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We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$to the nonlinear fractional diffusion equation, $\partial_t u +\mathcal{L}F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$,with appropriate homogeneous Dirichlet boundary conditions. As $\mathcal{L}$ wecan use a quite general class of linear operators that includes the two mostcommon versions of the fractional Laplacian $(-\Delta)^s$, $0<s<1$, in abounded domain with zero Dirichlet boundary conditions, but it also includesmany other examples since our theory only needs some basic properties that aretypical of "linear heat semigroups." The nonlinearity $F$ is assumed to beincreasing and is allowed to be degenerate, the prototype is the power case$F(u)=|u|^{m-1}u$, with $m>1$.  In this paper we propose a suitable class of solutions of the equation, andcover the basic theory: we prove existence, uniqueness of such solutions, andwe establish upper bounds of two forms (absolute bounds and smoothing effects),as well as weighted-$L^1$ estimates. The class of solutions is very well suitedfor that work. The standard Laplacian case $s=1$ is included and the linearcase $m=1$ can be recovered in the limit.  In a companion paper [12], we will complete the study with more advancedestimates, like the upper and lower boundary behaviour and Harnackinequalities, for which the results of this paper are needed.

Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains Part I. Existence, Uniqueness and Upper Bounds