Open AccessBoundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena
Author(s) -
José M. Arrieta,
Anı́bal Rodrı́guez-Bernal,
Philippe Souplet
Publication year - 2009
Publication title -
annali della scuola normale superiore di pisa. classe di scienze
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.444
H-Index - 37
eISSN - 2036-2145
pISSN - 0391-173X
DOI - 10.2422/2036-2145.2004.1.01
Subject(s) - bounded function , mathematics , nonlinear system , lyapunov function , mathematical analysis , norm (philosophy) , limit (mathematics) , compact space , time derivative , uniform boundedness , physics , political science , law , quantum mechanics
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