Regularity of Time-Periodic Solutions to Autonomous Semilinear Hyperbolic PDEs

TL;DR

This paper investigates the regularity of time-periodic solutions to autonomous semilinear hyperbolic PDEs, establishing conditions under which solutions are infinitely smooth in both time and space, and providing examples illustrating the necessity of these conditions.

Contribution

It proves that under certain smoothness and non-resonance conditions, solutions to these PDEs are infinitely smooth in time and space, and presents counterexamples when conditions are not met.

Findings

Solutions are $C^ abla$-smooth in $t$ and $x$ under specified conditions.

Counterexamples show solutions can lack smoothness without non-resonance.

Fredholm properties and equivariant equations are key tools in proofs.

Abstract

This paper concerns autonomous boundary value problems for 1D semilinear hyperbolic PDEs. For time-periodic classical solutions, which satisfy a certain non-resonance condition, we show the following: If the PDEs are continuous with respect to the space variable $x$ and $C^\infty$-smooth with respect to the unknown function $u$, then the solution is $C^\infty$-smooth with respect to the time variable $t$, and if the PDEs are $C^\infty$-smooth with respect to $x$ and $u$, then the solution is $C^\infty$-smooth with respect to $t$ and $x$. The same is true for appropriate weak solutions. Moreover, we show examples of time-periodic functions, which do not satisfy the non-resonance condition, such that they are weak, but not classical solutions, and such that they are classical solutions, but not $C^\infty$-smooth, neither with respect to $t$ nor with respect to $x$, even if the PDEs are $C^\infty$-smooth with respect to $x$ and $u$. For the proofs we use Fredholm solvability properties of linear time-periodic hyperbolic PDEs and a result of E. N. Dancer about regularity of solutions to abstract equivariant equations.