Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 3 (2004) no. 1, p. 1-15
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.
Classification:  35K60,  35K65,  35B45
@article{ASNSP_2004_5_3_1_1_0,
author = {Arrieta, Jos\'e M. and Rodriguez-Bernal, Anibal and Souplet, Philippe},
title = {Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 3},
number = {1},
year = {2004},
pages = {1-15},
zbl = {1072.35098},
mrnumber = {2064964},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2004_5_3_1_1_0}
}
Arrieta, José M.; Rodriguez-Bernal, Anibal; 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, Série 5, Volume 3 (2004) no. 1, pp. 1-15. http://www.numdam.org/item/ASNSP_2004_5_3_1_1_0/

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