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 --> in finity of the solutions of first-order Hamilton-Jacobi equations. The first one concerns the behavior of the Lax-Oleinik semigroup for equations set i n non-compact domains. We show that even for smooth, strictly convex Hamilt onians, the convergence, as t --> infinity, of solutions of such equations may fail, in contrast to what happens in the compact framework, were conver gence 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 conver gence to a space-time periodic solution is not true in general. Here we pro vide very simple explicit counterexamples of this fact. (C) 2000 Academie d es sciences Editions scientifiques et medicales Elsevier SAS.