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We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type    $$\begin{aligned} \partial ^2_tu(t,x)- a(x,\lambda )^2\partial _x^2u(t,x)= b(x,\lambda ,u(t,x),u(t-\tau ,x),\partial _tu(t,x),\partial _xu(t,x)), \; x \in (0,1) \end{aligned}$$  ∂ t 2 u ( t , x ) - a ( x , λ ) 2 ∂ x 2 u ( t , x ) = b ( x , λ , u ( t , x ) , u ( t - τ , x ) , ∂ t u ( t , x ) , ∂ x u ( t , x ) ) , x ∈ ( 0 , 1 ) with smooth coefficient functions  a  and  b  such that    $$a(x,\lambda )&gt;0$$  a ( x , λ ) &gt; 0 and    $$b(x,\lambda ,0,0,0,0) = 0$$  b ( x , λ , 0 , 0 , 0 , 0 ) = 0 for all  x  and    $$\lambda $$  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    $$\tau $$  τ and    $$\lambda $$  λ ) of small non-stationary time-periodic solutions, which bifurcate from the stationary solution    $$u=0$$  u = 0 , and we derive a formula which determines the bifurcation direction with respect to the bifurcation parameter    $$\tau $$  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    $$\tau $$  τ .

Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay