Equivalence and finite time blow-up of solutions and interfaces for two nonlinear diffusion equations

Abstract

In this work, we construct a transformation between the solutions to the following reaction-convection-diffusion equation u_t = (u^m)_{xx} + a(x)(u^m)_x + b(x)u^m, posed for x ∈ R, t ≥ 0 and m > 1, where a, b are two continuous real functions, and the solutions to the nonhomogeneous diffusion equation of porous medium type f(y)θ_τ = (θ^m)_{yy}, posed in the half-line y ∈ [0, ∞) with τ ≥ 0, m > 1 and suitable density functions f(y). We apply this correspondence to the case of constant coefficients a(x) = 1 and b(x) = K > 0. For this case, we prove that compactly supported solutions to the first equation blow up in finite time, together with their interfaces, as x → −∞. We then establish the large time behavior of solutions to a homogeneous Dirichlet problem associated to the first equation on a bounded interval. We also prove a finite time blow-up of the interfaces for compactly supported solutions to the second equation when f(y) = y^{−γ} with γ > 2.