Global bifurcation of periodic solutions in nonlinear evolution problems with periodic forcing Original Research Article

This paper discusses the global bifurcation of 2π2π-periodic solutions of ut=P(λ,x,∂)u+f(λ,t,u)ut=P(λ,x,∂)u+f(λ,t,u) with a homogeneous Dirichlet boundary condition, where P(λ,x,∂)P(λ,x,∂) is linear elliptic and the nonlinearity ff is 2π2π-periodic in tt. The main differences from existing theories devoted to this type of problem can roughly be summarized as follows: (i) the bifurcation analysis makes no use of evolution operators or related concepts (Poincaré maps, Floquet multipliers, etc.); (ii) the bifurcation/nonbifurcation points are characterized through an associated stationary problem; (iii) the functional setting allows for nonlinearities ff exhibiting time discontinuities. Among other things, the results include various partial generalizations of the “bifurcation from the principal eigenvalue” theorem, which, unlike the classical version, do not require linear parameter dependence.