Semigroup Growth Bounds

$L_\infty$-estimates for the torsion function and $L_\infty$-growth of semigroups satisfying Gaussian bounds

We investigate selfadjoint $C_0$-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the long time behaviour of the $L_\infty$ operator norm of the semigroup. As an application we prove a new $L_\infty$-bound for the torsion function of a Euclidean domain that is close to optimal.

Optimality of the quantified Ingham-Karamata theorem for operator semigroups with general resolvent growth

We prove that a general version of the quantified Ingham-Karamata theorem for $C_0$-semigroups is sharp under mild conditions on the resolvent growth, thus generalising the results contained in a recent paper by the same authors. It follows in particular that the well-known Batty-Duyckaerts theorem is optimal even for bounded $C_0$-semigroups whose generator has subpolynomial resolvent growth. Our proof is based on an elegant application of the open mapping theorem, which we complement by a crucial technical lemma allowing us to strengthen our earlier results.

解析半群族的一致界与平面波前的Lyapunov线性稳定性

Uniform bounds of families of analytic semigroups and Lyapunov linear stability of planar fronts

该研究利用半群生成元的谱性质与预解算子估计，得到解析半群族的一致范数界条件，并将其应用于证明反应扩散方程平面行波及电生理bidomain方程的Lyapunov线性稳定性。

On asymptotics for $C_0$-semigroups

We stretch the spectral bound equal growth bound condition along with a generalized Lyapunov stability theorem, known to hold for $C_0$-semigroups of normal operators on complex Hilbert spaces, to $C_0$-semigroups of scalar type spectral operators on complex Banach spaces. For such semigroups, we obtain exponential estimates with the best stability constants. We also extend to a Banach space setting a celebrated characterization of uniform exponential stability for $C_0$-semigroups on complex Hilbert spaces and thereby acquire a characterization of uniform exponential stability for scalar type…

From resolvent bounds to semigroup bounds

The purpose of this note is to revisit the proof of the Gearhardt-Prüss-Hwang-Greiner theorem for a semigroup S(t), following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on the norm of S(t) in terms of bounds on the resolvent of the generator.

离散增长-衰减-裂变方程的长期动力学

Long term dynamics of the discrete growth-decay-fragmentation equation

本文证明了对于一大类增长-衰减-裂变问题，解半群是解析且紧致的，从而具有非同步指数增长性质。