论文标题
半空间中的反应扩散方程
Reaction-diffusion equations in the half-space
论文作者
论文摘要
我们研究半空间中各种类型的反应扩散方程。对于具有差异边界条件的可动反应，我们证明了有条件的唯一性：有一个独特的非零界稳态稳态，超过了大球上的Bistable阈值。此外，从足够大的初始数据开始的解决方案以$ t \ to \ infty $收敛到该稳态。对于紧凑的初始数据，这种传播的渐近速度与一维行驶波的独特速度$ c _*$一致。我们此外，在速度$ c _*$的半平面中构建了一个行驶波。 同时，我们在Dirichlet和Robin边界条件下都显示出点火反应的类似结果。使用我们的点火结构，我们为具有相同边界条件的单一反应获得了更强的结果。对于此类反应，我们通常表明存在一个独特的非零界面稳态。此外，单稳态反应表现出发触发效果：每个具有非平凡初始数据的解决方案都将稳定状态收敛为$ t \ to \ infty $。给定紧凑的初始数据，这种干扰以$ c _*$等于一维行驶波的最低速度的速度传播。我们还以任何速度$ c \ geq c _*$构建了Dirichlet或Robin半平面的单个行进波。
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 $c_*$ of the one-dimensional traveling wave. We furthermore construct a traveling wave in the half-plane of speed $c_*$. 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 $c_*$ 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 $c \geq c_*$.
