Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

Arrieta, José M.¹ ; Rodriguez-Bernal, Anibal¹ ; Souplet, Philippe² 

¹ Departamento de Matemática Aplicada Universidad Complutense 28040 Madrid, Spain
² Département de Mathématiques INSSET Université de Picardie 02109 St-Quentin, France and Laboratoire de Mathématiques Appliquées UMR CNRS 7641 Université de Versailles 45 avenue des Etats-Unis 78035 Versailles, France

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 1, pp. 1-15

Résumé

We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions $u$: either the space derivative $u_x$ blows up in finite time (with $u$ itself remaining bounded), or $u$ is global and converges in $C^1$ norm to the unique steady state. The main difficulty is to prove $C^1$ boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out the method of Zelenyak. After deriving precise estimates on the solutions and on the Lyapunov functional, we proceed by contradiction by showing that any $C^1$ unbounded global solution should converge to a singular stationary solution, which does not exist. As a consequence of our results, we exhibit the following interesting situation: - the trajectories starting from some bounded set of initial data in $C^1$ describe an unbounded set, although each of them is individually bounded and converges to the same limit; - the existence time $T^{*}$ is not a continuous function of the initial data.

Détail

Classification : 35K60, 35K65, 35B45