C.J.K. Batty, R. Chill, Approximation and asymptotic behaviour of evolution families, Dierential Integral Equations 15 (2002), 477-512. 57  Home/Search   Document Details and Download   Summary   Related Articles   Check

....In view of Theorem 5.2 one has to show that kTU (1) Gamma T V (1)k = sup kU(t 1; t) Gamma V (t 1; t)k = ffi is smaller than a certain number ffi 0 0 depending on the dichotomy constants of U( Delta; Delta) see [126, Prop.2.3] So the assertion is a consequence of Theorem 4. 7 of [12] which says that ffi cq for some j 0 and a constant c independent of q. Looking at the results used in the above proof, one sees that only depends on the constants in (2.4) w, the type of A(t) Gamma w and B(t) Gamma w, and the dichotomy constants of U( Delta; Delta) Moreover, the ....

....to reduce this problem to case (a) see [78, p.180] or [126, 3.2) Such inheritance properties were established in [78, Chap.10] for bounded A(t) in [79] 89] for ordinary delay equations (see also [53, x6.6. 3] 108] 109] for the case of a dominant eigenvalue of the autonomous problem) in [12], 126] for parabolic problems (see also [50] 135, x5.8] for the case of exponential stability) and in [124] 127] for retarded parabolic equations. Further, the operators U(s t; s) and the dichotomy projections P (s) tend strongly to e and P as s 1, and P (s) and I Gamma P (s) inherit ....

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C.J.K. Batty, R. Chill, Approximation and asymptotic behaviour of evolution families, as a preprint in: Ulmer Seminare 5 (2000), 105--136.

....autonomous case, where A(t) A, and in the periodic case, where A(t) A(t p) see [3] 4] 6] 11] 12] 16] 21] and the references therein. Equations with almost periodic A( are treated in, e.g. 8] and [10] for X = C and in [13] for a certain class of parabolic problems, see also [5], 15] 17] For general evolution families U (but subject to an extra condition not assumed here) it is shown in [14] that U has an exponential dichotomy with an almost periodic Green s function if and only if there is a unique almost periodic mild solution u of (1.1) for each almost periodic f ....

C.J.K. Batty, R. Chill, Approximation and asymptotic behaviour of evolution families, Dierential Integral Equations 15 (2002), 477-512.

....variation of parameters formula, characteristic equation, evolution semigroup. 1 due to [30, Thm.4.1] These results extend a theorem by H. Tanabe, 32, Thm.5.6.1] see also [15] for closely related facts and [30] for further references. Very recently, C.J.K. Batty and R. Chill showed in [5] that one can allow for ff = 0 in (1.2) i.e. convergence in L(X) This paper extends [30] in several directions, e.g. the almost periodicity of U( Delta; Delta) is studied. We now complement (1.1) by a delay term and treat the retarded problem u(t) A(t)u(t) L(t)u t f(t) t s; u s = ....

....as we show in Section 5. There we also discuss a retarded parabolic partial differential equation. 2 To deduce the theorems sketched above, we will make extensive use of the regularity properties of the undelayed parabolic equation (1. 1) established in [1] 2] These facts were also used in [5] and [30] to derive convergence results for evolution families which are crucial to the present work. We further need the spectral theory of the associated evolution semigroup , cf. 6] or [10, xVI.9] to verify the exponential dichotomy of the homogeneous problem (1.5) with f = 0. Our approach ....

C.J.K. Batty, R. Chill, Approximation and asymptotic behaviour of evolution families, preprint.

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C.J.K. Batty, R. Chill, Approximation and asymptotic behaviour of evolution families, Dierential Integral Equations 15 (2002), 477-512. 57

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