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To reduce equation (2) to the form (1) one has to add a new variable p = 8,~). We assume that the functions ,f (71, s) and ,g(:z. s) satisfy some general conditions providing the solvability of the Cauchy problem for (2) (see [ 191). But, generally speaking, under these conditions, the corresponding solution IL(~) = ,(/,(.I:; t) need not be unique. A trajectory attractor A is constructed for the family of equations (1). We start from the fact that the attractor A may not change when the initial symbol (TO(S) is replaced by any shifted symbol CT"(S + h.). h > 0. This is why, together with the initial equation (1) having the symbol (T,~(s)~ we consider the family of equations (1) with shifted symbols "a(.~ + /L); h > 0. This family contains also any symbol 'T(S) that is a limit of some sequence {cru(s + h,) ] h

SynopsisThere is a large number of papers in which attractors of parabolic reaction-diffusion equations in bounded domains are investigated. In this paper, these equations are considered in the whole unbounded space, and a theory of attractors of such equations is built. While investigating these equations, specific difficulties arise connected with the noncompactness of operators, with the continuity of their spectra, etc. Therefore some new conditions on nonlinear terms arise. In this paper weighted spaces are widely applied. An important feature of this problem is worth mentioning: namely, properties of semigroups corresponding to equations with solutions in spaces of growing and of decreasing functions essentially differ.

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