Details

Autor(en) / Beteiligte

• Du, Yihong
• Gui, Changfeng
• Wang, Kelei
• Zhou, Maolin

Titel

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

Ist Teil von

• Proceedings of the American Mathematical Society, 2021-05, Vol.149 (5), p.2091-2104

Erscheinungsjahr

2021

Quelle

American Mathematical Society

Beschreibungen/Notizen

• We show that for a monostable, bistable or combustion type of nonlinear function f(u), the Stefan problem \displaystyle \left \{ \begin {aligned}&u_t-\Delta u=f(u),\; u>0 & &\text {for}... ...a _x u\vert^2 && \text {for}~~x\in \partial \Omega (t), \end{aligned} \right . has a traveling wave solution whose free boundary is \Lambda -shaped, and whose speed is \kappa , where \kappa can be any given positive number satisfying \kappa >\kappa _*, and \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 \alpha \in (0, \pi /2) is determined by \sin \alpha =\kappa _*/\kappa , then for any finite set of unit vectors \{\nu _i: 1\leq i\leq m\}\subset \mathbb{R}^n, there is a \Lambda -shaped semi-wave with traveling speed \kappa , with traveling direction -e_{n+1}=(0,...,0, -1)\in \mathbb{R}^{n+1}, and with free boundary given by a hypersurface in \mathbb{R}^{n+1} of the form \displaystyle x_{n+1}=\phi (x_1,..., x_n)=\Phi ^*(x_1,...,x_n))+O(1)\displaystyle \text { as }\vert(x_1,..., x_n)\vert\to \infty , where \displaystyle \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 \vert\nabla \Phi \vert=\cot \alpha on \mathbb{R}^n.