Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 08, pp. 1-26.

Title: Behavior near the extinction time for systems of differential equations with sublinear dissipation terms

Author: Luan Hoang (Texas Tech Univ., Lubbock, TX, USA)
  
Abstract:
This article focuses on the behavior near the extinction time of 
solutions to systems of ordinary differential equations with a 
sublinear dissipation term. Suppose the dissipation term is a product 
of a linear mapping $A$ and a positively homogeneous scalar  function 
$H$ of a negative degree $-\alpha$.  Then any solution  with an extinction 
time $T_*$ behaves like $(T_*-t)^{1/\alpha}\xi_*$ as time $t\to T_*^-$, 
where $\xi_*$ is an eigenvector of $A$. The result allows the higher 
order terms to be general and the nonlinear function $H$ to take very 
complicated forms. As a demonstration, our theoretical study is applied 
to an inhomogeneous population model.

Submitted April 26, 2024. Published January 17, 2025.
Math Subject Classifications: 34D05, 41A60.
Key Words: Sublinear dissipation; extinction time; extinction profile; 
 vanish in finite time; asymptotic behavior; asymptotic approximation.