Equivalence and finite time blow-up of solutions and interfaces for two nonlinear diffusion equations
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 $$ f(y)\partial_τθ=(θ^m)_{yy}, $$ posed in the half-line $y\in[0,\infty)$ with $τ\geq0$, $m>1$ 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 $f(y)=y^{-γ}$ with $γ>2$.