Hiroki Yagisita

Spiral wave behaviors in an excitable reaction-diffusion system on a sphere

Abstract The dynamics of chemical spiral waves in an excitable reaction-diffusion system on a sphere is numerically investigated by employing a spectral method using spherical harmonics as basis functions. A nearly antisymmetric spiral wave is produced even from a symmetric spiral wave in the presence of a little inhomogeneity of the medium. In a homogeneous medium, the tip of the nearly antisymmetric spiral wave at the source rotates steadily, but the other tip changes its shape in an almost periodic manner.

Nearly Spherically Symmetric Expanding Fronts in a Bistable Reaction-Diffusion Equation

We consider nearly spherically symmetric expanding fronts in the scalar bistable reaction-diffusion equation on RN. As t→∞, the front is known to look more and more like a sphere under the rescaling of the radius to unity. In this paper we prove that, if the initial state is spherically symmetric and approximated by a one-dimensional traveling wave with a sufficiently large radius, then the solution is approximated uniformly for all t≥0 without the rescaling of the radius by the one-dimensional traveling wave with the speed of V=c−(N−1)κ, where c>0 is the speed of the one-dimensional traveling wave solution and κ the mean curvature of the sphere. We further show that, if the initial state is a slightly perturbed one from the spherical front, the difference between the actual front and the expanding sphere hardly grows or decays for all t≥0, although the relative magnitude of the perturbation to the radius of the sphere decreases to zero.

Convergence of a three-dimensional crystalline motion to Gauss curvature flow

We introduce a three-dimensional crystalline motion whose Wulff shape is a convex polyhedron (Wk). We prove that this crystalline motion converges to the motion by Gauss curvature in ℝ3 under the assumptions that the polyhedra (Wk) converge to the unit ball B3 and are symmetric with respect to the origin.K. Ishii and H. M. Soner showed the convergence of the two-dimensional crystalline motion to the curve shortening flow by a kind of perturbed test function methods. We employ their method to prove our result under aid from the theory of Minkowski problem.