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 Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz.: Year: Volume: Issue: Page: Find

 Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 2018, Volume 152, Pages 34–45 (Mi into349)

Existence of Weak Solutions of Aggregation Equation with the $p(\cdot)$-Laplacian

V. F. Vil'danovaa, F. Kh. Mukminovb

a Bashkir State Pedagogical University, Ufa
b Institution of Russian Academy of Sciences Institute of Mathematics with Computer Center, Ufa

Abstract: We consider an aggregation elliptic-parabolic equation of the form
\begin{equation*} b(u)_t=\operatorname{div}( |\nabla u|^{p(x)-2}\nabla u-b(u)G(u))+\gamma(x,b(u)), \end{equation*}
where $b$ is a nondecreasing function and $G(u)$ is an integral operator. The condition on the boundary of a bounded domain $\Omega$ ensures that the mass of the population $\int u(x,t)dx=\operatorname{const}$ is preserved for $\gamma=0$. The existence of a weak solution of the problem with a nonnegative bounded initial function in the cylinder $\Omega\times(0,T)$ is proved. A formula for the guaranteed time $T$ for the existence of the solution is obtained.

Keywords: aggregation equation, $p(\cdot)$-Laplacian, existence of solution

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00428_à This work was partially supported by the Russian Foundation for Basic Research (project No. 18-01-00428-A).

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Bibliographic databases:
UDC: 517.956.45, 517.968.74
MSC: 35K20, 35K55, 35K65

Citation: V. F. Vil'danova, F. Kh. Mukminov, “Existence of Weak Solutions of Aggregation Equation with the $p(\cdot)$-Laplacian”, Mathematical physics, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 152, VINITI, Moscow, 2018, 34–45

Citation in format AMSBIB
\Bibitem{VilMuk18} \by V.~F.~Vil'danova, F.~Kh.~Mukminov \paper Existence of Weak Solutions of Aggregation Equation with the $p(\cdot)$-Laplacian \inbook Mathematical physics \serial Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. \yr 2018 \vol 152 \pages 34--45 \publ VINITI \publaddr Moscow \mathnet{http://mi.mathnet.ru/into349} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3903376}