Global attractor for a class of partial functional differential equations with infinite delay

Global Attractor for Some Partial Functional Differential Equations with Infinite Delay

“…With regard to the infinite delay case, the situation is di¤erent, since properties of solutions depend on the choice of the phase space. The fundamental theory related to functional di¤erential equations with infinite delay can be found in [25] and we refer the reader to works in [1,2,3,6,10,11,12,13,14,17,24] and the references therein, in which the results are about some quantitative and qualitative aspects of study.…”

“…The global attractor of Eq. (1.1) has been studied in, e.g., [19] with A ¼ 0, and [13] with A non-densely defined. It is well known that if A satisfies (H1), then it generates a locally Lipschitz continuous integrated semigroup SðtÞ, t b 0, see Lemma 2.1 below.…”

“…In [13], under the hypotheses (H1) and (H2), the authors showed existence of integral solutions of Eq. (1.1), which generates a strongly continuous semigroup UðtÞ, t b 0.…”

“…As the authors in [13] stated, the proof of Theorem 1.1 is based on the following proposition. Proposition 1.1.…”

“…In fact, from the proof of Theorem 1.1 in [13] and Proposition 1.1, we see that the authors in [13] only proved the asymptotic smoothness of the semigroup UðtÞ, t b 0, corresponding to Eq. From [19,Lemma 3.4.5] we know that, if we can give some conditions to guarantee that (b) holds, then so does (a).…”