2011/12/03 by Soyeun Jung, Jung, Soyeun
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.1112.0663
arxiv created 2011/12/03 · openalex publication_date 2011/12/03 · arxiv updated 2011/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
By working with the periodic resolvent kernel and Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction diffusion equations.With our linearized estimates together with a nonlinear iteration scheme developed by Johnson-Zumbrun, we obtain Lp- behavior(p ≥ 1) of a nonlinear solution to a perturbation equation of a reaction-diffusion equation with respect to initial data in L1 ∩ H1 recovering and slightly sharpening results obtained by Schneider using weighted energy and renormalization techniques. We obtain also pointwise nonlinear estimates with respect to two different initial perturbations |u0|≤ E0e-|x|2/M and |u0| ≤ E0(1+|x|)-3/2, respectively, E0>0 sufficiently small and M>1 sufficiently large, showing that behavior is that of a heat kernel. These pointwise bounds have not been obtained elsewhere, and do not appear to be accessible by previous techniques.