Decay estimates for nonlinear nonlocal diffusion problems in the whole space

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Title: Decay estimates for nonlinear nonlocal diffusion problems in the whole space
Authors: Ignat, Liviu I. | Pinasco, Damián | Rossi, Julio D. | San Antolín Gil, Ángel
Research Group/s: Curvas Alpha-Densas. Análisis y Geometría Local
Center, Department or Service: Universidad de Alicante. Departamento de Análisis Matemático
Keywords: Nonlocal diffusion | Eigenvalues
Knowledge Area: Análisis Matemático
Issue Date: 1-Apr-2014
Publisher: Springer | The Hebrew University Magnes Press
Citation: Journal d'Analyse Mathématique. 2014, 122(1): 375-401. doi:10.1007/s11854-014-0011-z
Abstract: In this paper, we obtain bounds for the decay rate in the L r (ℝ d )-norm for the solutions of a nonlocal and nonlinear evolution equation, namely, ut(x,t)=∫RdK(x,y)|u(y,t)−u(x,t)|p−2(u(y,t)−u(x,t))dy,x∈Rd,t>0. . We consider a kernel of the form K(x, y) = ψ(y−a(x)) + ψ(x−a(y)), where ψ is a bounded, nonnegative function supported in the unit ball and a is a linear function a(x) = Ax. To obtain the decay rates, we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T(u)=−∫RdK(x,y)|u(y)−u(x)|p−2(u(y)−u(x))dy,1⩽p<∞. . The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole space ℝ d : λ1,p(Rd)=2(∫Rdψ(z)dz)∣∣∣∣1|detA|1/p−1∣∣∣∣p. Moreover, we deal with the p = ∞ eigenvalue problem, studying the limit of λ 1,p 1/p as p→∞.
Sponsor: L. I. Ignat is partially supported by grants PN II-RU-TE 4/2010 and PCCE-55/2008 of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, MTM2011-29306-C02-00, MICINN, Spain and ERC Advanced Grant FP7-246775 NUMERIWAVES. D. Pinasco is partially supported by grants ANPCyT PICT 2011-0738 and CONICET PIP 0624. J. D. Rossi and A. San Antolin are partially supported by the grant MTM2011-27998 MICINN MICINN, Spain.
ISSN: 0021-7670 (Print) | 1565-8538 (Online)
DOI: 10.1007/s11854-014-0011-z
Language: eng
Type: info:eu-repo/semantics/article
Rights: The original publication is available at
Peer Review: si
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