Kinetic method for solving systems of differential equations (Q393841)

The system of ordinary differential equations \[ \frac{du}{dt}=f(u, t),\quad t>0, u\in \Omega_n \tag{1} \] is investigated, where the right-hand side \(f=\{f_i\}_{i=1}^n\), consists of polynomials in \(u=\{u_i\}_{i=1}^n\) at the interior points of \(\Omega_n\) with locally bounded coefficients for \(t\geq 0\), \(\Omega_n=\prod_{i=1}^n[0, 1]\subset \mathbb R_n\). The equations (1) are associated with a kinetic process involving interactions of particles of \(n\) different types and the number of the types coincides with the dimension of the system (1). The author gives conditions for the coefficients of \(f\) that ensure existence and uniqueness of a nonnegative solution of the Cauchy problem for (1) on the cube \(\Omega_n\).