Stability of travelling waves in a multidimensional free boundary problem

The multidimensional perturbations of the one-dimensional travelling waves (TW) solution of ut(t,ζ) = uζζ(t,ζ)+u(t,ζ)uζ(t,ζ), t≥0, ζ≠ξ(t), u(t,ξ(t)) = u*, [∂u/∂ζ](tξ(t)) = -1, t≥0, u(t,-∞) = 0, u(t∞) = u∞, t≥0. (1.1) was studied. Therefore, the following n-dimensional extension of problem (1.1), ut(t,ζ,y) = Δu(t,ζ,y)+u(t,ζ,y)uζ(t,ζ,y), t≥0, ζ≠ξ(t,y), y∈Ω̄, u(t,ξ(t,y),y) = u*, [∂u/∂v](t,ξ(t,y),y) = -1, t≥0, y∈Ω̄, ∂u/∂v(t,ζy) = 0, t≥, ζ≠ξ(t,y), y∈∂Ω, ∂ξ/∂v(t,y) = 0, t≥0, y∈∂Ω, u(t,-∞,y) = 0, u(t,∞,y) = u∞, t≥0, y∈Ω̄, (1.5) where Ω⊂Rn is a bounded open set with C2+α boundary ∂Ω, 0<α<1 was considered. The phenomenon of sharp exchange of stability occurs at u* = u*c even in problem (1.5) and also the existence of infinitely many branches of nonplanar TW at a sequence of bifurcation points u*k↓u*c were proven. This may contribute to a better understanding of the sharp exchange of stability at u*c.