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Maximal $L_q$-regularity of nonlocal parabolic equations in higher order Bessel potential spaces

Nikolaos Roidos, Yuanzhen Shao

Abstract

We consider fractional parabolic equations with variable coefficients and establish maximal $L_{q}$-regularity in Bessel potential spaces of arbitrary nonnegative order. As an application, we show higher order regularity and instantaneous smoothing for the fractional porous medium equation and for a nonlocal Kirchhoff equation.

Maximal $L_q$-regularity of nonlocal parabolic equations in higher order Bessel potential spaces

Abstract

We consider fractional parabolic equations with variable coefficients and establish maximal -regularity in Bessel potential spaces of arbitrary nonnegative order. As an application, we show higher order regularity and instantaneous smoothing for the fractional porous medium equation and for a nonlocal Kirchhoff equation.

Paper Structure

This paper contains 10 sections, 14 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces
  4. Functional analytic tools
  5. Imaginary powers of elliptic operators
  6. The Fractional Powers of $\mathscr{L}$
  7. $L_q$-Maximal regularity
  8. Applications
  9. Fractional porous medium equation
  10. Nonlocal Kirchhoff diffusion problem

Key Result

Proposition 2.1

Suppose that $0\leq s_0<s_1<\infty$ and $\theta\in (0,1)$. Then and where $s_\theta=(1-\theta)s_0 + \theta s_1$.

Theorems & Definitions (31)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3: Dissipativity
  • Definition 2.4: Sectoriality
  • Definition 2.5: Bounded imaginary powers
  • Definition 2.6: Bounded $H^{\infty}$-calculus
  • Definition 2.7: $R$-boundedness
  • Theorem 2.8: Kalton and Weis, KaW or Weis2
  • Theorem 3.1
  • ...and 21 more