Linear stability of the non-zero equilibrium state for the viscous Burgers equation with time delay

Abstract In this article, we consider the linear stability of the non-zero equilibrium state for the viscous Burgers equation with a delay effect. The linear stability is analyzed by using the characteristic equation of the corresponding eigenvalue problem. If our equation does not have a delay effect, the characteristic equation is given by a polynomial equation. On the other hand, if our equation has a delay effect, the characteristic equation becomes a transcendental equation, and it is difficult to analyze it. In this situation, we apply the useful known result concerned with the characteristic equation for the ordinary delay differential equations and try to get the sharp stability condition for the viscous Burgers equation with delay. Furthermore, we apply the obtained results to a model of the traffic flow.