This paper deals with the following growth model; let \(\{K_t\}\), \(t\geq t_0\), be a growing family of connecting sets, where \(K_{t_0}\) is the initial configuration, and \(K_s\subset K_t\) for \(s 1\}\to \Omega_t\qquad (\infty\mapsto\infty,\;\varphi_t'(\infty)> 0). \] The main goal of the paper is the stability analysis of stationary solutions. By definition chain \(\{K_t\}\) is called stationary of \(K_t= \lambda(t) K_{t_0}\) for some positive increasing function \(\lambda(t)\), which means that the shape of configurations \(K_t\) does not change.