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

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.