Finite time blow-up in higher dimensional two species problem in the Cauchy problem

In this paper, we study the blow-up radial solution of fully parabolic system with higher dimensional two species Cauchy problem for some initial condition. In addition, we show that the set of positive radial functions in $L^1(\mathbb{R})\cap BUC({\mathbb{R}}^N)\times L^1(\mathbb{R})\cap BUC({\mathbb{R}}^N)\times W^{1,1}({\mathbb{R}}^N)\cap W^{1,\mathrm{\infty }}({\mathbb{R}}^N)$ has a dense subset composed of positive radial initial data causing blow-up in finite time with respect to topology $L^p({\mathbb{R}}^N)\times L^p({\mathbb{R}}^N)\times H^1({\mathbb{R}}^N)\cap W^{1,1}({\mathbb{R}}^N)$ for $p\in [1,\frac{2N}{N+2})$.