Abstract
Let X be a Banach space and let A be a linear operator from D(A)⊂X into X. Given x ∈ X the abstract Cauchy problem for A with initial data x consists of finding a solution u(t) to the initial value problem
where by a solution we mean an X valued function u(t) such that u(t) is continuous for t ≥ 0, continuously differentiable and u(t) ∈ D(A) for t> 0 and (1.1) is satisfied. Note that since u(t) ∈ D(A) for t > 0 and u is continuous at t = 0, (1.1) cannot have a solution for x ∉ D(A).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Pazy, A. (1983). The Abstract Cauchy Problem. In: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol 44. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5561-1_4
Download citation
DOI: https://doi.org/10.1007/978-1-4612-5561-1_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-5563-5
Online ISBN: 978-1-4612-5561-1
eBook Packages: Springer Book Archive