Let xi(t, x) denote space-time white noise and consider a reaction-diffusion equation of the form

<(u) over dot>(t, x) = 1/2u ''(t, x) + b(u(t, x)) + sigma(u(t,x))xi(t,x)

on R+ x [0, 1], with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists epsilon > 0 such that vertical bar b(z)vertical bar >= vertical bar z vertical bar (log vertical bar z vertical bar)(1+epsilon) for all sufficiently-large values of vertical bar z vertical bar. When sigma equivalent to 0, it is well known that such PDEs frequently have nontrivial stationary solutions. By contrast, Bonder and Groisman [Phys. D 238 (2009) 209-215] have recently shown that there is finite-time blowup when sigma is a nonzero constant. In this paper, we prove that the Bonder-Groisman condition is unimprovable by showing that the reaction-diffusion equation with noise is "typically" well posed when vertical bar b(z)vertical bar =O(vertical bar vertical bar z vertical bar log(+) vertical bar z vertical bar) as vertical bar z vertical bar -> infinity. We interpret the word "typically" in two essentially-different ways without altering the conclusions of our assertions.