Optimal decay rates to diffusion wave for nonlinear evolution equations with ellipticity

Abstract

We derive the optimal convergence rates to diffusion wave for the Cauchy problem of a set of nonlinear evolution equations with ellipticity and dissipative effects { 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 , subject to the initial data with end states ( 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 < 4 α ( 1 − α ) . Introducing the auxiliary function to avoid the difference of the end states, we show that the solutions to the reformulated problem decay as t → ∞ with the optimal decay order. The decay properties of the solution in the L 2 -sense, which are not optimal, were already established in paper [C.J. Zhu, Z.Y. Zhang, H. Yin, Convergence to diffusion waves for nonlinear evolution equations with ellipticity and damping, and with different end states, Acta Math. Sinica (English ed.), in press]. The main element of this paper is to obtain the optimal decay order in the sense of L p space for 1 ⩽ p ⩽ ∞ , which is based on the application of Fourier analysis and interpolation inequality under some suitable restrictions on coefficients α and ν. Moreover, we discuss the asymptotic behavior of the solution to general system (1.1) at the end. However, the optimal decay rates of the solution to general system (1.1) remains unknown.