Existence and boundedness of solutions for a Keller-Segel system with gradient dependent chemotactic sensitivity

We consider the Keller-Segel system with gradient dependent chemotactic sensitivity uₜ = u- (u| v|^p-2 v), x, \; t>0, vₜ = v-v+u, x, \; t>0, u{ = v=0, x, \; t>0, u (x, 0) =u₀ (x), v (x, 0) =v₀ (x), x } in a smooth bounded domain \ (Rⁿ\), \ (n2\). We shown that for all reasonably regular initial data \ (u₀ 0\) and \ (v₀0\), the corresponding Neumann initial-boundary value problem possesses a global weak solution which is uniformly bounded provided that \ (1<p<n/ (n-1) \). For more information see https: //ejde. math. txstate. edu/Volumes/2020/122/abstr. html