Asymptotic behaviour of the solutions of non-linear elliptic and parabolic systems in tube domains

The paper is devoted to the study of the asymptotic behaviour of solutions of weakly non-linear elliptic and parabolic systems of second-order equations. In particular, the behaviour as $t\to+\infty$ of the solution of a second-order non-linear parabolic equation satisfying a Neumann boundary condition at the boundary of a bounded Lipschitz domain is studied. The proofs are based on a result on the asymptotic equivalence of two systems of ordinary differential equations.