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

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