On the behavior and cases of nonexistence of the free-boundary in a semibounded porous-medium

Abstract

The authors consider the Fokker-Planck equation ut=(um)xx+b(uλ)x, x>0, t>0, with initial and boundary data u(x,0)=u0(x), x>0, u(0,t)=u1(t), t>0, u0 having its support in a bounded interval. They concentrate on the case 0<λ<1, m≥1 with the aim of investigating the behavior of the free boundary, i.e. the moving boundary of suppu, in various different cases. When b>0 it is shown that if u1 tends to zero as t→∞, then the free boundary tends to zero. If u1 vanishes in a finite time, so does the free boundary. The possibility that the free boundary tends to infinity is also discussed. Moreover, conditions are found on m,λ and on u1 such that the free boundary can be estimated from above (localization) and from below by a positive constant. When b<0 it is shown that the free boundary never exists (for λ≥1, m>1 the free boundary is known to start from the right endpoint of suppu0).