R}})\cap L^{\infty}({\mathbb{R}})$ , f(u)=αu²/2 and |v₀|≤au₀. Under the sub‐characteristic condition we show that, as t→∞, the component u tends towards a fundamental solution of the convection‐diffusion equation u_t+f(u)_x=a²u_xx in the L^p norm, at a rate faster than t^{−(p−1)/2p}.

We study a singular perturbation problem for the nonlinear Poisson equation and we classify its solutions. This classification is based on their monotonicity properties, which mainly depend on the stationary points of the so‐called Sagdeev potential. For each class of solution, we give necessary and sufficient conditions for the resolution of the problem and provide a boundary layer analysis.