RT Journal Article
T1 Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena
A1 Arrieta Algarra, José María
A1 Rodríguez Bernal, Aníbal
A1 Souplet, Philippe
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., 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.
PB Scuola Normale Superiore
SN 0391-173X
YR 2004
FD 2004
LK https://hdl.handle.net/20.500.14352/50336
UL https://hdl.handle.net/20.500.14352/50336
LA eng
NO DGES (Spain)
DS Docta Complutense
RD 5 abr 2025