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 {psi(t) = -(1 - alpha)psi-theta(x) + alphapsi(xx), theta(t) = -(1 - alpha)theta +nupsi(x) + 2psitheta(x) + alphatheta(xx), (E) with initial data (psi,theta)(x, 0) = (psi(0)(x),theta(0)( x)) --> (psi+/-, theta+/-) as x --> +/-infinity, ( I) where alpha and nu are positive constants such that alpha < 1, ν < alpha( 1 - alpha). Through constructing a correct function (θ) over cap (x, t) defined by (2.13) and using the energy method, we show sup(xis an element ofR) (|(psi, theta)(x, t)|+ |(psi(x),theta(x))(x, t)|) --> 0 as t --> infinity and the solutions decay with exponential rates. The same problem is studied by Tang and Zhao [10] for the case of (psi+/-, theta+/-) = (0, 0).