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The Abstract Cauchy Problem

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Part of the Applied Mathematical Sciences book series (AMS,volume 44)

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

$$\left\{ {\begin{array}{*{20}{c}} {\frac{{du(t)}}{{dt}} = Au(t), t > 0} \hfill \\ {u(0) = x} \hfill \\ \end{array} } \right.$$
(1.1)

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).

Keywords

  • Banach Space
  • Linear Operator
  • Invariant Subspace
  • Strong Solution
  • Mild Solution

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1983 Springer-Verlag New York, Inc.

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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

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  • DOI: https://doi.org/10.1007/978-1-4612-5561-1_4

  • Publisher Name: Springer, New York, NY

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