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 $$ \partial^2_t u(t,x)- a(x,\lambda)^2\partial_x^2u(t,x)= b(x,\lambda,u(t,x),u(t-\tau,x),\partial_tu(t,x),\partial_xu(t,x)), \; x \in (0,1) $$ with smooth coefficient functions $a$ and $b$ such that $a(x,\lambda)>0$ and $b(x,\lambda,0,0,0,0) = 0$ for all $x$ and $\lambda$. 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 $\tau$ and $\lambda$) 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 $\tau$. To this end, we transform the wave equation into a system of partial integral equations by means of integration along characteristics, and then we apply a Lyapunov-Schmidt procedure and a generalized implicit function theorem to this system. 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 $\tau$.