Abstract: For a strongly degenerate parabolic equation,we first use the results of Ph.Blanc to get that the Neumann problem may not have a global classical solution.The reason is that the gradierts of some local classical solutions blow up in finite time.And under some conditions,we get the global integral solution by the use of seanigroup.Then,we give a sufficient condition under
which the Neumann problem has e global classical solution.Finally.we study the asymptotic be- haviour of the classical solution and prove that the solution converges to the mean of the initial value.

Key words: strongly degenerate parabolic equation, semigroup, comparison theorem ,