Strip-type operators and abstract Cauchy problems

TL;DR

This paper studies well-posedness of abstract linear Schrödinger and wave equations with operators of strip-type and parabola-type in Banach spaces, extending results using R-boundedness and applying to semilinear wave equations.

Contribution

It introduces new well-posedness results for abstract Schrödinger and wave equations with strip-type and parabola-type operators, including extensions with R-boundedness, and applies these to semilinear wave equations.

Findings

Established well-posedness of classical solutions in Sobolev-Slobodetskii spaces.

Extended results to R-bounded operators for broader applicability.

Proved existence and uniqueness of solutions for short-time semilinear wave problems.

Abstract

We consider the non-homogeneous abstract linear Schr\"odinger 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.