On a class of parabolic equations with variable density and absorption
Róbert Kersner, Guillermo Reyes, Alberto Tesei · Advances in Differential Equations · 2002
We investigate qualitative properties of solutions to the Cauchy problem for the equation $\rho(x)u_t=(u^m)_{xx}-c_0 u^p$, where $m>1$ and $c_0, p >0$; the initial data are nonnegative with compact support and the density $\rho(x)>0$ satisfies suitable decay conditions as $|x|\to\infty$. If $p \ge m$ and $\rho(x)$ decays not faster than $|x|^{-k}$, where $0 k^*$, the interfaces can disappear in finite time. It is also proved that solutions go to zero uniformly as $t \to\infty $, at variance from the case $c_0=0$.