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

TL;DR

This paper constructs a transformation linking solutions of two nonlinear diffusion equations, demonstrating finite-time blow-up of solutions and interfaces, and analyzing their long-term behavior under specific conditions.

Contribution

It introduces a novel transformation between reaction-convection-diffusion and porous medium type equations, revealing finite-time blow-up and interface behavior.

Findings

Solutions blow up in finite time as x→−∞ for constant coefficients.

Large time behavior of solutions on bounded intervals is characterized.

Interfaces of solutions also blow up in finite time under certain conditions.

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 $x\in\real$, $t\geq0$ 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 $\tau\geq0$, $m>1$ 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 $\gamma>2$.