Spreading and vanishing in nonlinear diffusion problems with free boundaries
The authors study classical solutions to the free boundary problem \[ \begin{cases} u_t = u_{xx} + f(u), \qquad & (t,x) \in (0,\infty) \times (g(t),h(t)), \\ u(t,g(t))= u(t,h(t))=0, \qquad & t \in (0,\infty), \\ g^\prime (t) = -\mu u_x(t,g(t)), \; h^\prime (t) = -\mu u_x(t,h(t)), \qquad & t \in (0,\infty), \\ -g(0) = h(0) = h_0, \; u(0,x) = u_0(x), \qquad & x \in [-h_0, h_0], \end{cases} \] for the semilinear heat equation, where \(\mu\) and \(h_0\) are positive, \(f \in C^1([0,\infty))\) with \(f(0)=0\), and the initial data \(u_0 \in C^2 ([-h_0,h_0])\) are positive and satisfy the Dirichlet boundary conditions along with \(u_0^\prime \neq 0\) on the boundary. The authors prove that any global bounded solution \(u\) converges to a single stationary solution. For \(f\) being of monostable, bistable or combustion type, the authors further classify the large time behavior of the solution and distinguish, in particular, between spreading and vanishing (i.e., \(u(t)\) converges to 1 or 0 as \(t \to \infty\), respectively). In the case of spreading, they further determine the asymptotic speed of the wave front described by the free boundary. Important ingredients of their proofs are semi-waves and waves of finite length which are used as sub- or supersolutions and allow to adapt the concept of traveling waves to the free boundary problem.
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