Exponential Bounds and Hyperbolicity of Evolution Families

In our thesis, 9], we consider the non{autonomous linear Cauchy problem _ u(t) = A(t)u(t); t s; t; s 2 I; u(s) = y 2 Y s ; (CP) on a Banach space X, where I 2 fa; b]; a; 1); IRg and Y s is a subspace of X. We call (CP) well{posed (on the spaces Y s) if Y s , s 2 I, is dense and there is a unique C 1 {solution u = u(; s; y) depending continuously on intial data, 6]. In this case there are bounded linear operators U(t; s) on X such that the solution u of (CP) is given by u(t) = U(t; s)y, (t; s) 2 D := f(t; s) 2 I 2 : t sg. The family U = (U(t; s)) (t;s)2D is an evolution family, that is, it satisses U(t; s) = U(t; r)U(r; s) and U(s; s) = Id for t r s and D 3 (t; s) 7 ! U(t; s) is strongly continuous, see e.g. 6]. In 9, x1.1] we discuss various conditions ensuring well{ posedness of (CP). Our main interest is directed to the asymptotic behaviour of the solutions u and, hence, of the evolution family U. In particular, U is called exponentially bounded if there are constants w 2 IR and M 1 such that kU(t; s)k Me w(t?s) for (t; s) 2 D. If w < 0 then U is called exponentially stable. We remark that U is always bounded if I is compact. If A(t) A generates a C 0 {semigroup e tA =: U(s+t; s), then we have ke tA k M " e (s(A)+")t , t 0, for " > 0 and s(A) := supfRe : 2 (A)g provided that, e.g., A has compact resolvent or (e tA) t0 is analytic. However, it is known that in the non{autonomous situation there is no analogue of this result even for nite dimensional X. A new example illustrating this phenomenon is presented in 9, x3.3]: There are generators (A(t); D(A(t))), t 0, of uniformly bounded, commuting, positive C 0 {semigroups on an L 1 {space such that (CP) is well{posed on a dense subspace Y Y t D(A(t)), A()y is continuous for y 2 Y , A(t k) = 0 for a sequence (t k) IR + , and s(A(t)) = ?1 for t 6 = t k. However, the related …