One of the most important systems for understanding chemotactic aggregation is the Keller-Segel system. We consider the time-fractional Keller-Segel system of order ∈ (0, 1). We prove an existence result with small initial data in a class of Besov-Morrey spaces. Self-similar solutions are obtained and we also show an asymptotic behaviour result.
K E Y W O R D SBesov-Morrey, chemotaxis aggregation, fractional derivative, Keller-Segel model, spaces M S C ( 2 0 1 0 ) 26A33, 35A01, 35B40, 35K45, 35K55, 35Q92, 35R11, 92C15, 92C17
We investigate the asymptotic periodicity, L p -boundedness, classical (resp., strong) solutions, and the topological structure of solutions set of strongly damped semilinear wave equations. The theoretical results are well complemented with a set of very illustrating applications.
KEYWORDSasymptotic behavior, boundedness, classical solutions, damped wave equations, strong solutions, topological structure of solutions set where X 1 2 is the fractional power space associated with A as in the work of Henry. 19 Equations like (1.1) have a lot of nontrivial and interesting features and appear in the literature under the name of strongly damped wave equations. An example of mathematical model represented in the form (1.1) is the wave equation with structural damping (see Carvalho and Cholewa 9 and Chen and Triggiani [20][21][22] ). This manuscript is a natural continuation of the work of Cuevas et al, 18 which investigates the existence and uniqueness of asymptotically almost-periodic mild solutions for strongly damped wave equations of type 1.1. Here we are concerned with the asymptotic periodicity, L p -boundedness properties, classical (resp., strong) solutions, and the topologicalProof. The proof requires straightforward modifications in the proof of Lemma 4.1 by Andrede et al. 27 We omit the details.Throughout this paper we always assume that ( , A) is an admissible pair.
Pseudo S-asymptotically -periodic functionsLet Y be an arbitrary Banach space. In this work C b ([0, ∞); Y) denotes the space consisting of the continuous and bounded functions from [0, ∞) into Y, endowed with the norm of the uniform convergence. Definition 2.3. (Pierri and Rolnik[23]) A function f ∈ C b ([0, ∞); Y) is called pseudo S-asymptotically periodic if there is > 0 such that lim h→∞
One of the most important models for understanding chemotactic aggregation is the Keller-Segel system. We analyze the time-fractional Keller-Segel system in a new framework; we derive the global well-posedness and the asymptotic behavior of solutions in critical Besov-weak-Herz spaces that consist in Besov spaces based on weak-Herz spaces.
<p style='text-indent:20px;'>We consider the fractional chemotaxis Navier-Stokes equations which are the fractional Keller-Segel model coupled with the Navier-Stokes fluid in the whole space, and prove the existence of global mild solutions with the small critical initial data in Besov-Morrey spaces. Our results enable us to obtain the self-similar solutions provided the initial data are homogeneous functions with small norms and considering the case of chemical attractant without degradation rate. Moreover, we show the asymptotic stability of solutions as the time goes to infinity and obtain a class of asymptotically self-similar ones.</p>
This paper is devoted to the study of the global well-posedness for a non-local-in-time Navier-Stokes equation. Our results recover in particular other existing well-posedness results for the Navier-Stokes equations and their time-fractional version. We show the appropriate manner to apply Kato's strategy and this context, with initial conditions in the divergence-free Lebesgue space L σ d (R d ). Temporal decay at 0 and ∞ are obtained for the solution and its gradient.
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