### A class of growth and bounds to solutions of a differential equation.

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Under suitable hypotheses on $\gamma \left(t\right)$, $\lambda \left(t\right)$, $q\left(t\right)$ we prove some stability results which relate the asymptotic behavior of the solutions of ${u}^{\text{'}\text{'}}+\gamma \left(t\right){u}^{\text{'}}+(q\left(t\right)+\lambda \left(t\right))u=0$ to the asymptotic behavior of the solutions of ${u}^{\text{'}\text{'}}+q\left(t\right)u=0$.

A stochastic system of particles is considered in which the sizes of the particles increase by successive binary mergers with the constraint that each coagulation event involves a particle with minimal size. Convergence of a suitably renormalized version of this process to a deterministic hydrodynamical limit is shown and the time evolution of the minimal size is studied for both deterministic and stochastic models.