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In this work, we construct a transformation between the solutions to thefollowing reaction-convection-diffusion equation $$ \partial_tu=(u^m)_{xx}+a(x)(u^m)_x+b(x)u^m, $$ posed for $x\in\real$, $t\geq0$ and $m>1$,where $a$, $b$ are two continuous real functions, and the solutions to thenonhomogeneous 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 thefirst equation blow up in finite time, together with their interfaces, as$x\to-\infty$. We then establish the large time behavior of solutions to ahomogeneous Dirichlet problem associated to the first equation on a boundedinterval. We also prove a finite time blow-up of the interfaces for compactlysupported solutions to the second equation when $f(y)=y^{-\gamma}$ with$\gamma>2$.

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