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arXiv · 2301.00605

Regularity of Time-Periodic Solutions to Autonomous Semilinear Hyperbolic PDEs

Irina Kmit·Lutz Recke
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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.

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Irina Kmit, Lutz Recke. 2023-01-02. Regularity of Time-Periodic Solutions to Autonomous Semilinear Hyperbolic PDEs. https://doi.org/10.1016/j.jmaa.2023.127562

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