On the finite time blow-ups for solutions of nonlinear differential equations

We study systems of nonlinear ordinary differential equations where the dominant term, with respect to large spatial variables, causes blow-ups and is positively homogeneous of a degree $1+\alpha$ for some $\alpha >0$. We prove that the asymptotic behavior of a solution $y(t)$ near a finite blow-up time $T_{\ast }$ is $(T_{\ast }-t{)}^{-1\mathrm{/}\alpha }{\xi }_{\ast }$ for some nonzero vector ${\xi }_{\ast }$. Specific error estimates for $\mathrm{\mid }(T_{\ast }-t{)}^{1\mathrm{/}\alpha }y(t)-{\xi }_{\ast }\mathrm{\mid }$ are provided. In some typical cases, they can be a positive power of $(T_{\ast }-t)$ or $1\mathrm{/}\mathrm{\mid }\ln (T_{\ast }-t)\mathrm{\mid }$. This depends on whether the decaying rate of the lower order term, relative to the size of the dominant term, is of a power or logarithmic form. Similar results are obtained for a class of nonlinear differential inequalities with finite time blow-up solutions. Our results cover larger classes of nonlinear equations, differential inequalities and error estimates than those in the previous work.