Attractors for non-autonomous semilinear parabolic equations with delays (Q2248795)

The present paper is devoted to the asymptotic behaviour of a family of non-autonomous semi-linear parabolic equations in a bounded domain \(\Omega\subset\mathbb R^n\). After a brief survey of earlier results on the attractor for partial differential equations with delays in the non-autonomous case, the axiomatic definition of the principal notions which will be used in the paper, and some preliminaries are presented. Concerning the equation, the non-linear term is supposed to satisfy the polynomial type growth condition of arbitrary order with general delays, and the time-depended external force with an exponential growth is a function in \(L^2_{\mathrm{loc}}(\mathbb R, L^2(\Omega))\). Under suitable additional conditions, the existence and uniqueness of a weak solution for the process associated to the problem is obtained and then using the asymptotic a priori estimate method, the existence of a pull-back attractor to the problem is proved. With a somewhat stronger condition, that is, if the time-dependent external force is a translation bounded function in \(L^2_{\mathrm{loc}}(\mathbb R, L^2(\Omega))\), the existence of a uniform attractor are provided, where the case of a translation compact external force was first known. After that finally the relation between the pull-back and the uniform attractor are discussed. The obtained results extend some existing results for non-autonomous parabolic equations. The recent trend has been for research to be more interested in the mathematical natural sciences. (Some confusion due to apparently incorrect notations was experienced by the reviewer; for example in the introduction the Remarks 3.1 and 4.1 seems to be Remarks 3.10 and 4.5. Also in the statement of Remark 3.10, a subscript Lemma 3.1 appears which does seem to be Lemma 3.4).