Nonlinear Stability for Dissipative Nonlinear Evolution Equations with Ellipticity

We study the Cauchy problem for a set of nonlinear evolution equations, which has been proposed for the investigation of nonlinear interaction between ellipticity and dissipation. The zero equilibrium is linearly stable for coefficients within certain range. We further establish the global existence, nonlinear stability, and optimal decay rate of the solution for coefficients within the same range, though, with suitable restriction to initial data.