  Consider classical solutions to the parabolic reaction diffusion equation $$
&u_t =Lu+f(x,u), (x,t)\in R^n\times(0,\infty); &u(x,0) =g(x)\ge0, x\in R^n;
&u\ge0, $$ where $$ L=\sum_{i,j=1}^na_{i,j}(x)\frac{\partial^2}{\partial x_i
\partial x_j}+\sum_{i=1}^nb_i(x)\frac\partial{\partial x_i} $$ is a
non-degenerate elliptic operator, $g\in C(R^n)$ and the reaction term $f$
converges to $-\infty$ at a super-linear rate as $u\to\infty$. We give a sharp
minimal growth condition on $f$, independent of $L$, in order that there exist
a universal, a priori upper bound for all solutions to the above Cauchy
problem--that is, in order that there exist a finite function $M(x,t)$ on
$R^n\times(0,\infty)$ such that $u(x,t)\le M(x,t)$, for all solutions to the
Cauchy problem. Assuming now in addition that $f(x,0)=0$, so that $u\equiv0$ is
a solution to the Cauchy problem, we show that under a similar growth
condition, an intimate relationship exists between two seemingly disparate
phenomena--namely, uniqueness for the Cauchy problem with initial data $g=0$
and the nonexistence of unbounded, stationary solutions to the corresponding
elliptic problem. We also give a generic condition for nonexistence of
nontrivial stationary solutions.

Analysis of PDEs