Finite-time blow-up of solution for a chemotaxis model with singular sensitivity and logistic source

This paper deals with the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, {u(t)=Delta u -chi del .(u/v(gamma)del v ) +lambda u -mu u(k), x is an element of Omega ,t >0 , 0 =Delta v -v +u , x is an element of Omega ,t >0 under homogeneous Neumann boundary conditions and suitable initial conditions where Omega =B-R(0 ) subset of R-n (n >= 3 , R >0 ), chi >0 , gamma >0 , lambda is an element of R , mu >0 , k >1 . In the following two cases: center dot {7/-root{37 } 2 <gamma <1/2 if n is an element of{3 ,4 }, 33/-root{969 } 4 <gamma <1/2 if n =6}, and 1/2 gamma <= k < {1 +1/-gamma 6 if n is an element of{3 ,4 }, 1 +3/-2 gamma 30 if n =6}; center dot 1/2 <=gamma <1 and 1 <k <1 +1/-gamma 6 with n is an element of{3 ,4 } , it is proved that the corresponding solution blows up in finite time, which extends the blow-up result of Winkler [44] to the case with singular sensitivity.