Time-Asymptotic Description of the Solution for an Abstract Cauchy Problem

Aref Jeribi
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Aref Jeribi: University of Sfax, Department of Mathematics

Chapter Chapter 4 in Spectral Theory and Applications of Linear Operators and Block Operator Matrices, 2015, pp 121-137 from Springer

Abstract: Abstract In this chapter, we give a description of the large time behavior of solutions to an abstract Cauchy problem on Banach spaces without restriction on the initial data. Let X be a Banach space and let T : 𝒟 ( T ) ⊆ X → X $$T: \mathcal{D}(T) \subseteq X\longrightarrow X$$ be the infinitesimal generator of a C 0-semigroup of bounded linear operators (U(t)) t ≥ 0 acting on X. We consider the Cauchy problem 4.0.1 ∂ ψ ∂ t = A ψ : = T ψ + F ψ ψ ( 0 ) = ψ 0 , $$\displaystyle{ \left \{\begin{array}{rcl} \frac{\partial \psi } {\partial t} & =&A\psi:= T\psi + F\psi \\ \psi (0)& =&\psi _{0}, \end{array} \right. }$$ where F is a bounded linear operator on X and ψ 0 ∈ X.

Keywords: Abstract Cauchy Problem; Large Time Behavior; Banach Space; Infinitesimal Generator; Positive Measure Space (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-17566-9_4

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DOI: 10.1007/978-3-319-17566-9_4

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