Regularity of time-periodic solutions to autonomous semilinear hyperbolic PDEs

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∞-smooth with respect to the unknown function u, then the solution is C∞-smooth with respect to the time variable t, and if the PDEs are C∞-smooth with respect to x and u, then the solution is C∞-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∞-smooth, neither with respect to t nor with respect to x, even if the PDEs are C∞-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.