In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations
where
is the half‐space
,
is a nonnegative continuous function, and
is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data
if and only if
for every
. In fact, we introduce a very special curve in
to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result.