  Superdiffusions corresponding to differential operators of the form $\LL
u+\beta u-\alpha u^{2}$ with large mass creation term $\beta$ are studied. Our
construction for superdiffusions with large mass creations works for the
branching mechanism $\beta u-\alpha u^{1+\gamma},\ 0<\gamma<1,$ as well.
  Let $D\subseteq\mathbb{R}^{d}$ be a domain in $\R^d$. When $\beta$ is large,
the generalized principal eigenvalue $\lambda_c$ of $L+\beta$ in $D$ is
typically infinite. Let $\{T_{t},t\ge0\}$ denote the Schr\"odinger semigroup of
$L+\beta$ in $D$ with zero Dirichlet boundary condition. Under the mild
assumption that there exists an $0<h\in C^{2}(D)$ so that $T_{t}h$ is
finite-valued for all $t\ge 0$, we show that there is a unique
$\mathcal{M}_{loc}(D)$-valued Markov process that satisfies a log-Laplace
equation in terms of the minimal nonnegative solution to a semilinear initial
value problem. Although for super-Brownian motion (SBM) this assumption
requires
  $\beta$ be less than quadratic, the quadratic case will be treated as well.
  When $\lambda_c = \infty$, the usual machinery, including martingale methods
and PDE as well as other similar techniques cease to work effectively, both for
the construction and for the investigation of the large time behavior of the
superdiffusions. In this paper, we develop the following two new techniques in
the study of local/global growth of mass and for the spread of the
superdiffusions: \begin{itemize} \item a generalization of the
Fleischmann-Swart `Poissonization-coupling,' linking superprocesses with
branching diffusions; \item the introduction of a new concept: the `{\it
$p$-generalized principal eigenvalue.}' \end{itemize} The precise growth rate
for the total population of SBM with $\alpha(x)=\beta(x)=1+|x|^p$ for
$p\in[0,2]$ is given in this paper.

Probability