Phase Transitions in a Relaxation Model of Mixed Type with Periodic Boundary Condition

We study the asymptotic behavior of solutions for a 2×2 relaxation model of mixed type with periodic initial and boundary conditions. We prove that the asymptotic behavior of the solutions and their phase transitions are dependent on the location of the initial data and the size of the viscosity. If the average of the initial data is in the hyperbolic region and the initial data does not deviate too much from its average,we prove that there exists a unique global solution and that it converges time-asymptotically to the average in the same hyperbolic region. No phase transition occurs after initial oscillations. If the average of the initial data is in the elliptic region and the initial data does not deviate too much from its average, and in addition if the viscosity is big, then the solution converges to the average in the same elliptic region, and does not exhibit phase transitions after initial oscillations. If, however, the viscosity is small, numerical evidence indicates that the solution oscillates across the hyperbolic and elliptic regions for all time, exhibiting phase transitions. In this case, we conjecture that the solution converges to an oscillatory standing wave (steady-state solution).