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

Abstract:

We show that for a monostable, bistable or combustion type of nonlinear function $f(u)$, the Stefan problem \[ \left \{ \begin {aligned} &u_t-\Delta u=f(u),\; u>0 & &\text {for}~~x\in \Omega (t)\subset \mathbb {R}^{n+1},\\ & u=0~\text {and}~u_t=\mu |\nabla _x u|^2 && \text {for}~~x\in \partial \Omega (t), \end {aligned} \right . \] has a traveling wave solution whose free boundary is $\Lambda$-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 \[ x_{n+1}=\phi (x_1,..., x_n)=\Phi ^*(x_1,...,x_n))+O(1)\text { as }|(x_1,..., x_n)|\to \infty , \] where \[ \Phi ^*(x_1,..., x_n)\colonequals - \left [\max _{1\leq i\leq m} \nu _i\cdot (x_1,..., x_n)\right ]\cot \alpha \] is a solution of the eikonal equation $|\nabla \Phi |=\cot \alpha$ on $\mathbb R^n$.