Periodic Solutions of Non-linear Anisotropic Partial Differential Equations

Periodic solutions of arbitrary period to semilinear partial differential equations of Zabusky or Boussinesq type are obtained. More generally, for a linear differential operator A(y,∂), the equation A(y,∂)u = (−1)^∣γ∣∂^γf(y,∂^γu), y = (t,x)∈ℝ^k×G is studied, where homogeneous boundary conditions on ∂G and periodicity conditions on t are imposed. The solutions are obtained by variational methods in anisotropic Sobolev spaces.