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 $${\partial }_{t}u=(u^m{)}_{xx}+a(x)(u^m{)}_x+b(x)u^m,$$ posed for $x\in \Re$, $t\ge 0$ and $m>1$, where $a$, $b$ are two continuous real functions, and the solutions to the nonhomogeneous diffusion equation of porous medium type $$f(y){\partial }_{\tau }\theta =({\theta }^m{)}_{yy},$$ posed in the half-line $y\in [0,\infty )$ with $\tau \ge 0$, $m>1$ 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 $x\to -\infty$. 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 $f(y)=y^{-\gamma }$ with $\gamma >2$.