The author studies the conditions for instantaneous support shrinking and sharp estimates for the support of the solution to the following Cauchy problem:
\[ \frac{\partial }{\partial t}(\left| u\right| ^{\beta -1}u(x,t))-\nabla (\left| \nabla u\right| ^{p-2}\nabla u)+\left| u\right| ^{r-1}u(x,t)=0,\text{ }x\in \mathbb{R}^{N},\text{ }t>0 \]
and
\[ \left| u\right| ^{\beta -1}u(x,0)=\left| u_{0}\right| ^{\beta -1}u_{0}(x),\text{ }x\in \mathbb{R}^{N}. \]
The case of slow diffusion and strong absorption i.e. \(\beta >0,p>\beta +1\) and \(r<\beta \) is considered in this work.
For a non negative (non positive) initial function, he gives a necessary and sufficient condition for instantaneous support shrinking in terms of the local behaviour of the mass of the initial data. Moreover, he estimates the size of the support. When the initial function changes sign, he gives only an upper estimates of the support. He uses the method of local integral estimates and estimates for the maximum of the absolute value in terms of the mass of the solution.