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

Superdiffusions with large mass creation --- construction and growth estimates