Pointwise asymptotic behavior of modulated periodic reaction–diffusion waves

Abstract

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 L p -behavior(p ≥ 1) of a nonlinear solution to a perturbation equation of a reaction-diffusion equation with respect to initial data in L 1 ∩ H 1 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 |u 0 | ≤ E 0 e −|x| 2 /M and |u 0 | ≤ E 0 (1 + |x|) −3/2 , respectively, E 0 > 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. : Research of S.J. was partially supported under NSF grant no. DMS-0300487. π −π e iξx e L ξ tĝ (ξ, x)dξ.By the translation invariance of (1.1), the functionū ′ (x) is a 1-periodic solution of the differential equation L 0 v = 0. Hence, it follows that λ = 0 is an eigenvalue of the Bloch operator L 0 . Define following [S1, S2, JZ2] the diffusive spectral stability conditions:for any fundamental matrix solution Φ ξ of the (1.10). Main resultWith these preparations, we now state our two main results. INTRODUCTION4 Theorem 1.2. The Green function G(x, t; y) for equation (1.2) satisfies the estimates:(1.12)uniformly on t ≥ 0, for some sufficiently large constants M > 0 and η > 0, where q and q are the periodic right and left eigenfunctions of L 0 , respectively, at λ = 0. In particular q(x, 0) =ū ′ (x).Theorem 1.3. Define the nonlinear perturbation u :=ũ −ū, whereũ satisfies (1.1). Then the asymptotic behavior of u with respect to three kinds of initial data(denoted by u 0 ):M , E 0 > 0 sufficiently small and M > 1 sufficiently large (3) |u 0 (x)| < E 0 (1 + |x|) −r , E 0 > 0 sufficiently small and r > 2 converges to a heat kernel with the following estimates, respectively, M ′′ > M and C > 0 sufficiently large and some constant U * (defined in Section 6).Remark 1.4. The 3 parts of Theorem 1.3 is established in Theorem 6.7, 6.23 and 7.13, respectively.Remark 1.5. The initial condition |u 0 | L 1 ∩H 1 , |xu 0 | L 1 sufficiently small is compared with Schneider's [S2] initial assumption. By Fourier transform, we can roughly consider |(1 + |x| 2 )u 0 | H 2 as Schneider's initial condition with weight (1 + |x| 2 )(See Schneider [S2], pp690-691). This implies that our initial data roughly satisfies |u 0 | |x| −2 whereas Schneider's initial data roughly satisfies |u 0 | |x| − 5 2 . Our L p bounds on asymptotic behavior for all p ≥ 1 are also compared with Schneider's L ∞ bound. In particular, our L ∞ bound t −1 ln(1 + t) is roughly equivalent to but slightly sharper than Schneider's L ∞ bound t −1+ε for ε > 0. Though Schneid...

L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves

“…As a starting point, we used this formula to estimate linear behaviors of in terms of the localized and nonlocalized data in [2] and [3], respectively. Indeed, pointwise estimates on Green function of , by plugging ( ) into the Dirac delta function ( ), have been obtained in [2] to estimate ( )V 0 for a localized data V 0 .…”

“…This is the reason why their stability analysis has been restricted to ≥ 2. In this paper, we extend their (R)-stability results ( ≥ 2) to 1 (R)-stability by using pointwise estimate of linear behaviors under localized data V 0 ( ) fl̃0( − ℎ 0 ( )) − ( ) and nonlocalized modulational data ℎ 0 ( ) established in [2,3].…”

“…Johnson, Zumbrun, and their collaborators also showed (R) ( ≥ 2) nonlinear modulational stability of periodic traveling waves of systems of reaction-diffusion equations and of conservation under both localized and nonlocalized perturbations in [1,[7][8][9]. By using pointwise linear estimates together with a nonlinear iteration scheme developed by Johnson-Zumbrun, pointwise nonlinear stability of such 2 Advances in Mathematical Physics waves has been also studied in [2,3,10]. For other related works on modulated periodic traveling waves, we refer readers to [11][12][13].…”

“…In this paper, by using the results of [1][2][3], we consider 1 (R)-nonlinear modulational stability of spatially periodic traveling waves in a system of reaction-diffusion equations:…”

“…To begin with, we first review the concept of stability of ( ) of (2). Roughly speaking, we say that ( ) is (boundedly) stable if any other solutions̃( , ) of (2) which are initially near ( ) stay near ( ) for all ≥ 0.…”