Regularity of the free boundary for a parabolic cooperative system

In this paper we study the following parabolic system $$\begin{aligned} \Delta \mathbf{u }-\partial _t \mathbf{u }=|\mathbf{u }|^{q-1}\mathbf{u }\,\chi _{\{ |\mathbf{u }|>0 \}}, \qquad \mathbf{u }= (u^1, \cdots , u^m) \ , \end{aligned}$$ Δ u - ∂ t u = | u | q - 1 u χ { | u | > 0 } , u = ( u 1 , ⋯ , u m ) , with free boundary $$\partial \{|\mathbf{u }| >0\}$$ ∂ { | u | > 0 } . For $$0\le q 0 ≤ q < 1 , we prove optimal growth rate for solutions $$\mathbf{u }$$ u to the above system near free boundary points, and show that in a uniform neighbourhood of any a priori well-behaved free boundary point the free boundary is $$C^{1, \alpha }$$ C 1 , α in space directions and half-Lipschitz in the time direction.