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 Mat. Sb., 2020, Volume 211, Number 2, Pages 74–105 (Mi msb9216)

Existence and uniqueness of a weak solution of an integro-differential aggregation equation on a Riemannian manifold

V. F. Vil'danova

Bashkir State Pedagogical University n. a. M. Akmulla, Ufa, Russia

Abstract: A class of integro-differential aggregation equations with nonlinear parabolic term $b(x,u)_t$ is considered on a compact Riemannian manifold $\mathscr M$. The divergence term in the equations can degenerate with loss of coercivity and may contain nonlinearities of variable order. The impermeability boundary condition on the boundary $\partial\mathscr M\times[0,T]$ of the cylinder $Q^T=\mathscr M\times[0,T]$ is satisfied if there are no external sources of ‘mass’ conservation, $\int_\mathscr Mb(x,u(x,t)) d\nu=\mathrm{const}$. In a cylinder $Q^T$ for a sufficiently small $T$, the mixed problem for the aggregation equation is shown to have a bounded solution. The existence of a bounded solution of the problem in the cylinder $Q^\infty=\mathscr M\times[0,\infty)$ is proved under additional conditions.
For equations of the form $b(x,u)_t=\Delta A(x,u)-\operatorname{div}(b(x,u)\mathscr G(u))+f(x,u)$ with the Laplace-Beltrami operator $\Delta$ and an integral operator $\mathscr G(u)$, the mixed problem is shown to have a unique bounded solution.
Bibliography: 26 titles.

Keywords: aggregation equation on a manifold, existence of a solution, uniqueness of a solution.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00428-a This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 18-01-00428-a).

DOI: https://doi.org/10.4213/sm9216

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English version:
Sbornik: Mathematics, 2020, 211:2, 226–257

Bibliographic databases:

UDC: 517.968.74+517.954
MSC: 35D40, 34C40

Citation: V. F. Vil'danova, “Existence and uniqueness of a weak solution of an integro-differential aggregation equation on a Riemannian manifold”, Mat. Sb., 211:2 (2020), 74–105; Sb. Math., 211:2 (2020), 226–257

Citation in format AMSBIB
\Bibitem{Vil20} \by V.~F.~Vil'danova \paper Existence and uniqueness of a~weak solution of an integro-differential aggregation equation on a~Riemannian manifold \jour Mat. Sb. \yr 2020 \vol 211 \issue 2 \pages 74--105 \mathnet{http://mi.mathnet.ru/msb9216} \crossref{https://doi.org/10.4213/sm9216} \elib{https://elibrary.ru/item.asp?id=43298494} \transl \jour Sb. Math. \yr 2020 \vol 211 \issue 2 \pages 226--257 \crossref{https://doi.org/10.1070/SM9216} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000529470500001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85085358779} 

• http://mi.mathnet.ru/eng/msb9216
• https://doi.org/10.4213/sm9216
• http://mi.mathnet.ru/eng/msb/v211/i2/p74

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. F. Vil'danova, “Existence of a solution to the Cauchy problem for the aggregation equation in hyperbolic space”, Russian Math. (Iz. VUZ), 64:7 (2020), 27–37
2. V. F. Vildanova, “Edinstvennost resheniya zadachi Koshi dlya uravneniya agregatsii v giperbolicheskom prostranstve”, Izv. vuzov. Matem., 2021, no. 8, 27–36
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