Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity

In this paper, we study the global existence and the asymptotic behavior of the solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects with initial data (ψ, θ)(x, 0) = (ψ0(x),θ0(x)) → (ψ±, θ±) as → ±∞, (I) where α and νare positive constants such that α < 1, ν < α(1 - α). Through constructing a correct function θ(x, t) defined by (2.13) and using the energy method, we show sup(|(ψ, θ)(x, t)|+ |(ψx, θx)(x, t)|) → 0 as t → ∞ and the solutions decay with exponential rates. The same problem is studied by Tang and Zhao [10] for the case of (ψ±, θ±) = (0, 0).