Time-Periodic Second-Order Hyperbolic Equations: Fredholmness, Regularity, and Smooth Dependence

TL;DR

This paper studies second-order hyperbolic equations with periodic time conditions, establishing Fredholm properties, regularity results, and how solutions depend smoothly on data, under specific non-resonance conditions.

Contribution

It proves the Fredholm alternative and regularity results for hyperbolic equations with periodic conditions, including the effects of coefficient perturbations on solution smoothness.

Findings

Fredholm alternative holds under non-resonance conditions.

Higher regularity of solutions follows from higher regularity of data.

Perturbations in the coefficient 'a' can cause loss of smoothness, unlike other coefficients.

Abstract

The paper concerns the general linear one-dimensional second-order hyperbolic equation $$ \partial^2_tu - a^2(x,t)\partial^2_xu + a_1(x,t)\partial_tu + a_2(x,t)\partial_xu + a_3(x,t)u=f(x,t), \quad x\in(0,1) $$ with periodic conditions in time and Robin boundary conditions in space. Under a non-resonance condition (formulated in terms of the coefficients $a$, $a_1$, and $a_2$) ruling out the small divisors effect, we prove the Fredholm alternative. Moreover, we show that the solutions have higher regularity if the data have higher regularity and if additional non-resonance conditions are fulfilled. Finally, we state a result about smooth dependence on the data, where perturbations of the coefficient $a$ lead to the known loss of smoothness while perturbations of the coefficients $a_1$, $a_2$, and $a_3$ do not.