Local boundedness for doubly degenerate quasi-linear parabolic systems

in the cylinder QT = R x (0, T), 0 < T < 00, where R is a bounded region in Rn, n > 2, z = (Xl,. . . ,Zn), 21 = (d,. . . ,uN), N > 1, IuI = [‘&?1(u”)2]1/2, 2 denotes the gradient, ( $$=I ,..., N; j-1 ,..., ,, of U, and C is a positive constant. For simplicity, in the sequel we shall assume that a pair of same indices will mean summation from 1 to n or from 1 to N. For p 2 2, system (1) is degenerate when u = 0, or when the gradient of u vanishes. We do not consider here the case when p < 2; this corresponds to a situation when the system can be degenerate and singular. We note that the case when a! = 0 is known to be of importance in fluid mechanics (11. DiBenedetto and Friedman [2] have studied the c8se cr = 0 and established a