In this work we obtain the complete asymptotic expansion of solutions, as time goes to infinity, of both the linearized Benjamin-Bona-Mahony-Burger equation and the linear Korteweg-de Vries-Burger equation. We also compute the second term in the asymptotic expansion of the solution to the two-dimensional Benjamin-Bona-Mahony-Burger equation, with quadratic nonlinear term.

In this article we study non classical singular perturbation problems involving boundary layers in the interior of the domain. As usual, these problems contain a small parameter which produces, when this parameter approaches zero, classical boundary layers located at the boundary. If we moreover consider a singular source function, we produce also boundary layers inside the domain. Our aim in this article is to study this kind of boundary layers. We consider, here, a model with a fourth-order differential operator, and the open set is a channel to avoid the technicalities due to the curvature of the boundary. Other stationary problems and time dependent problems will be considered elsewhere.

This paper is devoted to a complete classification of the radial singular and possibly sign changing solutions of a semilinear elliptic equation with critical or subcritical growth.

We consider a linear quadratic regulator problem in a general Banach space. The control is by an operator with range in the extrapolated Favard class. The observation operator is unbounded. We prove the existence of a Riccati operator which describes the value function for the optimal control and can be used to synthesize optimal feedback, similarly as in Hilbert spaces.

In this paper we study the global existence for Dirac--Klein--Gordon equations with small data. We apply the contraction mapping principle and use the Strichartz estimate.

For $m\ge 3$, let $M$, with dimension $m$, and $N$ be compact Riemannian manifolds without boundaries. We prove the energy identity (1.2) for a sequence of weakly convergent $m$-harmonic maps in $C^1(M,N)$. We also generalize the result to certain regular approximated $m$-harmonic maps whose tension fields are bounded in $L^{m\over m-1}$.