The author studies the equation (*) \(u_ t=\Delta \phi (u)+f(x)u\), \(x\in {\mathbb{R}}^ N\), \(t\geq 0\) where \(\phi\) is an increasing function, and \(f(x_ 0)>0\) for some \(x_ 0\) and \(f>0\) for \(| x|\) large. It is shown that under appropriate conditions the corresponding steady state problem admits a positive solution of compact support and \(u\in C^{1,\lambda}({\mathbb{R}}^ N)\) for all \(\lambda \in (0,1).\) The solution of the nonstationary problem (*) is shown to exist also under appropriate conditions. Some important regularity properties are then proved in the nonstationary case as well.