  In this paper, we study existence and uniqueness of strong as well as weak
solutions for general time fractional Poisson equations. We show that there is
an integral representation of the solutions of time fractional Poisson
equations with zero initial values in terms of semigroup for the infinitesimal
spatial generator ${\cal L}$ and the corresponding subordinator associated with
the time fractional derivative. This integral representation has an integral
kernel $q(t, x, y)$, which we call the fundamental solution for the time
fractional Poisson equation, if the semigrou for ${\cal L}$ has an integral
kernel. We further show that $q(t, x, y)$ can be expressed as a time fractional
derivative of the fundamental solution for the homogenous time fractional
equation under the assumption that the associated subordinator admits a
conjugate subordinator. Moreover, when the Laplace exponent of the associated
subordinator satisfies the weak scaling property and its distribution is
self-decomposable, we establish two-sided estimates for the fundamental
solution $q(t,x, y)$ through explicit estimates of transition density functions
of subordinators.
