A necessary-sufficient condition for the existence or nonexistence of global solutions to the following reaction-diffusion equations { u t = Δ u + ψ ( t ) u p , in R N ×( 0 , t ∗ ) , u ( ⋅ , 0 )= u 0 ≥ 0 , in R N , has not been known and remained as an open problem for a few decades. The purpose of this paper is to resolve this problem completely, even for more general source ψ( t) f( u) as follows: There is a global solution to the equation if and only if ∫ 0 ∞ ψ ( t ) f ( ‖ S ( t ) u 0 ‖ ∞ ) ‖ S ( t ) u 0 ‖ ∞ dt < ∞ for some nonnegative and nontrivial u 0 ∈ C 0 ( R N ) . Here, ( S ( t ) ) t ≥ 0 is the heat semigroup on R N .