Spiral traveling wave solutions of nonlinear diffusion equations related to a model of spiral cyrstal growth (Q1884463)

The subject of the paper is a nonlinear diffusion equation \[ u_t = \Delta u + f(u-\sigma \theta) \qquad (x \in \Omega, t>0) \] with boundary condition \[ u_r = 0 \qquad (x \in \partial \Omega, t>0). \] Here \(\Omega\) is a two-dimensional annulus with center at the origin, \((r, \theta)\) are the polar coordinates, \(\sigma\) is a positive integer and \(f\) is a periodic function. The above boundary value problem is related to a model of spiral crystal growth [\textit{I. Sunagawa, K. Narita, P. Bennema} and \textit{B. van der Hoek}, J. Crystal Growth, 42, 121--126 (1977)]. A solution \(u(x,t)\) of the form \[ u(x,t) = \varphi (r, \theta - \omega_1 t) + \omega_2 t \] with constant \(\omega_1, \, \omega_2\) is called is called a spiral traveling wave solution. Under suitable assumptions the existence, uniqueness and asymptotic stability of such solutions is proved. The paper is closely related to the results obtained by R. Kobayashi (to be published).