We investigate the dynamics of a two-dimensional axial next-nearest- neighbor Ising model following a quench to zero temperature. The Hamiltonian is given by H=- J0 i,j=1 L Si,j Si+1,j - J1 i,j=1 (Si,j Si,j+1 -κ Si,j Si,j+2). For κ<1, the system does not reach the equilibrium ground state but slowly evolves to a metastable state. For κ>1, the system shows a behavior similar to that of the two-dimensional ferromagnetic Ising model in the sense that it freezes to a striped state with a finite probability. The persistence probability shows algebraic decay here with an exponent θ=0.235±0.001 while the dynamical exponent of growth z=2.08±0.01. For κ=1, the system belongs to a completely different dynamical class; it always evolves to the true ground state with the persistence and dynamical exponent having unique values. Much of the dynamical phenomena can be understood by studying the dynamics and distribution of the number of domain walls. We also compare the dynamical behavior to that of a Ising model in which both the nearest and next-nearest-neighbor interactions are ferromagnetic. © 2008 The American Physical Society.