Solving a system of nonlinear ODE in matrix form

Question

I am interested in the following equation for $v(t) \in \mathbb{R}_+^n$ and $m \in \mathbb{R}$, $m > 1$:

$v' = -A \cdot v^m,$

where $A$ is a positive-definite and symmetric matrix, and $v^m$ denotes componentwise exponentiation. Under what conditions may this system of nonlinear ODEs be solved explicitly, and what can be said about the asymptotics of the solutions?

Answer 1

If $A$ is a diagonal matrix. The system is decoupled. Each component is a separable ordinary differential equation. You can determine the solution analytically.