where c is an arbitrary nonnegative parameter. In some cases this maximum principle may be used to derive explicit exponential decay bounds for lul and luxl. Some extensions in N space dimensions are indicated. This work may be considered as a continuation of previous works by Payne and Philippin (Mathematical Models and Methods in Applied Sciences, 5 (1995), 95-110; Decay bounds in quasilinear parabolic problems, In: Nonlinear Problems in Applied Mathematics, Ed. by T.S. Angell, L. Pamela, Cook, R.E., SIAM, 1997).