论文标题
一般1D半线性波方程的HOPF分叉延迟
Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay
论文作者
论文摘要
我们考虑了1D自主抑制和延迟的半连续波方程的边界价值问题，$ \ partial^2_t u（t，x）-a（x，λ）^2 \ partial_x^2u（t，x，x）= b（x，λ，λ，U（t- u（t- t- x），u（t-τ，x），u（t-τ，x），x _ partial xu（x） \; x \ in（0,1）$$具有平滑系数功能$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a（x，λ）> 0 $和$ b（x，λ，0,0,0,0）= 0 $ x $ and $ x $和$λ$。 We state conditions ensuring Hopf bifurcation, i.e., existence, local uniqueness (up to time shifts), regularity (with respect to $t$ and $x$) and smooth dependence (on $τ$ and $λ$) of small non-stationary time-periodic solutions, which bifurcate from the stationary solution $u=0$, and we derive a formula which determines the bifurcation direction with respect to the分叉参数$τ$。 为此，我们通过沿特征的集成将波方程转换为部分积分方程的系统，然后将Lyapunov-Schmidt过程应用于该系统。必须管理的主要技术困难对于双曲线PDE（有无延迟）是典型的：小除数和“衍生物的损失”属性。 我们不使用相应的初始值问题的任何属性。特别是，对于负延迟$τ$，我们的结果也是如此。
We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type $$ \partial^2_t u(t,x)- a(x,λ)^2\partial_x^2u(t,x)= b(x,λ,u(t,x),u(t-τ,x),\partial_tu(t,x),\partial_xu(t,x)), \; x \in (0,1) $$ with smooth coefficient functions $a$ and $b$ such that $a(x,λ)>0$ and $b(x,λ,0,0,0,0) = 0$ for all $x$ and $λ$. 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 $τ$ and $λ$) 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 $τ$. 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 $τ$.
