Approximation and asymptotic behaviour of evolution families. (English) Zbl 1161.34336

Batty, Charles J. K.; Chill, Ralph

Differ. Integral Equ. 15, No. 4, 477-512 (2002).

In this paper, the authors consider the long-time asymptotic behavior of non-autonomous Cauchy problems of the form: \(\frac{d}{dt}u(t)=A(t)u(t)\), \((t\geq s\geq 0)\), \(u(s)=x\) on a Banach space \(X\). Let \((A(t))_{t\geq 0}\) and \((B(t))_{t\geq 0}\) be two families of closed operators on \(X\), and let \((U(t,s))_{t\geq s\geq 0}\) and \((V(t,s))_{t\geq s\geq 0}\) be the associated evolution families. Then the authors obtain estimates for \(| | U(t,s)-V(t,s)| | \) in terms of \(| | A(\tau )^{-1}-B(\tau )^{-1}| | \) for \(s\leq \tau \leq t,\) in several different cases, namely when \(\{A(t)\}\) and \(\{B(t)\}\) satisfy the Acquistapace-Terreni conditions, or the Kato-Tanabe conditions, or the \(L^{p}\)-maximal regularity conditions for some \(1<p<\infty.\) Their results extend previous results by P. Acquistapace and B. Terreni [Rend. Semin. Mat. Univ. Padova 78, 47–107 (1987; Zbl 0646.34006)], by T. Kato and H. Tanabe [Osaka Math. J. 14, 107–133 (1962; Zbl 0106.09302)], by R. Schnaubelt [J. Evol. Equ. 1, No. 1, 19–37 (2001; Zbl 1098.34551)] and by M. Hieber and S. Monniaux [Proc. Am. Math. Soc. 128, No. 4, 1047–1053 (2000; Zbl 0937.35195)].

Reviewer: Behzad Djafari-Rouhani (El-Paso)

MSC:

34D05	Asymptotic properties of solutions to ordinary differential equations
47D06	One-parameter semigroups and linear evolution equations
34G10	Linear differential equations in abstract spaces
47B38	Linear operators on function spaces (general)

Keywords:

evolution family; asymptotic behaviour; closed operator; Cauchy problem; exponential dichotomy; Acquistapace-Terreni conditions; Kato-Tanabe conditions; Maximal regularity assumptions

Citations:

Zbl 0646.34006; Zbl 0106.09302; Zbl 1098.34551; Zbl 0937.35195