Remarks on the large time behavior of viscosity solutions of   quasi-monotone weakly coupled systems of Hamilton--Jacobi equations
  We investigate the large-time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton--Jacobi equations on the $n$-dimensional torus. We establish a convergence result to asymptotic solutions as time goes to infinity under rather restricted assumptions. 
Hiroyoshi Mitake, Hung V. Tran
math.AP