On the positivity of the fundamental solutions for parabolic systems (Q1124048)

The author considers the Cauchy problem for parabolic systems of the first order in t, \[ Lu\equiv (\partial /\partial t)u(t,x)-\sum_{| \alpha | \leq 2m}A_{\alpha}(t,x)(\partial /\partial x)^{\alpha}u(t,x)=0, \] \[ u(t_ 0,x)=u_ 0(x),\quad t_ 0<t\leq T,\quad x\in R^ n, \] where \(u(t,x)=^ t(u_ 1(t,x),u_ 2(t,x),...,u_ N(t,x))\) are n-dimensional column vectors and \(A_{\alpha}(t,x)=(a_{\alpha,j,k}(t,x))_{1\leq j,k\leq N}\) are \(N\times N\) matrices, and then, gives a necessary and sufficient condition for the positivity of fundamental solution.