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Dissipative parabolic equations in locally uniform spaces

✍ Scribed by J. M. Arrieta; J. W. Cholewa; Tomasz Dlotko; A. Rodríguez–Bernal

Publisher
John Wiley and Sons
Year
2007
Tongue
English
Weight
290 KB
Volume
280
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

The Cauchy problem for a semilinear second order parabolic equation u~t~ = Δ__u__ + f (x, u,∇u), (t, x) ∈ ℝ^+^ × ℝ^N^ , is considered within the semigroup approach in locally uniform spaces $ {\dot W}^{s,p}_U $ (ℝ^N^ ). Global solvability, dissipativeness and the existence of an attractor are established under the same assumptions as for problems in bounded domains. In particular, the condition sf (s, 0) < 0, |s | > s~0~ > 0, together with gradient's “subquadratic” growth restriction, are shown to guarantee the existence of an attractor for the above mentioned equation. This result cannot be located in the previous references devoted to reaction‐diffusion equations in the whole of ℝ^N^ . (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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