Reaction–diffusion equations in the half-space

Abstract

We study reaction–diffusion equations of various types in the half-space. For bistable reactions with Dirichlet boundary conditions, we prove conditional uniqueness: there is a unique nonzero bounded steady state which exceeds the bistable threshold on large balls. Moreover, solutions starting from sufficiently large initial data converge to this steady state as $t\to \infty$. For compactly supported initial data, the asymptotic speed of this propagation agrees with the unique speed $c_{\ast }$ of the one-dimensional traveling wave. We furthermore construct a traveling wave in the half-plane of speed $c_{\ast }$.

In parallel, we show analogous results for ignition reactions under both Dirichlet and Robin boundary conditions. Using our ignition construction, we obtain stronger results for monostable reactions with the same boundary conditions. For such reactions, we show in general that there is a unique nonzero bounded steady state. Furthermore, monostable reactions exhibit the hair-trigger effect: every solution with nontrivial initial data converges to this steady state as $t\to \infty$. Given compactly supported initial data, this disturbance propagates at a speed $c_{\ast }$ equal to the minimal speed of one-dimensional traveling waves. We also construct monostable traveling waves in the Dirichlet or Robin half-plane with any speed $c\ge c_{\ast }$.