Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity

We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space equation presented, with 0 < ν² < 4 α (1 - α), 0 < α < 1. S. Q. Tang and H. Zhao [4] have considered the problem and obtained the optimal decay property for suitably small data. In this paper we derive the asymptotic profile using the Gauss kernel G(t, x), which shows the precise behavior of solution as time tends to infinity. In fact, we will show that the asymptotic formula equation presented, holds, where D ₀, β₀ are determined by the data. It is the key point to reformulate the system to the nonlinear parabolic one by suitable changing variables.