The canonical ensemble relaxation function of a particle in a symmetric anharmonic potential well in D = 1 is known to exhibit slow algebraic behavior [S. Sen, R. S. Sinkovits and S. Chakravarti, Phys. Rev. Lett. 77, 4855 (1996); R. S. Sinkovits, S. Sen, J. C. Phillips, and S. ChakravartiPhys. Rev. E 59, 6497 (1999)]. In the present work, we report a study of relaxation of a particle in symmetric and asymmetric quartic anharmonic potential wells of the form V(x,y)= 1/2(x2 + Cy2)+ 1/4 (x2+ Cy2)2 in D = 2. The relaxation in the above system is identical to that in D= 1 wells when C=0 (since it is then a D= 1 system) and C= 1. However, for 0 <C<1 and for C≫1, the frequencies associated with well dynamics are strongly affected and hence the power spectra are altered as a function of C. Our calculations suggest that the exponents of the long-time tails associated with the relaxation processes are insensitive to D. In closing, we comment on the consequences of our analysis for the study of slow dynamics in interacting many-particle systems that are connected by harmonic springs with the individual particles in anharmonic potential wells. ©2001 The American Physical Society.