Asymptotic expansions for nonlocal diffusion equations in Lq-norms for 1 ≤ q ≤ 2

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:362

Article

Ignat, Liviu I.¹  Rossi, Julio D.² 

[1] Acad Romana, Inst Math, RO-014700 Bucharest, Romania

[2] IMDEA Matemat, Madrid, Spain

关键词: Nonlocal diffusion;    Asymptotic behavior;    Fractional Laplacian;   

DOI  :  10.1016/j.jmaa.2009.08.025

来源: Elsevier

【 摘 要 】

We study the asymptotic behavior for solutions to nonlocal diffusion models of the form u(t) (x, t) = J * u(x, t) - u(x, t) = integral(Rd) J (x - y)u(y, t) dy - u (x, t) in the whole R-d with an initial condition u(x, 0) = u(0)(x). Under suitable hypotheses on J (involving its Fourier transform) and u(0), it is proved an expansion of the form parallel to u(x,t) - Sigma(|alpha|<= k)(-1)(|alpha|)/alpha!(integral u(0)(x)x(alpha) dx)partial derivative(alpha) K-t parallel to(Lq(Rd)) <= Ct(-A), where K-t is the regular part of the fundamental solution and the exponent A depends on J, q, k and the dimension d. Moreover, we can obtain bounds for the difference between the terms in this expansion and the corresponding ones for the expansion of v(t)(x, t) = -(-Delta)s/2 v(x, t). Here we deal with the case 1 <= q <= 2. The case 2 <= q <= infinity was treated previously, by other methods, in LI. Ignat and J.D. Rossi (2008) [11]. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

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