Sharp growth rates for semigroups using resolvent bounds

Abstract

Abstract. We study growth rates for strongly continuous semigroups. We prove that a growth rate for the resolvent on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the semigroup is asymptotically analytic, or if the semigroup is positive and the underlying space is an L p -space or a space of continuous functions. We also prove variations of the main results on fractional domains; these are valid on more general Banach spaces. In the second part of the article, we apply our main theorem to prove optimality in a classical example by Renardy of a perturbed wave equation which exhibits unusual spectral behavior.

Positive Semigroups Using Resolvent Estimate Bounds on Sharp Growth Rates

“…It is known that (1) implies ‖∑ (1 + )‖ ≤ (1 + ) if is -dimensional. Furthermore, as was shown in Jan and Mark (2018), if is a Hilbert space and (1) holds then ‖ (1 + )‖ grows at most linearly in (1 + ), while there exist semigroups on general Banach spaces which satisfy (1) but grow exponentially.…”

Semi-uniform stability of operator semigroups and energy decay of damped waves

“…Another line of counterexamples to the SMT stemming from examples due to Wolff and Greiner, Voigt and Wolff is based on the study of translation semigroups like ( T ( t ) f )( s ) = f (e t s ) on intersections of weighted L p -spaces or on the Sobolev space H 1 (1, ∞) [1, Example 5.3.2]; see [1] for a thorough discussion and further references. Renardy’s paper [22] is a standard reference for failure of the WSMT in the case of a semigroup originating from the simple hyperbolic equation u tt − u xx − u yy = e 2 πxi u y on false(0,1false)×false(0,1false)×R with periodic boundary conditions; see also [23,24]. Other important counterexamples from the point of view of applications are given in Lebeau [25], Schenck [26] and Jin [27] in the context of the damped wave equation u tt + b u t − u xx − u yy = 0.…”

“…If there are C , k > 0 such that N ( ω ) ≤ C ( ω − ω ( T )) − k for ω − ω ( T ) > 0 sufficiently small, then, as shown in [42], one gets supt≥1t−k e−ωfalse(Tfalse)t||Tfalse(tfalse)||<normal∞. In fact, one may associate with every blow-up rate of the resolvent an appropriate correction of exponential growth of the semigroup; see [24] for details. Recently, using the techniques of [42], the following interesting result was proved by Wei in [43, Theorem 1.3].…”

“…Some Banach space counterparts of Theorem 2.3 (as well as additional references) may be found in [24,44], while several interesting and relevant examples of applications are considered e.g. in [42,43] and [45, Remark 3.13].…”

Bounds on the Phillips calculus of abstract first order differential operators

“…See [13] by Frey and Portal for fairly specific examples of groups generated by operators with rough coefficients in L p . Exponential growth of a group can be obtained by perturbation but polynomial boundedness is quite difficult to establish as can be seen in the paper by Rozendaal and Veraar (Theorem 1.1, [25]). Coifman-Weiss and Kriegler's transference results thus have assumptions that are difficult to establish while general transference results do not give a large functional calculus.…”