A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions II: Asymptotic profiles of solutions and radial terrace solution

This paper is a continuation of our previous paper (Kaneko-Matsuzawa-Yamada, Discrete Contin. Dyn. Syst., 2022), where we have classified all large-time behaviors of radially symmetric solutions to a free boundary problem of reaction diffusion equation $u_t=\Delta u+f\left(u\right)$ with positive bistable nonlinearity f in high space dimensions. The positive bistable nonlinearity means that $f\left(u\right)=0$ has exactly two positive stable equilibria. Among the classified solutions, we are interested in a spreading solution, that is a solution $\left(u\right(t,\left|x\right|),h(t\left)\right)$ for $x\in R^N$ with free boundary $\left|x\right|=h\left(t\right)$ such that, as $t\to \infty$, $\left|x\right|\le h\left(t\right)$ expands to the whole space $R^N$ and $u(t,\cdot )$ converges to a positive stable equilibrium for $f\left(u\right)$ uniformly in any compact set of $R^N$. When we discuss whole asymptotic profiles of spreading solutions, it has been known that they are generally described with use of a semi-wave obtained from the corresponding semi-wave problem.

Our main purpose is to study precise asymptotic estimates for any spreading solution whose profile accompanies a propagating terrace with two different types of propagating speeds. Such a spreading phenomenon occurs when $u(t,\cdot )$ converges to the largest equilibrium of f and the related semi-wave problem does not have a solution. We will prove that the propagating terrace consists of two functions; one is a semi-wave corresponding to a smaller positive equilibrium of f and the other is a traveling wave connecting two positive equilibria of f. In order to give sharp estimates for a radial terrace solution with the above properties, we need so called logarithmic shiftings, which are revealed by Uchiyama (1983) and Du-Matsuzawa-Zhou (2015) in higher dimensional cases. Moreover, we will see that two types of the logarithmic shiftings appear in the estimates. The proof for the higher dimensional case is very different from the one-dimensional case established by Kaneko-Matsuzawa-Yamada (2020). To derive exact logarithmic terms, we have to construct a series of upper and lower solutions to make estimates more and more accurate.