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A note on the Cahn-Hilliard equation in H 1 ( N ) involving critical exponent

Jan W. CholewaAníbal Rodríguez-Bernal — 2014

Mathematica Bohemica

We consider the Cahn-Hilliard equation in H 1 ( N ) with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as | u | and logistic type nonlinearities. In both situations we prove the H 2 ( N ) -bound on the solutions and show that the individual solutions are suitably attracted by the set of equilibria. This complements the results in the literature; see J. W. Cholewa, A. Rodriguez-Bernal (2012).

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

José M. ArrietaAnibal Rodriguez-BernalPhilippe Souplet — 2004

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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...

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