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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Existence and stability for partial functional differential equations


Authors: C. C. Travis and G. F. Webb
Journal: Trans. Amer. Math. Soc. 200 (1974), 395-418
MSC: Primary 34G05; Secondary 35R10, 47H15
MathSciNet review: 0382808
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Abstract: The existence and stability properties of a class of partial functional differential equations are investigated. The problem is formulated as an abstract ordinary functional differential equation of the form $ du(t)/dt = Au(t) + F({u_t})$, where $ A$ is the infinitesimal generator of a strongly continuous semigroup of linear operators $ T(t),t \geqslant 0$, on a Banach space $ X$ and $ F$ is a Lipschitz operator from $ C = C([ - r,0];X)$ to $ X$. The solutions are studied as a semigroup of linear or nonlinear operators on $ C$. In the case that $ F$ has Lipschitz constant $ L$ and $ \vert T(t)\vert \leqslant {e^{\omega t}}$, then the asymptotic stability of the solutions is demonstrated when $ \omega + L < 0$. Exact regions of stability are determined for some equations where $ F$ is linear.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0382808-3
PII: S 0002-9947(1974)0382808-3
Article copyright: © Copyright 1974 American Mathematical Society