Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay

TL;DR

This paper establishes conditions for Hopf bifurcation in 1D delayed semilinear wave equations, demonstrating existence, uniqueness, and smoothness of bifurcating periodic solutions, using integral transforms and bifurcation theory.

Contribution

It provides a novel framework for analyzing Hopf bifurcation in hyperbolic PDEs with delay, including formulas for bifurcation direction and handling technical difficulties.

Findings

Existence of bifurcating periodic solutions from stationary states.

Conditions ensuring local uniqueness and regularity of solutions.

Formula for bifurcation direction with respect to delay parameter.

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 $\tau$.