Blow-up solutions in a Cauchy problem of parabolic equations with spatial coefficients

This paper deals with a Cauchy problem of the parabolic equations $$\begin{aligned} u_t =\Delta u + a_{1}(x) u^{p_{1}} + b_{1}(x) v^{q_{1}},\ v_t =\Delta v + a_{2}(x) u^{p_{2}} + b_{2}(x) v^{q_{2}}, \end{aligned}$$ where the exponents $$p_{i}$$ , $$q_{i}$$ $$(i=1,2)$$ are positive constants; the coefficients $$a_{i}(x)\sim |x|^{\alpha _{i}}$$ and $$b_{i}(x)\sim |x|^{\beta _{i}}$$ as $$|x|\rightarrow +\infty $$ with the parameters $$\alpha _{i}$$ , $$\beta _{i}\in R$$ . For $$\alpha _{i}$$ , $$\beta _{i}\ge 0$$ , we determine the exponent regions where all of the solutions blow up for any nonnegative nontrivial initial data. For at least one negative parameter, we find different conditions on global existence of solutions according to different classifications of the parameters.