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oai:docta.ucm.es:20.500.14352/503362023-08-25T16:13:42Zcom_20.500.14352_14col_20.500.14352_15

Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena Arrieta Algarra, José María Rodríguez Bernal, Aníbal Souplet, Philippe 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. 2023-06-20T09:46:26Z 2023-06-20T09:46:26Z 2004 journal article 0391-173X 10.2422/2036-2145.2004.1.01 https://hdl.handle.net/20.500.14352/50336 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 http://www.numdam.org/?lang=en eng BFM2000-0798 restricted access Scuola Normale Superiore