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Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species

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  • This paper deals with two-species quasilinear parabolic-parabolic Keller-Segel system $ u_{it}=\nabla\cdot(\phi_i(u_i)\nabla u_i)-\nabla\cdot(\psi_i(u_i)\nabla v)$, $i=1,2$, $v_t=\Delta v-v+u_1+u_2$ in $\Omega\times (0,T)$, subject to the homogeneous Neumann boundary conditions, with bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$. We prove that if $\frac{\psi_i(u_i)}{\phi_i(u_i)}\leq C_iu_i^{\alpha_i}$ for $u_i>1$ with $0<\alpha_i<\frac{2}{n}$ and $C_i>0$, $i=1,2$, then the solutions are globally bounded, while if $\frac{\psi_1(u_1)}{\phi_1(u_1)}\geq C_1u_1^{\alpha_1}$ for $u_1>1$ with $\Omega=B_R$, $\alpha_1>\frac{2}{n}$, then for any radial $u_{20}\in C^0(\overline{\Omega})$ and $m_1>0$, there exists positive radial initial data $u_{10}$ with $\int_\Omega u_{10}=m_1$ such that the solution blows up in a finite time $T_{\max}$ in the sense $\lim_{{t\rightarrow T_{\max}}} \|u_1(\cdot,t)+u_2(\cdot,t)\|_{L^{\infty}(\Omega)}=\infty$. In particular, if $\alpha_1>\frac{2}{n}$ with $0<\alpha_2<\frac{2}{n}$, the finite time blow-up for the species $u_1$ is obtained under suitable initial data, a new phenomenon unknown yet even for the semilinear Keller-Segel system of two species.
    Mathematics Subject Classification: 92C17, 35K55, 35B40.


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