Critical growth velocity in diffusion-controlled aggregation

The coupled nonlinear partial differential equations for diffusion-controlled aggregation are solved analytically for dimension $d>\mathrm{}2\mathrm{}$ by a series expansion in kinks with propagation velocity $v>~v_c$. It is shown that the critical velocity $v_c={\epsilon }_{c}a$, where $a$ is a parameter in these equations, and ${\epsilon }_c$ is a constant which depends on $d$, and is evaluated for $d=3-10\mathrm{}$. A bounded initial seed density always grows asymptotically at this critical velocity.