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

• Published in: Journal of mathematical analysis and applications Vol. 319; no. 2; pp. 740 - 763
• Main Author: Wang, Zhian
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 15.07.2006; Elsevier
• Subjects: Evolution equation; Exact sciences and technology; Fourier analysis; Fourier transform; Global analysis, analysis on manifolds; Interpolation inequality; Mathematical analysis; Mathematics; Optimal decay rate; Partial differential equations; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Optimal decay rate; Evolution equation; Fourier transform; Interpolation inequality; Damping; Fourier transformation; Asymptotic behavior; Fourier analysis; Optimal convergence; Cauchy problem; Ellipticity; Non linear equation; Mathematical analysis; Convergence rate; End; Non linear wave
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2005.06.046

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.