Semi-waves with Λ-shaped free boundary for nonlinear Stefan problems: Existence

We show that for a monostable, bistable or combustion type of nonlinear function f ( u ) f(u) , the Stefan problem \[ { a m p ; u t − Δ u = f ( u ) , u > 0 a m p ; a m p ; for     x ∈ Ω ( t ) ⊂ R n + 1 , a m p ; u = 0   and   u t = μ | ∇ x u | 2 a m p ; a m p ; for     x ∈ ∂ Ω ( t ) , \left \{ \begin {aligned} &u_t-\Delta u=f(u),\; u>0 & &\text {for}~~x\in \Omega (t)\subset \mathbb {R}^{n+1},\\ & u=0~\text {and}~u_t=\mu |\nabla _x u|^2 && \text {for}~~x\in \partial \Omega (t), \end {aligned} \right . \] has a traveling wave solution whose free boundary is Λ \Lambda -shaped, and whose speed is κ \kappa , where κ \kappa can be any given positive number satisfying κ > κ ∗ \kappa >\kappa _* , and κ ∗ \kappa _* is the unique speed for which the above Stefan problem has a planar traveling wave solution. To distinguish it from the usual traveling wave solutions, we call it a semi-wave solution. In particular, if α ∈ ( 0 , π / 2 ) \alpha \in (0, \pi /2) is determined by sin ⁡ α = κ ∗ / κ \sin \alpha =\kappa _*/\kappa , then for any finite set of unit vectors { ν i : 1 ≤ i ≤ m } ⊂ R n \{\nu _i: 1\leq i\leq m\}\subset \mathbb R^n , there is a Λ \Lambda -shaped semi-wave with traveling speed κ \kappa , with traveling direction − e n + 1 = ( 0 , . . . , 0 , − 1 ) ∈ R n + 1 -e_{n+1}=(0,...,0, -1)\in \mathbb {R}^{n+1} , and with free boundary given by a hypersurface in R n + 1 \mathbb {R}^{n+1} of the form \[ x n + 1 = ϕ ( x 1 , . . . , x n ) = Φ ∗ ( x 1 , . . . , x n ) ) + O ( 1 )  as  | ( x 1 , . . . , x n ) | → ∞ , x_{n+1}=\phi (x_1,..., x_n)=\Phi ^*(x_1,...,x_n))+O(1)\text { as }|(x_1,..., x_n)|\to \infty , \] where \[ Φ ∗ ( x 1 , . . . , x n ) ≔ − [ max 1 ≤ i ≤ m ν i ⋅ ( x 1 , . . . , x n ) ] cot ⁡ α \Phi ^*(x_1,..., x_n)\colonequals - \left [\max _{1\leq i\leq m} \nu _i\cdot (x_1,..., x_n)\right ]\cot \alpha \] is a solution of the eikonal equation | ∇ Φ | = cot ⁡ α |\nabla \Phi |=\cot \alpha on R n \mathbb R^n .