Dissipative parabolic equations in locally uniform spaces (original) (raw)

2007, Mathematische Nachrichten

Keywords Reaction-diffusion equations, Cauchy problem in R N ,dissipativeness, global attractor MSC (2000) 35K15, 35K57, 35B40, 35B41 TheCauchyproblem forasemilinear second orderparabolic equation u t =∆u + f ( x, u, ∇ u ) , ( t, x ) ∈ R + × R N ,i sc onsidered within the semigroup approach in locally uniform spacesẆ s,p U`R N´. G lobals olvability, dissipativeness and thee xistence of an attractor are established under thes ame assumptions as for problems in boundedd omains. In particular,t he condition sf( s, 0 ) < 0 , | s | >s 0 > 0 ,t ogether with gradient's "subquadratic"growthrestriction, are showntoguarantee theexistence of an attractor for theabove mentioned equation. This resultcannot be located in theprevious references devotedtoreaction-diffusion equations in the wholeof R N .