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

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Arrieta Algarra, José María and Rodríguez Bernal, Aníbal and Souplet, Philippe (2004) Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, 3 (1). pp. 1-15. ISSN 0391-173X

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Official URL: http://archive.numdam.org/ARCHIVE/ASNSP/ASNSP_2004_5_3_1/ASNSP_2004_5_3_1_1_0/ASNSP_2004_5_3_1_1_0.pdf


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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., 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 tame limit; the existence time T* is not a continuous function of the initial data.


Item Type:Article
Uncontrolled Keywords:Heat-equations; Spaces; Bounds; Time
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:18083
Deposited On:31 Jan 2013 09:32
Last Modified:12 Dec 2018 15:07

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