Decay rates of solutions to the Cauchy problem for dissipative nonlinear evolution equations

Abstract

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 (E) { ψ t = − ( 1 − α ) ψ − θ x + α ψ x x , θ t = − ( 1 − α ) θ + ν ψ x + ( ψ θ ) x + α θ x x , with initial data (I) ( ψ , θ ) ( x , 0 ) = ( ψ 0 ( x ) , θ 0 ( x ) ) → ( ψ ± , θ ± ) as x → ± ∞ , where α and ν are positive constants such that α < 1 , ν < α ( 1 − α ) . Through constructing a correct function θ ˆ ( x , t ) defined by (2.13) and using the energy method, we show sup x ∈ R ( | ( ψ , θ ) ( x , t ) | + | ( ψ x , θ x ) ( x , t ) | ) → 0 as t → ∞ and the solutions decay with exponential rates. The same problem was studied by Tang and Zhao [S.Q. Tang, H.J. Zhao, Nonlinear stability for dissipative nonlinear evolution equations with ellipticity, J. Math. Anal. Appl. 233 (1999) 336–358] for the case of ( ψ ± , θ ± ) = ( 0 , 0 ) .