Abstract:
In the domain $D=\{0<x<X,\,0<y<\infty\}$, the following problem with free boundary $\Gamma:y=y_*(x)>0$, $x\in[0,X]$ is studied:
$$
u_iu_{ix}+v_iu_{iy}=\nu\bigl(|u_{iy}|^{n_i-1}u_{iy}\bigr)_y+U(x)U_x(x), \quad u_{ix}+v_{iy}=0,
$$
where $i=1$ for $0\le y\le y_*(x)$, $i=2$ for $y_*(x)<y<\infty$; $\nu_i>0$, $n_1=n>1$, $n_2=1$; $u_i(0,y)=u_{i0}(y)$, $u_1(x,0)=0$, $v_1(x,0)=v_0(x)>0$, $u_2(x,y)\to U(x)>0$ as $y\to\infty$;
$$
\frac{\partial y_*(x)}{\partial x}=\frac{v_i(x,y_*(x))}{u_i(x,y_*(x))}
$$ $y_*(0)>0$ is a given point; and $u_1=u_2$, $\nu_1|u_{1y}|^{n-1}u_{1y}=\nu_2u_{2y}$ on the curve $\Gamma$. Under some conditions, existence and uniqueness for a solution to the problem are proved.
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