Geodesic
Existence of Weak Solutions of Aggregation Equation with the $p(\cdot)$-Laplacian
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical physics, Tome 152 (2018), pp. 34-45

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We consider an aggregation elliptic-parabolic equation of the form \begin{equation*} b(u)_t=\operatorname{div}\Big( |\nabla u|^{p(x)-2}\nabla u-b(u)G(u)\Big)+\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
Mots-clés : existence of solution. 
@article{INTO_2018_152_a3,
     author = {V. F. Vil'danova and F. Kh. Mukminov},
     title = {Existence of {Weak} {Solutions} of {Aggregation} {Equation} with the $p(\cdot)${-Laplacian}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {34--45},
     publisher = {mathdoc},
     volume = {152},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2018_152_a3/}
}
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V. F. Vil'danova; F. Kh. Mukminov. Existence of Weak Solutions of Aggregation Equation with the $p(\cdot)$-Laplacian. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical physics, Tome 152 (2018), pp. 34-45. http://geodesic.mathdoc.fr/item/INTO_2018_152_a3/