Cauchy problem of some doubly degenerate parabolic equations with initial datum a measure
The author studies the existence of nonnegative solutions of the Cauchy problem \(u_t=\nabla\cdot(| \nabla u^m| ^{p-2}\nabla u^m)-u^q\) in \(\mathbb R^N\times(0,\infty)\), \(u(\cdot,0)=\mu\) in \(\mathbb R^N\), where \(p>1\), \(m>0\), \(q>0\) and \(\mu\) is a nonnegative bounded Radon measure. The existence is proved provided \(m(p-1)+p/N>1\) and \(q<m(p-1)+p/N\). It is also shown that these conditions are essentially optimal. The existence proof is based on the approximation of \(\mu\) by smooth functions.
scientific article; zbMATH DE number 2130973