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

• José M. Arrieta ^[1] ; Anibal Rodríguez-Bernal ^[2] ; Philippe Souplet ^[3]
  1. [1] Universidad Complutense de Madrid

     Universidad Complutense de Madrid

     Madrid, España

     
  2. [2] University of Picardie Jules Verne

     University of Picardie Jules Verne

     Arrondissement d’Amiens, Francia

     
  3. [3] Université de Versailles, France

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• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 3, Nº 1, 2004, págs. 1-15
• Idioma: inglés
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• Resumen

  • 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 ux blows up in finite time (with u itself remaining bounded), or u is global and converges in C1 norm to the unique steady state.

    The main difficulty is to proveC1 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 C1 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 C1 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.