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Semi-waves with \(\Lambda\)-shaped free boundary for nonlinear Stefan problems: existence. (English) Zbl 1461.35249

Summary: We show that for a monostable, bistable or combustion type of nonlinear function \(f(u)\), the Stefan problem \[ \begin{cases} u_t-\Delta u=f(u),\; u>0\quad & \text{for }x\in\Omega(t)\subset\mathbb{R}^{n+1},\\ u=0\text{ and }u_t=\mu|\nabla_xu|^2\quad & \text{for }x\in\partial\Omega(t), \end{cases} \] 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 \(-e_{n+1}=(0,\dots,0, -1)\in \mathbb{R}^{n+1} \), and with free boundary given by a hypersurface in \(\mathbb{R}^{n+1}\) of the form \[ x_{n+1}=\phi(x_1,\dots, x_n)=\Phi^*(x_1,\dots,x_n))+O(1)\text{ as }\vert(x_1,\dots, x_n)\vert\to\infty, \] where \[ \Phi^*(x_1,\dots,x_n):=-\left[\max\limits_{1\leq i\leq m}\nu_i\cdot (x_1,\dots, x_n)\right]\cot\alpha \] is a solution of the eikonal equation \(\vert\nabla\Phi\vert=\cot\alpha\) on \(\mathbb{R}^n\).

MSC:

35R35 Free boundary problems for PDEs
35C07 Traveling wave solutions
35K20 Initial-boundary value problems for second-order parabolic equations
35K58 Semilinear parabolic equations
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