Some counterexamples on the asymptotic behavior of the solutions of Hamilton-Jacobi equations

In this Note, we provide two counterexamples about the behavior as t --> infinity of the solutions of first-order Hamilton-Jacobi equations. The first one concerns the behavior of the Lax-Oleinik semigroup for equations set in non-compact domains. We show that even for smooth, strictly convex Hamiltonians, the convergence, as t --> infinity, of solutions of such equations may fail, in contrast to what happens in the compact framework, were convergence results where proved recently by Fathi, Namah and Roquejoffre and the authors. The second counterexample concerns the behavior of space periodic solutions in the case of space-rime periodic Hamiltonians. Recently Fathi and Mather showed, using dynamical systems types arguments, that the convergence to a space-time periodic solution is not true in general. Here we provide very simple explicit counterexamples of this fact. (C) 2000 Academie des sciences Editions scientifiques et medicales Elsevier SAS.