Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay

Abstract

We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type ∂2tu(t,x)−a(x,λ)2∂2xu(t,x)=b(x,λ,u(t,x),u(t−τ,x),∂tu(t,x),∂xu(t,x)),x∈(0,1) with smooth coefficient functions a and b such that a(x,λ)>0 and b(x,λ,0,0,0,0)=0 for all x and λ. We state conditions ensuring Hopf bifurcation, i.e., existence, local uniqueness (up to time shifts), regularity (with respect to t and x) and smooth dependence (on τ and λ) of small non-stationary time-periodic solutions, which bifurcate from the stationary solution u=0 , and we derive a formula which determines the bifurcation direction with respect to the bifurcation parameter τ . To this end, we transform the wave equation into a system of partial integral equations by means of integration along characteristics and then apply a Lyapunov-Schmidt procedure and a generalized implicit function theorem. The main technical difficulties, which have to be managed, are typical for hyperbolic PDEs (with or without delay): small divisors and the “loss of derivatives” property. We do not use any properties of the corresponding initial-boundary value problem. In particular, our results are true also for negative delays τ.