Curvature effect on asymptotic profiles of spiral curves

Abstract

We study the shape of spiral curves in an annulus which is governed by curvature flow equations with a driving force term. We establish that as the model parameter μ (which is the coefficient of the curvature) approaches ∞, the profile of the spiral curve tends to a line segment, while as μ approaches 0+, the limiting profile of the spiral curve is the involute of the inner circle of the annulus and the associated limiting rotational speed is the ratio of a constant c, which is the propagation speed of the planar wave, to the inner radius of the annulus. Hence the model parameter μ can be viewed as a twisted parameter. Finally, the spiral curve under consideration is shown to be with sign-changing curvature and exponentially stable.