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Direct and inverse abstract Cauchy problems with fractional powers of almost sectorial operators

Joel E. Restrepo

Abstract

We derive the explicit solution operator of an abstract Cauchy problem involving a time-variable coefficient and a fractional power of an almost sectorial operator. The time-variable coefficient is recovered by solving the inverse abstract Cauchy problem using the solution operator representation. As a complement, we also study similar problems by considering almost sectorial operators that depend on a time-variable.

Direct and inverse abstract Cauchy problems with fractional powers of almost sectorial operators

Abstract

We derive the explicit solution operator of an abstract Cauchy problem involving a time-variable coefficient and a fractional power of an almost sectorial operator. The time-variable coefficient is recovered by solving the inverse abstract Cauchy problem using the solution operator representation. As a complement, we also study similar problems by considering almost sectorial operators that depend on a time-variable.
Paper Structure (7 sections, 7 theorems, 50 equations)

This paper contains 7 sections, 7 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Direct abstract Cauchy problem
  4. Inverse abstract Cauchy problems
  5. Non-autonomous Cauchy problems
  6. Direct abstract Cauchy problem
  7. Inverse abstract Cauchy problems

Key Result

Theorem 2.4

Suppose that $A\in\Theta_\omega^\gamma$ for some $0<\omega<\frac{\pi}{2}$ and $0<\alpha<\frac{\pi}{2\omega}.$ Then the family $\{\mathscr{T}_\alpha(t):t\in S_{\frac{\pi}{2}-\alpha\omega}^0\}$ is an analytic semigroup of growth order $\frac{1+\gamma}{\alpha}.$ So the following assertions are true.

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Example 4.1
  • Theorem 4.2
  • ...and 7 more