ABSTRACT
A classical result of Liapunov asserts that the solution of the initial value problem u ′ ( t ) = A u ( t ) , u ( 0 ) = u 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/equ_chapter9_121_1.tif"/> where A is an n × n matrix, is exponentially stable provided that all eigenvalues of A have negative real part. On the other hand, it is well known that, in general, the asymptotic behavior of a C 0-semigroup T acting on a. Banach space X cannot be described adequately by the location of the spectrum σ (A) of its generator A. Indeed, set s ( A ) : = sup { R e λ; λ ∈ σ ( A ) and ω 0 ( T ) : = inf { ω ∈ ℝ ; sup t ≥ 0 ‖ e − ω t T ( t ) ‖ < ∞ } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/eqn9_1_1.tif"/>
