Regularity of the free boundary for a parabolic cooperative system

Abstract

In this paper we study the following parabolic system Δu-∂tu=|u|q-1uχ{|u|>0},u=(u1,⋯,um),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Delta \\mathbf{u }-\\partial _t \\mathbf{u }=|\\mathbf{u }|^{q-1}\\mathbf{u }\\,\\chi _{\\{ |\\mathbf{u }|>0 \\}}, \\qquad \\mathbf{u }= (u^1, \\cdots , u^m) \\ , \\end{aligned}$$\\end{document}with free boundary partial {|mathbf{u }| >0}. For 0le q<1, we prove optimal growth rate for solutions mathbf{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 } in space directions and half-Lipschitz in the time direction.