Finite Speed Of Propagation Of The Support Of Solutions To A Thin Film Equation In One Dimension

In this thesis, we consider the following thin film equation in one dimension space ht+(hhxxx)x+(h3hx)x=0, x∈(-L, L), t> 0, h(x,0)=ho(x),x∈(-L,L), where initial data satisfy ho(x)> 0, ho(x)∈L1(-L,L)∩H1(-L,L).Here unknown function h(x, t) represents the hight of the liquor thin film. There are two nonlinear terms:one is the fourth order derivative term and it is a stabilizing term; the second one is a second derivative term and it is a long-wave destabilizing term. The equation can be used to describe the flow of liquid membrane. In this paper, we mainly prove the Holder continuity and the finite speed of propagation of the support for solutions to the thin film equation. In particular, this thesis will be separated into three parts. The first one is the introduction and the key preliminary results. In the second part, we prove that solutions possess the Holder continuity with the index of 1/2 respect to the space variable x, and the Holder continuity with the index of 1/8 respect to the time variable t. Finally, we prove that the support of the solution has the finite speed of propagation.