In this paper, the parabolic problem $$u_t - div(\omega (x) \nabla u)= h(t) f(u) + l(t) g(u)$$ with non-negative initial conditions pertaining to $$C_b({\mathbb {R}}^N)$$ , will be studied, where weight $$\omega $$ is an appropriate function that belongs Muckenhoupt class $$A_{1 \frac{2}{N}}$$ and functions f, g, h l are continuous. The main goal establish global non-global existence of solutio...