Asymptotic Profiles of Solutions and Propagating Terrace for a Free Boundary Problem of Nonlinear Diffusion Equation with Positive Bistable Nonlinearity
Yuki Kaneko, Hiroshi Matsuzawa, Yoshio Yamada
Abstract
We will study a free boundary problem of nonlinear diffusion equation $u_t=u_{xx}+f(u),t>0, 0<x<h(t)$, with free boundary $h(t)$, which describes the spreading of a new or invasive species. When $f$ is positive bistable, that is, $f$ has two positive stable equilibrium points, Kawai and Yamada [J. Differential Equations, 261 (2016), pp. 538--572] have shown that asymptotic behaviors of solutions (except for vanishing) as $t\to\infty$ can be mainly classified into two types of spreading phenomena, which correspond to two positive stable equilibrium points. The purpose of this paper is to investigate whole asymptotic profiles of solutions as $t\to\infty$ when spreading occurs. In particular, we will prove that, under certain conditions, any spreading solution which corresponds to a larger positive stable equilibrium point of $f$ approaches a so-called propagating terrace connecting 0 and the larger positive stable equilibrium point. This terrace function consists of a semi-wave corresponding to a smaller positive stable equilibrium point and a traveling wave connecting two positive stable equilibrium points with speed slower than that of the semi-wave.