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 $${\partial }_t^{2}u(t,x)-a(x,\lambda {)}^2{\partial }_x^{2}u(t,x)=b(x,\lambda ,u(t,x),u(t-\tau ,x),{\partial }_{t}u(t,x),{\partial }_{x}u(t,x)),x\in (0,1)$$ 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 $\tau$.

36 pages

References in corpus (3)

• Periodic Solutions to Dissipative Hyperbolic Systems. II: Hopf Bifurcation for Semilinear Problems
• Solution regularity and smooth dependence for abstract equations and applications to hyperbolic PDEs
• Time-Periodic Second-Order Hyperbolic Equations: Fredholmness, Regularity, and Smooth Dependence

Cited by in corpus (1)

• Forced Frequency Locking for Semilinear Dissipative Hyperbolic PDEs