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Strip-type operators and abstract Cauchy problems

Nikolaos Roidos

TL;DR

This work develops an operator-theoretic framework for non-homogeneous abstract linear Schrödinger and wave equations driven by strip-type and parabola-type operators in Banach spaces. By leveraging sectorial calculus, $R$-boundedness, and the Da Prato-Grisvard formula, it establishes well-posedness of classical solutions in vector-valued Sobolev-Slobodetskii spaces and extends the analysis to $R$-bounded operator families. It further analyzes closures of the associated sums and provides a robust inversion theory via operator-valued Fourier multipliers, enabling short-time existence results for a class of abstract semilinear wave equations. The results unify linear and semilinear theories in UMD spaces and offer a pathway to quasilinear extensions and Hilbert-space specializations. Overall, the paper advances the understanding of evolution equations with strip-type and parabola-type spectral geometry in Banach spaces, with explicit solution representations and stability estimates.

Abstract

We consider the non-homogeneous abstract linear Schrödinger and wave equations with zero initial conditions, defined by operators of strip-type and parabola-type in Banach spaces, respectively, and establish the well-posedness of classical solutions in appropriate vector-valued Sobolev-Slobodetskii spaces. We obtain analogous results for two extensions of these equations by replacing the previously mentioned boundedness properties of the associated operators with $R$-boundedness. As an application, we consider an abstract semilinear wave equation and establish the existence and uniqueness of classical solutions to this problem for short times.

Strip-type operators and abstract Cauchy problems

TL;DR

This work develops an operator-theoretic framework for non-homogeneous abstract linear Schrödinger and wave equations driven by strip-type and parabola-type operators in Banach spaces. By leveraging sectorial calculus, -boundedness, and the Da Prato-Grisvard formula, it establishes well-posedness of classical solutions in vector-valued Sobolev-Slobodetskii spaces and extends the analysis to -bounded operator families. It further analyzes closures of the associated sums and provides a robust inversion theory via operator-valued Fourier multipliers, enabling short-time existence results for a class of abstract semilinear wave equations. The results unify linear and semilinear theories in UMD spaces and offer a pathway to quasilinear extensions and Hilbert-space specializations. Overall, the paper advances the understanding of evolution equations with strip-type and parabola-type spectral geometry in Banach spaces, with explicit solution representations and stability estimates.

Abstract

We consider the non-homogeneous abstract linear Schrödinger and wave equations with zero initial conditions, defined by operators of strip-type and parabola-type in Banach spaces, respectively, and establish the well-posedness of classical solutions in appropriate vector-valued Sobolev-Slobodetskii spaces. We obtain analogous results for two extensions of these equations by replacing the previously mentioned boundedness properties of the associated operators with -boundedness. As an application, we consider an abstract semilinear wave equation and establish the existence and uniqueness of classical solutions to this problem for short times.
Paper Structure (6 sections, 12 theorems, 208 equations)

This paper contains 6 sections, 12 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. sectorial and strip-type operators
  3. The derivation operator in $L^{p}$
  4. the abstract schrödinger and wave equations
  5. On the closure of $\pm iA+B$
  6. An abstract semilinear wave equation

Key Result

Lemma 2.6

Let $\theta\in (0,1)$, $X$ be a complex Banach space, and let $A\in \mathcal{P}(0)$ in $X$. For any $\lambda\in \rho(-A)\backslash [0,\infty)$, we have where Moreover, the map is holomorphic, and for any $c>0$ we have Finally, for any $\phi\in (0,\pi]$ and $\rho>0$, there exists a $C>0$ such that

Theorems & Definitions (33)

  • Definition 2.1: sectorial operators
  • Definition 2.2: bounded imaginary powers
  • Definition 2.3: $R$-boundedness
  • Definition 2.4: strip-type operators
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Lemma 2.8
  • proof
  • ...and 23 more