Cauchy Problem of Some Doubly Degenerate Parabolic Equations with Initial Datum a Measure

This paper discusses the Cauchy problem of the equation1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_{t} = \nabla \cdot {\left( {{\left| \nabla \right|}_{u} ^{m} \left| {^{{p - 2}} } \right.\nabla _{u} ^{m} } \right)} - u^{q} $$\end{document} with initial datum a measure. Under the assumption of the parameters, one proves the existence and non-existence of the non-negative generalized solution.