Qualitative Properties Of A Kind Of Singular Diffusion Equations

There are two parts in this thesis. In the first part we discuss the Cauchy problem of a singular diffusion equation with absorption, and in the second part we discuss the Cauchy problem of a singular diffusion equation with heat source.1. Consider the Cauchy problem for singular diffusion equation with absorption: Conclusion:We first give the expression of the large time behavior, gradient estimates and some other important estimates of solutions. And then, we discuss the geometrical properties of the solution. At last we prove the continuous dependence of solutions with the initial data.2. Consider the Cauchy problem for singular diffusion equation with heat source: Conclusion:The expression of the large time behavior, gradient estimates and some other important estimates of solutions are given, and the geometrical properties of the solution are discussed.