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

Abstract

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 $T_*$ is $(T_*-t)^{-1/\alpha}\xi_*$ for some nonzero vector $\xi_*$. Specific error estimates for $|(T_*-t)^{1/\alpha}y(t)-\xi_*|$ are provided. In some typical cases, they can be a positive power of $(T_*-t)$ or $1/|\ln(T_*-t)|$. 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.