We show that for a monostable, bistable or combustion type of nonlinear function f (u), the Stefan problem
| { | ut -∆ u = f(u), u > 0 | for x ∈ Ω(t) ⊂ ℝn+1, |
|
| | u = 0 and ut = μ | ∇xu|2 | for x ∈ Ωt, |
has a traveling wave solution whose free boundary is Λ-shaped, and whose speed is κ, where κ can be any given positive number satisfying κ > κ*, and κ* 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) is determined by sin α = κ(*/)κ, then for any finite set of unit vectors (νi : 1 <= i <= m} ⊂ ℝn, there is a Λ-shaped semi-wave with traveling speed κ, with traveling direction -en+1n+1 , and with free boundary given by a hypersurface in ℝn+1 of the form
xn+1 = π(x1, ..., xn) = ɸ* (x1, ..., xn) + O(1) as |(x1, ..., xn)| -> ∞,
where
ɸ* (x1, ..., xn) \ colonequals - [max(1 <= i <= m) νi . (x1, ..., xn)] cot α
is a solution of the eikonal equation | ∇ ɸ | = cot α on ℝn.