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Ahn, Jaewook,Yoon, Changwook Institute of Physics and the London Mathematical S 2019 Nonlinearity Vol.32 No.4
<P>This paper deals with a Keller–Segel type parabolic–elliptic system involving nonlinear diffusion and chemotaxis <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn001.gif'/> in a smoothly bounded domain <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn002.gif'/>, <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn003.gif'/>, under no-flux boundary conditions. The system contains a Fokker–Planck type diffusion with a motility function <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn004.gif'/>, <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn005.gif'/>.</P> <P>The global existence of the unique bounded classical solutions is established without smallness of the initial data neither the convexity of the domain when <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn006.gif'/>, <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn007.gif'/> or <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn008.gif'/>, <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn009.gif'/>. In addition, we find the conditions on parameters, <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn010.gif'/> and <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn011.gif'/>, that make the spatially homogeneous equilibrium solution globally stable or linearly unstable.</P>