On the behaviour and cases of nonexistence of the free boundary in a semibounded porous medium
Consider the Cauchy-Dirichlet problem for the nonlinear Fokker-Planck equation \[ u_ t=(u^ m)_{xx}+b_ 0(u^{\lambda})_ x\quad in\quad {\mathbb{R}}^ 2_+,\quad u(x,0)=u_ 0(x),\quad x>0,\quad u(0,t)=u_ 1(t),\quad t>0, \] where \(m\geq 1\), \(\lambda >0\), \(b_ 0\in {\mathbb{R}}\), \({\mathbb{R}}^ 2_+=\{(x,t):\) \(x>0\), \(t>0\}\). This paper studies qualitative properties of the solution of this problem and, especially, the occurrence or not of the free boundaries, or interfaces, separating the sets where \(u>0\) and \(u=0\).
scientific article; zbMATH DE number 4101727