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Zbl 0691.35019
Nicolaenko, B.; Scheurer, B.; Temam, R.
Some global dynamical properties of a class of pattern formation equations. (English)
[J] Commun. Partial Differ. Equations 14, No.2, 245-297 (1989). ISSN 0360-5302; ISSN 1532-4133/e

The authors consider the equations $$ \align \partial u/\partial t + \Delta\sp2 u + \Delta u + (1/2) \vert\nabla u\vert\sp2 &= 0 \tag{0.1} \\ \partial u/\partial t + \Delta\sp2 u + \Delta u - \beta\Delta(u\sp3) &= 0 \quad (\beta >0). \tag{0.2} \endalign $$ 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.
[Y.Ebihara]

MSC 2000:: *35B45 A priori estimates
35K55 Nonlinear parabolic equations
35K25 Higher order parabolic equations, general
76F99 Turbulence
35B35 Stability of solutions of PDE

Keywords: Kuramoto-Sivashisky equation; Cahn-Hilliard equation; phase transition; global attractor; absorbing sets

Cited in: Zbl 0806.58032

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