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Consider the following space-time fractional heat equation withRiemann-Liouville derivative of non-homogeneous time-fractional Poisson process\begin{eqnarray*} \partial^\beta_t u(x,t) =-\kappa(-\Delta)^{\alpha/2} u(x,t) +I_t^{1-\beta}[\sigma(u)D_t^\vartheta N^\nu_\lambda(t)], \,\, t\geq 0, \,x \in\mathbb{R}^d, \end{eqnarray*} where $\kappa>0, \,\,\beta,\,\vartheta\in(0,1),\,\,\nu\in(0,1],\,\alpha\in(0,2].$ The operator $D_t^\vartheta N^\nu_\lambda(t)= \frac{\rm d}{\mathrm{d} t} I_t^{1-\vartheta} N_\lambda^\nu(t) = \frac{\rmd}{\mathrm{d} t} \mathcal{N}_\lambda^{1-\vartheta,\nu}(t)$ with$\mathcal{N}_\lambda^{1-\vartheta,\nu}(t)$ the Riemann-Liouvillenon-homogeneous fractional integral process, $\partial^\beta_t$ is the Caputofractional derivative, $-(-\Delta)^{\alpha/2}$ is the generator of an isotropicstable process, $I^\beta_t$ is the fractional integral operator, and $\sigma :\mathbb{R} \rightarrow \mathbb{R}$ is Lipschitz continuous. The above timefractional stochastic heat type equations may be used to model sequence ofcatastrophic events with thermal memory. The mean and variance for the process$\frac{\rm d}{\mathrm{d} t}\mathcal{N}^{1-\vartheta,\nu}_\lambda(t)$ for somespecific rate functions were computed. Consequently, the growth moment boundsfor the class of heat equation perturbed with the non-homogeneous fractionaltime Poisson process were given and we show that the solution growsexponentially for some small time interval $t\in [t_0,T], \,\,T<\infty$ and$t_0>1$; that is, the result establishes that the energy of the solution growsatleast as $ c_4(t+t_0)^{(\vartheta-a\nu)}\exp(c_5 t)$ and at most as $c_1t^{(\vartheta- a\nu)}\exp(c_3 t)$ for different conditions on the initial data,where $c_1,\,c_3,\,c_4$ and $c_5$ are some positive constants depending on $T$.Existence and uniqueness result for the mild solution to the equation was givenunder linear growth condition on $\sigma$.

On Space-Time Fractional Heat Type Non-Homogeneous Time-Fractional Poisson Equation