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

Arrieta, José M.; Rodriguez-Bernal, Anibal; Souplet, Philippe

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

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