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- W4313484265 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." @default.
- W4313484265 created "2023-01-06" @default.
- W4313484265 creator A5049300409 @default.
- W4313484265 creator A5062942964 @default.
- W4313484265 date "2023-01-02" @default.
- W4313484265 modified "2023-09-26" @default.
- W4313484265 title "Regularity of Time-Periodic Solutions to Autonomous Semilinear Hyperbolic PDEs" @default.
- W4313484265 doi "https://doi.org/10.48550/arxiv.2301.00605" @default.
- W4313484265 hasPublicationYear "2023" @default.
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