Page 195 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 195
OPERATOR SEMIGROUPS AND EVOLUTION EQUATIONS (MS-29)
Spectral Multiplier Theorems in Lp For Abstract Differential Operators
Himani Sharma, Himani.sharma@anu.edu.au
Australian National University, Australia
For an operator generating a group on Lp spaces transference results give bounds on the Phillips
functional calculus also known as spectral multiplier estimates. In this talk, we will consider
specific group generators which are abstraction of first order differential operators and show
similar spectral multiplier estimates assuming only that the group is bounded on L2 rather than
Lp. We will also show some R-bounded Hörmander calculus results. Firstly for the square
of a perturbed Hodge-Dirac operator, by assuming an abstract Sobolev embedding property.
Secondly for an abstract Harmonic oscillator obtained using Weyl pairs.
Strong and Polynomial Stability for Delay Semigroups
Sachi Srivastava, sachi_srivastava@yahoo.com
University of Delhi, India
In this talk we will discuss strong and polynomial stability for semigroups associated with delay
differential equations. In particular we will study some conditions on the delay operator Φ and
the generator B of the underlying semigroup that ensure strong and polynomial stability of the
delay semigroup associated with the abstract delay differential equation
u (t) = Bu(t) + Φut, t > 0,
u(0) = x,
u0 = f,
where X is a Banach space, ut(σ) = u(t + σ), −1 ≤ σ ≤ 0, x ∈ X, f lies in an appropriate
space, (B, D(B)) generates a C0-semigroup on X and Φ is the delay operator.
On decay rates of the solutions of parabolic Cauchy problems
Jari Taskinen, jari.taskinen@helsinki.fi
University of Helsinki, Finland
Coauthors: José Bonet, Wolfgang Lusky
We consider the Cauchy problem in the Euclidean space RN x for the parabolic equation
∂tu(x, t) = Au(x, t), where the operator A (e.g. the Laplacian) is assumed, among other
things, to be a generator of a C0 semigroup in a weighted Lp-space Lwp (RN ) with 1 ≤ p < ∞
and a fast growing weight w. We show that there is a Schauder basis (en)n∞=1 in Lwp (RN ) with
the following property: given an arbitrary positive integer m there exists nm > 0 such that, if
the initial data f belongs to the closed linear span of en with n ≥ nm, then the decay rate of
the solution of the problem is at least t−m for large times t. In other words, the Banach space
of the initial data can be split into two components, where the data in the infinite-dimensional
component leads to decay with any pre-determined speed t−m, and the exceptional component
is finite dimensional.
We discuss in detail the needed assumptions of the integral kernel of the semigroup etA.
193
Spectral Multiplier Theorems in Lp For Abstract Differential Operators
Himani Sharma, Himani.sharma@anu.edu.au
Australian National University, Australia
For an operator generating a group on Lp spaces transference results give bounds on the Phillips
functional calculus also known as spectral multiplier estimates. In this talk, we will consider
specific group generators which are abstraction of first order differential operators and show
similar spectral multiplier estimates assuming only that the group is bounded on L2 rather than
Lp. We will also show some R-bounded Hörmander calculus results. Firstly for the square
of a perturbed Hodge-Dirac operator, by assuming an abstract Sobolev embedding property.
Secondly for an abstract Harmonic oscillator obtained using Weyl pairs.
Strong and Polynomial Stability for Delay Semigroups
Sachi Srivastava, sachi_srivastava@yahoo.com
University of Delhi, India
In this talk we will discuss strong and polynomial stability for semigroups associated with delay
differential equations. In particular we will study some conditions on the delay operator Φ and
the generator B of the underlying semigroup that ensure strong and polynomial stability of the
delay semigroup associated with the abstract delay differential equation
u (t) = Bu(t) + Φut, t > 0,
u(0) = x,
u0 = f,
where X is a Banach space, ut(σ) = u(t + σ), −1 ≤ σ ≤ 0, x ∈ X, f lies in an appropriate
space, (B, D(B)) generates a C0-semigroup on X and Φ is the delay operator.
On decay rates of the solutions of parabolic Cauchy problems
Jari Taskinen, jari.taskinen@helsinki.fi
University of Helsinki, Finland
Coauthors: José Bonet, Wolfgang Lusky
We consider the Cauchy problem in the Euclidean space RN x for the parabolic equation
∂tu(x, t) = Au(x, t), where the operator A (e.g. the Laplacian) is assumed, among other
things, to be a generator of a C0 semigroup in a weighted Lp-space Lwp (RN ) with 1 ≤ p < ∞
and a fast growing weight w. We show that there is a Schauder basis (en)n∞=1 in Lwp (RN ) with
the following property: given an arbitrary positive integer m there exists nm > 0 such that, if
the initial data f belongs to the closed linear span of en with n ≥ nm, then the decay rate of
the solution of the problem is at least t−m for large times t. In other words, the Banach space
of the initial data can be split into two components, where the data in the infinite-dimensional
component leads to decay with any pre-determined speed t−m, and the exceptional component
is finite dimensional.
We discuss in detail the needed assumptions of the integral kernel of the semigroup etA.
193
