Large-Time Behavior for a Fully Nonlocal Heat Equation
- PDF / 345,777 Bytes
- 14 Pages / 439.642 x 666.49 pts Page_size
- 21 Downloads / 261 Views
Large-Time Behavior for a Fully Nonlocal Heat Equation ´ 1 · Fernando Quiros ´ 2,3 Carmen Cortazar
· Noem´ı Wolanski4
Received: 19 May 2020 / Accepted: 24 August 2020 / © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract We study the large-time behavior in all Lp norms and in different space-time scales of solutions to a nonlocal heat equation in RN involving a Caputo α-time derivative and a power of the Laplacian (−)s , s ∈ (0, 1), extending recent results by the authors for the case s = 1. The initial data are assumed to be integrable, and, when required, to be also in Lp . The main novelty with respect to the case s = 1 comes from the behaviour in fast scales, for which, thanks to the fat tails of the fundamental solution of the equation, we are able to give results that are not available neither for the case s = 1 nor, to our knowledge, for the standard heat equation, s = 1, α = 1. Keywords Fully nonlocal heat equation · Caputo derivative · Fractional Laplacian · Asymptotic behavior Mathematics Subject Classification (2010) 35B40 · 35R11 · 35R09 · 45K05
1 Introduction The aim of this paper is to study the large-time behavior of solutions to the fully nonlocal Cauchy problem ∂tα u + (−)s u = 0 in RN × (0, ∞),
u(·, 0) = u0 ≥ 0 in RN ,
Dedicated to Enrique Zuazua on the occasion of his 60th birthday. Fernando Quir´os
[email protected] Carmen Cort´azar [email protected] Noem´ı Wolanski [email protected] 1
Departamento de Matem´atica, Pontificia Universidad Cat´olica de Chile, Santiago, Chile
2
Departamento de Matem´aticas, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain
3
Instituto de Ciencias Matem´aticas ICMAT (CSIC-UAM-UCM-UC3M), 28049 Madrid, Spain
4
IMAS-UBA-CONICET, Ciudad Universitaria, Pabell´on I, (1428) Buenos Aires, Argentina
(1.1)
C. Cort´azar et al.
where ∂tα , α ∈ (0, 1), denotes the Caputo α-derivative, introduced in [3], defined for smooth functions by t 1 u(x, τ ) − u(x, 0) ∂t dτ, ∂tα u(x, t) = (1 − α) (t − τ )α 0 s and (−) , s ∈ (0, 1), is the usual s power of the Laplacian, defined for smooth functions by v(x) − v(y) dy. (−)s v(x) = cN,s P.V. N+2s RN |x − y| Here cN,s is a positive normalization constant, chosen so that (−)s = F −1 (| · |2s F ), where F denotes Fourier transform. The initial data are always assumed to be non-negative, integrable and non-trivial, and when required, to belong also to Lp (RN ) for some p ∈ (1, ∞]. Fully nonlocal heat equations, like (1.1), nonlocal both in space and time, are useful to model situations with long-range interactions and memory effects, and have been proposed for example to describe plasma transport see [8, 9]; see also [4, 6, 15, 16] for further models that use such equations. Despite their wide number of applications, the mathematical study of fully nonlocal heat equations started only quite recently. In the interesting paper [13], Kemppainen, Siljander and Zacher proved, among several other things, that if both u0 and F (u0 ) belong to L1 (RN ), then proble
Data Loading...
