Diffusion, Cross-Diffusion, and Their Spike-Layer Steady States

where ut = ∂u ∂t and ∆ =∑ni=1 ∂2 ∂xi is the usual Laplace operator, Ω is a bounded smooth domain in Rn, υ is the unit outer normal to ∂Ω, and u0 is a realvalued continuous function (not identically zero) representing the initial heat distribution. Here the boundary condition implies that (1) is an isolated system. It is well known that the solution u(x, t) of (1) becomes smooth as soon as t becomes positive and eventually converges to the constant 1 |Ω| ∫Ω u0(x)dx as t tends to ∞. In other words, in an isolated system, no matter what the initial heat distribution is, eventually the heat distribution becomes homogeneous. A similar result holds when a source/sink term (or a reaction term) is present. That is, if we replace the linear heat equation in (1) by ut = ∆u + f (u),