Global decaying solution to dissipative nonlinear evolution equations with ellipticity

We establish the global existence and decaying results for the Cauchy problem of nonlinear evolution equations:equation(E)ψt=-(1-α)ψ-θx+ψψx+αψxx,θt=-(1-α)θ+νψx+2ψθx+αθxx,forinitial data with different end states,equation(I)(ψ(x,0),θ(x,0))=(ψ0(x),θ0(x))→(ψ±,θ±),asx→±∞,which displays the complexity in between ellipticity and dissipation. Although the nonlinear term ψψx appears in equation (E)1, which makes calculations more complicated, due to smoothing effect of the parabolic operator, we detail its regularity property and decay estimates when t > 0 for the higher order spatial derivatives despite its relatively lower regularity of the initial data, and we also discuss the decay estimates. Furthermore, we do not restrict L1 bound on the initial data (ψ0(x), ϕ0(x)) as in [2].