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Some global dynamical properties of a class of pattern formation equations. (English) Zbl 0691.35019

The authors consider the equations

$\begin{array}{cc}\hfill \partial u/\partial t+{{\Delta }}^{2}u+{\Delta }u+\left(1/2\right){|\nabla u|}^{2}& =0\hfill \\ \hfill \partial u/\partial t+{{\Delta }}^{2}u+{\Delta }u-\beta {\Delta }\left({u}^{3}\right)& =0\phantom{\rule{1.em}{0ex}}\left(\beta >0\right)·\hfill \end{array}$

The mathematical study of (0.1) and (0.2) and their natural generalization is achieved by conceiving them as infinite dimensional dynamical systems. In this setting they are mainly concerned with the description of the asymptotic behavior of the solutions of (0.1) and (0.2). This amounts to perform a nonlinear stability analysis, to describe the structure of the associated global attractor, to compute an upper bound for its dimension, to construct absorbing sets.

Reviewer: Y.Ebihara

MSC:
 35B45 A priori estimates for solutions of PDE 35K55 Nonlinear parabolic equations 35K25 Higher order parabolic equations, general 76F99 Turbulence 35B35 Stability of solutions of PDE