Differential and Integral Equations

Linearized stability for semilinear non-autonomous evolution equations with applications to retarded differential equations

Gabriele Gühring, Wolfgang M. Ruess, and Frank Räbiger

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Abstract

We consider the semilinear non-autonomous evolution equation $\frac{d}{dt}u(t)=Au(t)+G(t,u(t))$, $t\geq s\geq 0,$ where $(A,D(A))$ is a Hille-Yosida operator on a Banach space $X$ and $G$ is a continuous function on $\mathbb R_+\times \overline{D(A)}$ with values in the extrapolated Favard class corresponding to $A$. In our main results we present principles of linearized stability and instability for a solution of such an equation. Our approach is based on the theory of extrapolation spaces. We apply the results to non-autonomous semilinear retarded differential equations.

Article information

Source
Differential Integral Equations Volume 13, Number 4-6 (2000), 503-527.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356061237

Mathematical Reviews number (MathSciNet)
MR1750038

Zentralblatt MATH identifier
0990.34068

Subjects
Primary: 34G20: Nonlinear equations [See also 47Hxx, 47Jxx]
Secondary: 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx] 35B35: Stability 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

Citation

Gühring, Gabriele; Räbiger, Frank; Ruess, Wolfgang M. Linearized stability for semilinear non-autonomous evolution equations with applications to retarded differential equations. Differential Integral Equations 13 (2000), no. 4-6, 503--527. https://projecteuclid.org/euclid.die/1356061237.


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