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Semilinear Cauchy Problems with Almost Sectorial Operators

100%
Dlotko T.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2007
|
tom 55
|
nr 4
333-346
EN
Existence of a mild solution to a semilinear Cauchy problem with an almost sectorial operator is studied. Under additional regularity assumptions on the nonlinearity and initial data we also prove the existence of a classical solution to this problem. An example of a parabolic problem in Hölder spaces illustrates the abstract result.
2
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Hyperbolic Equations in Uniform Spaces

64%
Cholewa J., Dlotko T.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2004
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tom 52
|
nr 3
249-263
EN
The paper is devoted to the Cauchy problem for a semilinear damped wave equation in the whole of ℝ ⁿ. Under suitable assumptions a bounded dissipative semigroup of global solutions is constructed in a locally uniform space $Ḣ¹_{lu}(ℝ ⁿ) × L̇²_{lu}(ℝ ⁿ)$. Asymptotic compactness of this semigroup and the existence of a global attractor are then shown.
3
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Partly dissipative systems in uniformly local spaces

64%
Carvalho A., Dlotko T.
Colloquium Mathematicum
|
2004
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tom 100
|
nr 2
221-242
EN
We study the existence of attractors for partly dissipative systems in ℝⁿ. For these systems we prove the existence of global attractors with attraction properties and compactness in a slightly weaker topology than the topology of the phase space. We obtain abstract results extending the usual theory to encompass such two-topologies attractors. These results are applied to the FitzHugh-Nagumo equations in ℝⁿ and to Field-Noyes equations in ℝ. Some embeddings between uniformly local spaces are also proved.
4
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Abstract parabolic problem with non-Lipschitz nonlinearity

64%
Cholewa J. W., Dlotko T.
Banach Center Publications
|
2000
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tom 52
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nr 1
73-81
EN
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.
5
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Abstract parabolic problems in ordered Banach spaces

51%
Carvalho A., Cholewa J., Dlotko T.
Colloquium Mathematicum
|
2001
|
tom 90
|
nr 1
1-17
EN
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 resolvents and integral operators having positive kernels.
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