Interface dynamics of the Fisher equation with degenerate diffusion
The authors study the initial-boundary value problem \[ \begin{alignedat}{2} u_t&= \varepsilon\Delta u^m + \frac{1}{\varepsilon}u(1-u) &&\quad \text{in } \Omega\times [0,T],\\ \frac{\partial(u^m)}{\partial\nu}&= 0, &&\quad \text{on } \partial\Omega\times [0,T],\\ u(x,0)&= g(x), &&\quad x \in \Omega, \end{alignedat} \] where \(\Omega\subset\mathbb R^N\) \((N\geq 1)\) is a smooth bounded domain, \(\nu\) is the outward normal vector to \(\partial\Omega\), \(m \geq 2\) and \(\varepsilon\) is a small parameter. The initial data \(g\geq 0\) has a convex compact support in \(\Omega\). This problem arises in population dynamics where it is called population density-dependent dispersal of individuals. The results of the article are focused on the asymptotic behavior of the solution \(u^{\varepsilon}\) of the above-mentioned problem as \(\varepsilon \to 0\). The main result is that for \(m\geq 2\) the solution \(u^{\varepsilon}\) converges to \(0\) or \(1\) on both sides of an interface moving with constant normal velocity \(c_m\) for compactly supported initial data.
scientific article; zbMATH DE number 5280781