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Analytic semigroups in weighted $L_1$-spaces on the half-line generated by singular or degenerate operators

Patrick Guidotti, Philippe Laurençot, Christoph Walker

Abstract

Ranges of the real-valued parameters $α$, $a$, $b$, and $m$ are identified for which the operator $$\mathcal{A}_α(a,b)f(x):=x^α\left(f''(x)+\frac{a}{x}f'(x)+\frac{b}{x^2}f(x)\right), \quad x>0,$$ generates an analytic semigroup in $L_1((0,\infty),x^m\mathrm{d}x)$.

Analytic semigroups in weighted $L_1$-spaces on the half-line generated by singular or degenerate operators

Abstract

Ranges of the real-valued parameters , , , and are identified for which the operator generates an analytic semigroup in .

Paper Structure

This paper contains 7 sections, 20 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Semigroup and its Generator
  4. General Operators
  5. The Absorption Semigroup
  6. Estimates for Certain Integral Operators
  7. Auxiliary Results

Key Result

Theorem 1.1

Let $\kappa<1$ and $\kappa-2<m\le 1$. Define the operator where $\mathrm{dom}(A_{1,m})$ depends on the parameters $m$ and $\kappa$: Then $A_{1,m}$ generates a bounded and positive analytic semigroup $S_{1,m}$ on $X_m$ of angle $\pi/2$. In addition, $S_{1,m}$ is a semigroup of contractions on $X_m$ when $m\in (\kappa,1]$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 2.1: MNS_JDE22
  • proof
  • Remark 2.2
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • ...and 29 more