Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains

Dissipative parabolic equations in locally uniform spaces

“…Note that several different versions of (1.1), including the case of systems, have been considered by a number of authors, e.g. [1,6,9,12,13,18,19,23,27,32,33,34], where attractors have been constructed under suitable functional settings and suitable growth sign and structure conditions on the nonlinear term f (x, u, ∇u).…”

“…On the other hand for any nonnegative smooth compactly supported initial data, the solution converges to 1 in L ∞ loc R N ; see e.g. [6]. Hence to account for all this dynamics one must choose a large space, containing bounded functions, but whose norm is not too strong.…”

“…The space of bounded and uniformly continuous functions BU C R N , was used in [32]. More recently Bessel potentials spaces H s p R N have been used in [6]. The so-called locally uniform spaces have been used in [12,13,33,34] and a systematic study of linear equations in such spaces was developed in [7].…”

“…Slightly weaker assumptions on the linear part are used in [32] and even weaker ones have been considered in [6]. On the other hand assuming monotonicity conditions on f 0 , conditions on the linear term are dropped in [13,22], which implies that the bistable equation can be handled within these frameworks.…”

“…Therefore, using the techniques in [26], we can solve 6) constructing the solution as a continuous function from [0, τ u0 ) intoẆ 2α,p U R N satisfying the variation of constants formula…”