Duan, Renjun; Lin, Saipan; Zhu, Changjiang Optimal \(L^{p}(1\leq p \leq \infty)\) rates of decay to linear diffusion waves for nonlinear evolution equations with ellipticity and dissipation. (English) Zbl 1113.35088 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 11, 2335-2344 (2007). Summary: We are concerned with the large time decay estimates of solutions to the Cauchy problem of nonlinear evolution equations with ellipticity and damping. Different end states of initial data are considered. The optimal rate of decay to the linear diffusion waves is obtained. The optimal rate of decay of solutions to the linearized system plays a crucial role in the analysis. Cited in 3 Documents MSC: 35K45 Initial value problems for second-order parabolic systems 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations Keywords:evolution equations; diffusion waves; optimal decay rate; large time decay estimates PDFBibTeX XMLCite \textit{R. Duan} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 11, 2335--2344 (2007; Zbl 1113.35088) Full Text: DOI References: [1] Duan, R. J.; Zhu, C. J., Asymptotics of dissipative nonlinear evolution equations with ellipticity, J. Math. Anal. Appl., 302, 15-35 (2005) · Zbl 1063.35069 [2] Hsieh, D. Y.; Tang, S. Q.; Wang, X. P., On hydrodynamic instability, chaos, and phase transition, Acta. Mech. Sinica, 12, 1-14 (1996) · Zbl 0854.76040 [3] Hsieh, D. Y.; Tang, S. Q.; Wang, X. P.; Wu, L. X., Dissipative nonlinear evolution equations and chaos, Stud. Appl. Math., 101, 233-266 (1998) · Zbl 1020.37050 [4] Keefe, L. R., Dynamics of perturbed wavetrain solutions to the Ginzburg-Landau equation, Stud. Appl. Math., 73, 91-153 (1985) · Zbl 0575.76055 [5] Kuramoto, Y.; Tsuzuki, T., On the formation of dissipative structures in reaction-diffusion systems, Progr. Theoret. Phys., 54, 687-699 (1975) [6] Sivashinsky, G., Nonlinear analysis of hydrodynamic instability in laminar flames, I. derivation of basic equations, Acta Astronaut., 4, 1177-1206 (1977) · Zbl 0427.76047 [7] Tang, S. Q.; Zhao, H. J., Nonlinear stability for dissipative nonlinear evolution equations with ellipticity, J. Math. Anal. Appl., 233, 336-358 (1999) · Zbl 0932.35022 [8] Zhao, H. J., Large time decay estimates of solutions of nonlinear parabolic equations, Discrete Contin. Dyn. Syst., 8, 69-114 (2002) · Zbl 1136.35406 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.