On the porous medium equations with lower order singular nonlinear terms (Q1069041)

The problem \[ -u_ t+(u^ m)_{xx}-b(u^{\lambda})_ x-cu^ p=0\quad in\quad {\mathbb{R}}\times (0,\infty),\quad u(x,0)=u_ 0(x),\quad x\in {\mathbb{R}}, \] is studied, where \(m\geq 1\), \(\lambda >0\), \(p>0\), \(b>0\), \(c>0\) are constants and \(u_ 0\) is nonnegative continuous function with compact support. First the regularity of the solution is proven in the form: \(| (u^{\sigma})_ x| \leq cons\tan t\), where the constant \(\sigma\) is given by m, \(\lambda\), p. Finally some propagation properties, localization of the support and the behavior of interfaces are considered.