# A. F. Lavriks truncated equations by Kaufman R. M.

By Kaufman R. M.

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**Example text**

As stated in Zheng [450], the study of asymptotic behavior of the solution to a nonlinear evolutionary equation as time goes to inﬁnity can be divided into two categories. The ﬁrst category is to investigate the asymptotic behavior of solutions for any given initial datum. The second category is to investigate the asymptotic behavior of all solutions when initial data vary in any bounded set. There are essentially ﬁve apparently distinct properties that a semigroup may possess and such that each of them together with the existence of a bounded absorbing set leads to the existence of a global attractor.

15 is a C m domain, m ≥ 1, 1 ≤ p < ∞, then Note that in the previous embedding theorems, we always assume that m is a nonnegative integer in the deﬁnition of W m, p ( ). The embedding theorems may not be optimal. In fact, when = Rn , we may use the Bessel potential or the Riesz potential to deﬁne the Sobolev spaces of fractional order. 32) with which we can deﬁne Banach spaces H α, p (Rn ) = Jα L p (Rn ), H˙ α, p (Rn ) = Pα L p (Rn ), α ∈ R. 33) H α, p (Rn ) = W α, p (Rn ). 34) = Rn , we can still deﬁne the Sobolev spaces of fractional order as Let s > 0 be a non-integer, s = [s] + λ, λ ∈ (0, 1).

Some Inequalities in Analysis 19 then for any t ≥ t0 , t u(t0 ) exp − b(s)u(s)ds ≤ u(t) ≤ u(t0 ) exp t0 t b(s)u(s)ds . 3) 0 and is one of the basic tools in the theory of differential equations. On the basis of various motivations it has been extended and used considerably in various contexts. For instance, in the Picard-Cauchy type of iteration for establishing existence and uniqueness of solutions, this inequality and its variants play a signiﬁcant role. 1) are also encountered frequently in the perturbation and stability theory of differential equations.