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Asymptotics of parabolic equations with possible blow-up

Radosław Czaja — 2004

Colloquium Mathematicae

We describe the long-time behaviour of solutions of parabolic equations in the case when some solutions may blow up in a finite or infinite time. This is done by providing a maximal compact invariant set attracting any initial data for which the corresponding solution does not blow up. The abstract result is applied to the Frank-Kamenetskii equation and the N-dimensional Navier-Stokes system with small external force.

Remarks on the Fractal Dimension of Bi-Space Global and Exponential Attractors

Jan W. Cholewa;  Radoslaw Czaja;  Gianluca Mola — 2008

Bollettino dell'Unione Matematica Italiana

Bi-space global and exponential attractors for the time continuous dynamical systems are considered and the bounds on their fractal dimension are discussed in the context of the smoothing properties of the system between appropriately chosen function spaces. The case when the system exhibits merely some partial smoothing properties is also considered and applications to the sample problems are given.

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