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).
The gradient flow of bending energy for plane curve is studied. The flow is considered under two kinds of constraints; one is under the area and total-length constraints; the other is under the area and local-length constraints. The fundamental results (the local existence and uniqueness) were obtained independently by Kurihara and the second author for the former one; by Okabe for the later one. For the former one the global existence was shown for any smooth initial curves, but the asymptotic...
The aim of this paper is to study the effect of vibrations on convective instability of reaction fronts in porous media. The model contains reaction-diffusion equations coupled with the Darcy equation. Linear stability analysis is carried out and the convective instability boundary is found. The results are compared with direct numerical simulations.