On an evolution problem of thermocapillarity convection

We consider a free boundary problem of incompressible viscous flow governing the motion of an isolated liquid mass. The liquid is subjected to capillary forces at the boundary and the coefficient of the surface tension depends on the temperature satisfying the heat equation with convection and dissipation terms. It is shown that this problem is solvable in a certain finite time interval, however, if the data are close to the rest state the solution can be extended to the interval t>0. In the case when the temperature satisfies the heat equation without dissipation term a local existence theorem was proved by M.V. Lagunova and V.A. Solonnikov [2], and global result was obtained by V.A.Solonnikov [7].