We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type

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 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 \(\tau \).

我们考虑一维自主阻尼和延迟半线性波动方程的边值问题

具有平滑系数函数a和b使得\(a(x,\lambda )>0\)和\(b(x,\lambda ,0,0,0,0) = 0\)对于所有x和\( λ \)。我们陈述了确保 Hopf 分岔的条件，即存在性、局部唯一性（直到时移）、规律性（关于t和x）和平滑依赖（对\(\tau \)和\(\lambda \)）的小非平稳时间周期解，从平稳解\(u=0\)分岔，我们推导出一个公式，该公式确定关于分岔参数\(\tau \)的分岔方向 . 为此，我们通过沿特征积分的方式将波动方程转化为偏积分方程组，然后应用Lyapunov-Schmidt过程和广义隐函数定理。必须解决的主要技术难题是双曲线 PDE（有或没有延迟）的典型难题：小除数和“导数损失”属性。我们不使用相应初始边界值问题的任何性质。特别是，我们的结果对于负延迟\(\tau \) 也是正确的。