Higher order asymptotic expansion of solutions to abstract linear hyperbolic equations

TL;DR

This paper develops higher order asymptotic expansions for solutions to abstract hyperbolic equations in Hilbert spaces, explicitly expressing profiles using the semigroup generated by the operator, under regularity assumptions.

Contribution

It introduces a method to explicitly write asymptotic profiles of arbitrary order for hyperbolic equations using semigroup theory and maximal regularity assumptions.

Findings

Explicit formulas for asymptotic profiles of solutions.

Use of maximal regularity for the semigroup $e^{-tA}$.

Extension to arbitrary order asymptotic expansions.

Abstract

The paper concerned with higher order asymptotic expansion of solutions to the Cauchy problem of abstract hyperbolic equations of the form $u''+Au+u'=0$ in a Hilbert space, where $A$ is a nonnegative selfadjoint operator. The result says that by assuming the regularity of initial data, asymptotic profiles (of arbitrary order) are explicitly written by using the semigroup $e^{-tA}$ generated by $-A$. To prove this, a kind of maximal regularity for $e^{-tA}$ is used.