  We consider non-linear time-fractional stochastic heat type equation
$$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2}
u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h)
\stackrel{\cdot}{\tilde N }(t,x,h)\bigg]$$ and $$\frac{\partial^\beta
u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t
\bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{N }(t,x,h)\bigg]$$
  in $(d+1)$ dimensions, where $\alpha\in (0,2]$ and
$d<\min\{2,\beta^{-1}\}\alpha$, $\nu>0$, $\partial^\beta_t$ is the Caputo
fractional derivative, $-(-\Delta)^{\alpha/2} $ is the generator of an
isotropic stable process, $I^{1-\beta}_t$ is the fractional integral operator,
${N}(t,x)$ are Poisson random measure with $\tilde{N}(t,x)$ being the
compensated Poisson random measure. $\sigma:{\mathbb{R}}\to{\mathbb{R}}$ is a
Lipschitz continuous function. We prove existence and uniqueness of mild
solutions to this equation. Our results extend the results in the case of
parabolic stochastic partial differential equations obtained in "M. Foondun and
D. Khoshnevisan. Intermittence and nonlinear parabolic stochastic partial
differential equations. \emph{ Electron. J. Probab.} {\bf14} (2009), 548--568"
and " J. B. Walsh. An Introduction to Stochastic Partial Differential
Equations, \'Ecoled'\'et\'e de Probabilit\'es de Saint-Flour, XIV|1984, Lecture
Notes in Math., vol. 1180, Springer, Berlin, (1986), 265--439". Under the
linear growth of $\sigma$, we show that the solution of the time fractional
stochastic partial differential equation follows an exponential growth with
respect to the time. We also show the nonexistence of the random field solution
of both stochastic partial differential equations when $\sigma$ grows faster
than linear.
