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

Abstract

<p>We show that for a monostable, bistable or combustion type of nonlinear function <i>f</i> (<i>u</i>), the Stefan problem</p> <table><tr><td>{</td><td>u<sub>t</sup> -∆ <i>u = f</i>(<i>u</i>), <i>u</i> > 0</td><td> for <i>x</i> ∈ Ω(<i>t</i>) ⊂ ℝⁿ⁺¹,</td><tr><tr><td> </td><td><i>u</i> = 0 and <i>u</i>_t = μ | ∇_x<i>u</i>|²</td><td>for x ∈ Ω<sub>t</sup>,</td></tr></table> <p>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 (ν_i : 1 <= <i>i</i> <= <i>m</i>} ⊂ ℝⁿ, there is a Λ-shaped semi-wave with traveling speed κ, with traveling direction -e<sub>n+1</sub = (0, ..., 0, -1) ∈ ℝⁿ⁺¹ , and with free boundary given by a hypersurface in ℝⁿ⁺¹ of the form</p> <p><i>x</i>_n+1 = π(<i>x</i>₁, ..., <i>x</i>_n) = ɸ* (<i>x</i>₁, ..., <i>x</i>_n) + <i>O</i>(1) as |(<i>x</i>₁, ..., <i>x</i>_n)| -> ∞,</p> <p>where</p> <p>ɸ* (<i>x</i>₁, ..., <i>x</i>_n) \ colonequals - [max(1 <= <i>i</i> <= <i>m</i>) ν_i . (<i>x</i>₁, ..., <i>x</i>_n)] cot α</p> <p>is a solution of the eikonal equation | ∇ ɸ | = cot α on ℝⁿ.</p>