Reactions-Diffusion Equations

where t ≥ 0 denotes time and x ∈ Ω ⊂ R position typical within a d−dimensional bounded domain Ω with “sufficiently smooth” boundary or Ω = R. The states u = (u1, . . . , um), i = 1, . . . , m describe the concentrations or densities of substances/populations ui = ui(t, x). The functions di(t, x) are called diffusion coefficients or diffusivities. On a bounded domain Ω the eqs. (1) have to be complemented with boundary conditions. Typically examples are Dirichlet, Neumann, and mixed Robin boundary conditions: Dirichlet conditions prescribe the value at the boundary ui(t, x) = fD(t, x) , x ∈ ∂Ω . The case ρD(t) = 0 is called homogeneous Dirichlet condition. Neumann conditions prescribe the diffusion-flux though the boundary ∂Ω di(t, x)∇x ui(t, x) = fN(t, x) , x ∈ ∂Ω . Note that homogeneous Neumann conditions fN = 0 prevent particles/individuals from passing through the boundary ∂Ω. Robin conditions mix Dirichlet and Neumann conditions (α ≥ 0, β ≥ 0, α+ β > 0) α di(t, x)∇x ui(t, x) + β ui(t, x) = fR(t, x) , x ∈ ∂Ω .