Global solutions of semilinear parabolic equations are studied in the case when some weak a priori estimate for solutions of the problem under consideration is already known. The focus is on the rapid growth of the nonlinear term for which existence of the semigroup and certain dynamic properties of the considered system can be justified. Examples including the famous Cahn-Hilliard equation are finally discussed.
An abstract parabolic equation with sectorial operator and continuous nonlinearity is studied in this paper. In particular, the asymptotic behavior of solutions is described within the framework of the theory of global attractors. Examples included in the final part of the paper illustrate the presented ideas.
Under natural regularity assumptions on the data the powers of regular elliptic boundary value problems (e.b.v.p.) are shown to be higher order regular e.b.v.p.. This result is used in description of the domains of fractional powers of elliptic operators which information is in order important in regularity considerations for solutions of semilinear parabolic equations. Presented approach allows to avoid C-smoothness assumption on the data that is typical in many references.
Global solvability and asymptotics of semilinear parabolic Cauchy problems in are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over , . In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.
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 abstract parabolic problems in ordered Banach spaces and give conditions under which they have global attractors. Our approach is via comparison of solutions. Within this approach abstract comparison principles are obtained and bounds on the attractors are given by order intervals in Banach spaces. These results are applied to ordinary differential equations and to parabolic equations for which the main part is given by a sum of fractional powers of sectorial operators having increasing...
L'esistenza di attrattori globali per equazioni paraboliche semilineari è stata estensivamente studiata da molti autori mentre il caso quasilineare è stato meno considerato e ancora esistono molti problemi aperti. L'obiettivo di questo lavoro è di studiare, da un punto di vista astratto, l'esistenza di attrattori globali per equazioni paraboliche quasilineari con parte principale monotona. I risultati ottenuti vengono applicati a problemi parabolici degeneri del secondo ordine e di ordine superiore....
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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