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

new materials were added. To appear in Evolution Equations and Control Theory (31 pages)

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