Non-uniform continuity of the semiflow map associated to the porous medium equation

We prove that the semiflow map associated to the evolution problem for the porous medium equation (PME) is real-analytic as a function of the initial data in $H^s(\mathbb{S})$, $s>7/2,$ at any fixed positive time, but it is not uniformly continuous. More precisely, we construct two sequences of exact positive solutions of the PME which at initial time converge to zero in $H^s(\mathbb{S})$, but such that the limit inferior of the difference of the two sequences is bounded away from zero in $H^s(\mathbb{S})$ at any later time.