Interfaces in diffusion–absorption processes in nonhomogeneous media

We study the Cauchy problem for the nonlinear parabolic equation ρ(x)ut=(a(x)ϕx(u))x−b(x)h(u)in  R×(0,T] with nonnegative coefficients ρ(x)ρ(x), a(x)a(x) and b(x)b(x). It is assumed that ϕ(0)=0ϕ(0)=0, ϕ′(s)>0ϕ′(s)>0, ϕ′(s)/s∈L1(0,δ)ϕ′(s)/s∈L1(0,δ) for some δ>0δ>0, h(s)≥0h(s)≥0 and h(s)/sh(s)/s is nondecreasing for s≥0s≥0. The solution of this problem may possess the property of finite speed of propagation of disturbances from the data, which leads to formation of interfaces that bound the support of the solution. It is proved that the behavior of interfaces can be characterized in terms of convergence or divergence of the integrals ∫x0xρ(s)(∫x0sdza(z))ds,Jx0(x)=b(x)ρ(x)∫x0x(∫0sρ(z)a(z)dz)ds,b(x)ρ(x)Jx0(x),∫x0xρ(s)ds as x→∞x→∞ and ∫ϵdsh(s),∫ϵψ(s)h(s)dsas  ϵ→0+. We derive two-sided a priori bounds for the interface location, establish sufficient and necessary conditions for disappearance of interfaces in a finite time (the interface blow-up), and derive the integral equation for the interface.