The boundary degeneracy theory of a strongly degenerate parabolic equation

A kind of strongly degenerate parabolic equations, partial derivative u/partial derivative t = partial derivative/partial derivative x(i) (a(ij)(u, x, t) partial derivative u/partial derivative x(j)) + partial derivative b(i)(u, x, t)/partial derivative x(i), (x, t) is an element of Omega x (0, T), is considered. The paper first shows that the solution of the equation may be free from the limitation of the boundary value condition. The key is to determine the portion of the boundary on which we can impose the homogeneous boundary value. By introducing a new kind of entropy solution matching the partial boundary condition, the existence of the solution is obtained by the parabolic regularization method, and the stability of the solutions is obtained by Kruzkov's bi- variables method combined with an elegant partition technique.