In this paper we give a survey of recent developments in the spectral theory of bounded functions on the half line, and its applications to study the asymptotic behavior of solutions of evolution equations of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u'(t)=A(t)u(t)+f(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where A(t) is periodic in t. For the applications we consider spectra of the evolution semigroups associated with the evolution equations <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$u'(t)=A(t)u(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in various spectral invariant function spaces and their generators. The main results presented in the paper are concerned with the existence and uniqueness within decaying functions with specific spectral properties. These asymptotic behaviors are related to the Katznelson–Tzafriri Theorem and Massera Theorem. When the operator A(t) is independent of t the method of sums of commuting operators is used to give sufficient conditions in terms of spectral conditions of A instead of the monodromy operator.

Asymptotic Behavior of Evolution Equations via Spectral Theory of Functions on the Half Line, Sums of Commuting Operators and Evolution Semigroups

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2025年7月26日

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Nguyen Van Minh,
Vu Trong Luong

摘要

In this paper we give a survey of recent developments in the spectral theory of bounded functions on the half line, and its applications to study the asymptotic behavior of solutions of evolution equations of the form $u'(t)=A(t)u(t)+f(t)$ $u^{'} (t) = A (t) u (t) + f (t)$ , $t\ge 0$ $t \geq 0$ , where A(t) is periodic in t. For the applications we consider spectra of the evolution semigroups associated with the evolution equations $u'(t)=A(t)u(t)$ $u^{'} (t) = A (t) u (t)$ in various spectral invariant function spaces and their generators. The main results presented in the paper are concerned with the existence and uniqueness within decaying functions with specific spectral properties. These asymptotic behaviors are related to the Katznelson–Tzafriri Theorem and Massera Theorem. When the operator A(t) is independent of t the method of sums of commuting operators is used to give sufficient conditions in terms of spectral conditions of A instead of the monodromy operator.