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Fine scales of decay of operator semigroups. (English) Zbl 1418.34120

Many linear evolution equations can be rewritten in the form of an abstract Cauchy problem \(\dot{u}(t) + Au(t) = 0\) with initial condition \(u(0) = u_0\), where \(-A\) is the generator of a \(C_0\)-semigroup. The exponential stability of such equations is a classical and well-understood topic: the equation is exponentially stable if and only if \(\| e^{-At} \| \to 0\) as \(t \to \infty\).
In contrast, the study of finer decay properties of solutions is far more subtle. First, by the above characterization of exponentially stability such a rate cannot hold for the whole underlying Banach space, but still for a dense subset. A natural and typical question is to determine the decay of classical solutions, i.e., the decay of solutions with initial values \(u_0 \in D(A)\), the domain of \(A\). It is known from a celebrated result of A. Borichev and Y. Tomilov [Math. Ann. 347, No. 2, 455–478 (2010; Zbl 1185.47044)] that on Hilbert spaces the polynomial decay of classical solutions can be completely characterized by the growth of the resolvent of \(A\) on the imaginary axis. More precisely, for \(\alpha > 0\) one has \(\| T(t) A^{-1} \| = \mathcal{O}(| s |^{\alpha})\) if and only if \(\| T(t) A^{-1} \| = \mathcal{O}(t^{-1/\alpha})\) as \(t, | s | \to \infty\). Following the usual theme that working with resolvents is easier than working with the semigroups directly, the past years have shown that this result is extremely useful for determining the exact asymptotics of concrete evolution equations. It is important to note that this precise characterization is only available in the important Hilbert space setting. It is known that in general Banach spaces similar results can only be valid up to logarithmic corrections and this is the best one can achieve with purely function-theoretical approaches.
The article under review significantly broadens our understanding of optimal decay rates of evolution equations. In fact, it is very desired to generalize Borichev and Tomilov’s result to finer growth scales and the article under review provides such results for the scale of slowly varying functions, i.e., products of polynomials and slowly varying functions. The operator-theoretic approach used by the authors in particular gives the following characterization.
Theorem. Let \((e^{-tA})_{t \geq 0}\) be a bounded \(C_0\)-semigroup on a Hilbert space with \(\sigma(A) \cap i \mathbb{R} = \emptyset\). Then for \(\alpha > 0\) and \(\beta \geq 0\) the following are equivalent:
(i)
\(\| (is + A)^{-1} \| = \mathcal{O}(| s |^{\alpha} (\log | s |)^{-\beta})\) as \(| s | \to \infty\),
(ii)
\(\| T(t)A^{-1} \| = \mathcal{O}(t^{-1/\alpha} (\log t)^{-\beta / \alpha})\) as \(t \to \infty\).
An analogous, but slightly weaker characterization holds for positive logarithmic powers as well. Further, with the same methods the authors can also characterize the decay of the derivatives \(AT(t)x\) of solutions for \(x \in D(A)\), i.e., the decay of \(\| T(t)A(\mathrm{Id} + A)^{-1} \|\) and the decay of \(\| T(t)A (\mathrm{Id} + A)^{-2} \|\), i.e., uniform decay rates on \(D(A) \cap \mathrm{rg}(A)\).

MSC:

34G10 Linear differential equations in abstract spaces
34D05 Asymptotic properties of solutions to ordinary differential equations
47D06 One-parameter semigroups and linear evolution equations
47A10 Spectrum, resolvent
47D03 Groups and semigroups of linear operators

Citations:

Zbl 1185.47044
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