Critical mass phenomena in higher dimensional quasilinear Keller–Segel systems with indirect signal production

In this paper, we deal with quasilinear Keller–Segel systems indirect signal production, u t = ∇ · ( + 1 ) m − v , x ∈ Ω > 0 Δ μ w $$ \left\{\begin{array}{ll}{u}_t&#x0003D;\nabla \cdotp \left({\left(u&#x0002B;1\right)}&#x0005E;{m-1}\nabla u\right)-\nabla \left(u\nabla v\right),& x\in \Omega, t>0,\\ {}0&#x0003D;\Delta v-\mu (t)&#x0002B;w,& {}{w}_t&#x0002B;w&#x0003D;u,& t>0,\end{array}\right. complemented homogeneous Neumann boundary conditions and suitable initial conditions, where ⊂ ℝ n \Omega \subset {\mathrm{\mathbb{R}}}&#x0005E;n ≥ 3 \left(n\ge 3\right) is a bounded smooth domain, m\ge : ⨏ for . \mu (t):&#x0003D; {\fint}_{\Omega}w\left(\cdotp, t\right)\kern2em \mathrm{for}\kern0.5em t>0. We show that in the case 2 2-\frac{2}{n} there exists M c {M}_c>0 such if either m>2-\frac{2}{n} or ∫ < {\int}_{\Omega}{u}_0<{M}_c then solution globally remains bounded, ≤ m\le m<2-\frac{2}{n} ω M>{2}&#x0005E;{\frac{n}{2}}{n}&#x0005E;{n-1}{\omega}_n exist radially symmetric data {\int}_{\Omega}{u}_0&#x0003D;M blows up finite infinite time, blow-up time m&#x0003D;2-\frac{2}{n} particular, critical mass phenomenon sense inf ∃ corresponding {\displaystyle \begin{array}{cc}\hfill & \hfill \operatorname{inf}\left\{M>0:\exists {u}_0\kern0.5em \mathrm{with}\kern0.5em {\int}_{\Omega}{u}_0&#x0003D;M\kern0.5em \mathrm{such}\ \mathrm{that}\ \mathrm{the}\ \mathrm{corresponding}\right.\\ {}\hfill \kern2em \left.\mathrm{solution}\ \mathrm{blows}\ \mathrm{up}\ \mathrm{in}\ \mathrm{infinite}\ \mathrm{time}\right\}\hfill \end{array}} positive number.