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 |
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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. |
URI: | http://hdl.handle.net/10045/36559 |
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 www.springerlink.com |
Peer Review: | si |
Publisher version: | http://dx.doi.org/10.1007/s11854-014-0011-z |
Appears in Collections: | INV - CADAGL - Artículos de Revistas INV - GAM - Artículos de Revistas |
Files in This Item:
File | Description | Size | Format | |
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2014_Ignat_etal_JAM.pdf | Preprint (acceso abierto) | 230,93 kB | Adobe PDF | Open Preview |
2014_Ignat_etal_JAM-final.pdf | Versión final (acceso restringido) | 217,31 kB | Adobe PDF | Open Request a copy |
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