Solution Estimates for Semilinear Non-autonomous Evolution Equations with Differentiable Linear Parts

Michael Gil’

doi:10.1007/s12591-019-00493-9

Solution Estimates for Semilinear Non-autonomous Evolution Equations with Differentiable Linear Parts

Michael Gil’

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In a Hilbert space we consider the equation du(t) / dt= A(t) u(t) + F(t, u(t)) , where A(t) is an unbounded operator, having a bounded strong derivative, and F is a continuous mapping. We derive norm estimates for solutions of the considered equation. These estimates give us the L²-stability and absolute stability conditions. To the best of our knowledge an absolute stability test for nonautonomous evolution equations has been obtained for the first time. Our main tool is the norm estimate for the derivative of a solution of the time-dependent Lyapunov equation.

Original language	English
Pages (from-to)	59-68
Number of pages	10
Journal	Differential Equations and Dynamical Systems
Volume	29
Issue number	1
DOIs	https://doi.org/10.1007/s12591-019-00493-9
State	Published - 1 Jan 2021

Keywords

Absolute stability
Hilbert space
L-stability
Non-autonomous equation
Quasi-linear equation
Semi-linear differential equation

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1007/s12591-019-00493-9

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title = "Solution Estimates for Semilinear Non-autonomous Evolution Equations with Differentiable Linear Parts",

abstract = "In a Hilbert space we consider the equation du(t) / dt= A(t) u(t) + F(t, u(t)) , where A(t) is an unbounded operator, having a bounded strong derivative, and F is a continuous mapping. We derive norm estimates for solutions of the considered equation. These estimates give us the L2-stability and absolute stability conditions. To the best of our knowledge an absolute stability test for nonautonomous evolution equations has been obtained for the first time. Our main tool is the norm estimate for the derivative of a solution of the time-dependent Lyapunov equation.",

keywords = "Absolute stability, Hilbert space, L-stability, Non-autonomous equation, Quasi-linear equation, Semi-linear differential equation",

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N2 - In a Hilbert space we consider the equation du(t) / dt= A(t) u(t) + F(t, u(t)) , where A(t) is an unbounded operator, having a bounded strong derivative, and F is a continuous mapping. We derive norm estimates for solutions of the considered equation. These estimates give us the L2-stability and absolute stability conditions. To the best of our knowledge an absolute stability test for nonautonomous evolution equations has been obtained for the first time. Our main tool is the norm estimate for the derivative of a solution of the time-dependent Lyapunov equation.

AB - In a Hilbert space we consider the equation du(t) / dt= A(t) u(t) + F(t, u(t)) , where A(t) is an unbounded operator, having a bounded strong derivative, and F is a continuous mapping. We derive norm estimates for solutions of the considered equation. These estimates give us the L2-stability and absolute stability conditions. To the best of our knowledge an absolute stability test for nonautonomous evolution equations has been obtained for the first time. Our main tool is the norm estimate for the derivative of a solution of the time-dependent Lyapunov equation.

KW - Absolute stability

KW - Hilbert space

KW - L-stability

KW - Non-autonomous equation

KW - Quasi-linear equation

KW - Semi-linear differential equation

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JO - Differential Equations and Dynamical Systems

JF - Differential Equations and Dynamical Systems

IS - 1

ER -

Solution Estimates for Semilinear Non-autonomous Evolution Equations with Differentiable Linear Parts

Abstract

Keywords

ASJC Scopus subject areas

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