Initial trace for a porous medium equation: II. The critical absorption term

We consider the following non linear diffusion equation with critical absorption exponent: $u_{t}-\Delta u^{m}+u^{m}=0$ , $m>1$ , in a bounded set $\varOmega $ and in $\mathbb{R}^{N}$ . In the whole space, we give optimal conditions on measure initial data to have existence and uniqueness, and we study singular solutions, which explode in $\varOmega $ when $t\rightarrow0$ . The problem of the initial trace is also completely addressed, and we derive estimates for the initial trace of continuous solutions.