Semi-waves with Λ-shaped free boundary for nonlinear Stefan problems: Existence
Yihong Du, Changfeng Gui, Kelei Wang, Maolin Zhou · Proceedings of the American Mathematical Society · 2021
We show that for a monostable, bistable or combustion type of nonlinear function f ( u ) f(u) , the Stefan problem \[ { a m p ; u t − Δ u = f ( u ) , u > 0 a m p ; a m p ; for x ∈ Ω ( t ) ⊂ R n + 1 , a m p ; u = 0 and u t = μ | ∇ x u | 2 a m p ; a m p ; for x ∈ ∂ Ω ( t ) , \left \{ \begin {aligned} &u_t-\Delta u=f(u),\; u>0 & &\text {for}~~x\in \Omega (t)\subset \mathbb {R}^{n+1},\\ & u=0~\text {and}~u_t=\mu