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A heat equation with memory: Large-time behavior

Publicated to:JOURNAL OF FUNCTIONAL ANALYSIS. 281 (9): 109174- - 2021-11-01 281(9), DOI: 10.1016/j.jfa.2021.109174

Authors: Cortazar, Carmen; Quiros, Fernando; Wolanski, Noemi

Affiliations

CSIC-UAM-UC3M-UCM - Instituto de Ciencias Matematicas (ICMAT) - Author
IMAS UBA CONICET, Ciudad Univ,Pab 1, RA-1428 Buenos Aires, DF, Argentina - Author
Inst Ciencias Matemat ICMAT CSIC UAM UCM UC3M, Madrid 28049, Spain - Author
Pontificia Univ Catolica Chile, Dept Matemat, Santiago, Chile - Author
Pontificia Universidad Católica de Chile - Author
Univ Autonoma Madrid, Dept Matemat, Madrid 28049, Spain - Author
Universidad Autónoma de Madrid - Author
Universidad de Buenos Aires - Author
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Abstract

We study the large-time behavior in all Lp norms of solutions to a heat equation with a Caputo α-time derivative posed in RN (0<α<1). These are known as subdiffusion equations. The initial data are assumed to be integrable, and, when required, to be also in Lp. We find that the decay rate in all Lp norms, 1≤p≤∞, depends greatly on the space-time scale under consideration. This result explains in particular the so called “critical dimension phenomenon” (cf. [21]). Moreover, we find the final profiles (that strongly depend on the scale). The most striking result states that in compact sets the final profile (in all Lp norms) is a multiple of the Newtonian potential of the initial datum. Our results are very different from the ones for classical diffusion equations and show that, in accordance with the physics they have been proposed for, these are good models for particle systems with sticking and trapping phenomena or fluids with memory.

Keywords

asymptotic behaviorasymptotic-behaviorcaputo derivativerandom-walkssubdiffusion equationsAsymptotic behaviorCaputo derivativeHeat equation with nonlocal time derivativeNonlocal diffusion equation

Quality index

Bibliometric impact. Analysis of the contribution and dissemination channel

The work has been published in the journal JOURNAL OF FUNCTIONAL ANALYSIS due to its progression and the good impact it has achieved in recent years, according to the agency WoS (JCR), it has become a reference in its field. In the year of publication of the work, 2021, it was in position 47/333, thus managing to position itself as a Q1 (Primer Cuartil), in the category Mathematics.

From a relative perspective, and based on the normalized impact indicator calculated from World Citations provided by WoS (ESI, Clarivate), it yields a value for the citation normalization relative to the expected citation rate of: 1.66. This indicates that, compared to works in the same discipline and in the same year of publication, it ranks as a work cited above average. (source consulted: ESI Nov 14, 2024)

This information is reinforced by other indicators of the same type, which, although dynamic over time and dependent on the set of average global citations at the time of their calculation, consistently position the work at some point among the top 50% most cited in its field:

  • Field Citation Ratio (FCR) from Dimensions: 4.28 (source consulted: Dimensions Jun 2025)

Specifically, and according to different indexing agencies, this work has accumulated citations as of 2025-06-27, the following number of citations:

  • WoS: 4
  • Scopus: 4

Impact and social visibility

From the perspective of influence or social adoption, and based on metrics associated with mentions and interactions provided by agencies specializing in calculating the so-called "Alternative or Social Metrics," we can highlight as of 2025-06-27:

  • The use of this contribution in bookmarks, code forks, additions to favorite lists for recurrent reading, as well as general views, indicates that someone is using the publication as a basis for their current work. This may be a notable indicator of future more formal and academic citations. This claim is supported by the result of the "Capture" indicator, which yields a total of: 4 (PlumX).

It is essential to present evidence supporting full alignment with institutional principles and guidelines on Open Science and the Conservation and Dissemination of Intellectual Heritage. A clear example of this is:

  • The work has been submitted to a journal whose editorial policy allows open Open Access publication.

Leadership analysis of institutional authors

This work has been carried out with international collaboration, specifically with researchers from: Argentina; Chile.