Rotating waves in neutral partial functional differential equations. (English) Zbl 0939.35188

The local existence and global continuation of rotating waves for partial neutral functional differential equations \[ \frac{\partial }{\partial t}D(\alpha, u_t)=d\frac{\partial^2}{\partial x^2}D(\alpha,u_t)+f(\alpha,u_t)\tag{1} \] defined on the unit circle \(x\in S^1\) is investigated; where \(d>0\) is a given constant; \(D,\;f:\mathbb{R}\times X\rightarrow C(S^1;\mathbb{R}),\;X=C([-\tau, 0];C(S^1;\mathbb{R}))\cong C([-\tau, 0]\times S^1,\mathbb{R}),\;u_t\in X\) is defined as \(u_t(\theta,x)=u(t+\theta,x)\) for \((\theta, x)\in [-\tau, 0]\times S^1.\) The authors seek Hopf bifurcations of time-periodic solutions (rotating waves) \(u:\mathbb{R}\times S^1\rightarrow \mathbb{R}\) which satisfy (1) and \[ u(t,x)=u\biggl(t+\frac{p}{2\pi}x,0\biggr),\quad u(t+p,x)=u(t,x)\quad \text{for}\quad (t,x)\in \mathbb{R}\times S^1 \] (\(p>0\) is a constant) and reduce (1) into a second order ordinary neutral differential equation whose \(2\pi\)-periodic solutions give rise to rotating waves of (1). They then show that finding a \(2\pi\)-periodic solution of the second order ordinary neutral differential equation can be formulated as finding a fixed point of an \(S^1-\)equivalent set-condensing mapping depending on two parameters, by using the compact resolvent of a Fredholm operater arising from the linear part of the second order ordinary neutral differential equation. This abstract formulation allows the authors to apply the general global Hopf bifurcation theorem of ordinary neutral differential equations developed by W. Krawcewicz, J. Wu and H. Xia [Can. Appl. Math. Q. 1, No. 2, 167-220 (1993; Zbl 0801.34069)] to investigate the existence and maximal continuation of rotating waves for (1). The general results mentioned above are illustrated by an example arising from a continuous circular array of transmission lines.


35R10 Partial functional-differential equations
34K40 Neutral functional-differential equations
35K57 Reaction-diffusion equations


Zbl 0801.34069
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