In this paper we analyze the long time behavior of a phase-field model by showing the existence of global compact attractors in the strong norm of high order Sobolev spaces.
We consider the Cahn-Hilliard equation in with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as and logistic type nonlinearities. In both situations we prove the -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).
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 : either the space derivative blows up in finite time (with itself remaining bounded), or is global and converges in norm to the unique steady state. The main difficulty is to prove 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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