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. Chakravarti, Phys. 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)=12 (x2+Cy2)+14 (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.