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

José M. Arrieta[1]; Anibal Rodriguez-Bernal[1]; Philippe Souplet[2]

  • [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 (2004)

  • Volume: 3, Issue: 1, page 1-15
  • ISSN: 0391-173X

Abstract

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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.

How to cite

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Arrieta, José M., Rodriguez-Bernal, Anibal, and Souplet, Philippe. "Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.1 (2004): 1-15. <http://eudml.org/doc/84526>.

@article{Arrieta2004,
abstract = {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.},
affiliation = {Departamento de Matemática Aplicada Universidad Complutense 28040 Madrid, Spain; Departamento de Matemática Aplicada Universidad Complutense 28040 Madrid, Spain; 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},
author = {Arrieta, José M., Rodriguez-Bernal, Anibal, Souplet, Philippe},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {1-15},
publisher = {Scuola Normale Superiore, Pisa},
title = {Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena},
url = {http://eudml.org/doc/84526},
volume = {3},
year = {2004},
}

TY - JOUR
AU - Arrieta, José M.
AU - Rodriguez-Bernal, Anibal
AU - Souplet, Philippe
TI - Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2004
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 1
SP - 1
EP - 15
AB - 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.
LA - eng
UR - http://eudml.org/doc/84526
ER -

References

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