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

Consider the following space-time fractional heat equation with Riemann-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 $D_t^\vartheta N^\nu_\lambda(t) = \frac{\rm d}{\mathrm{d} t} I_t^{1-\vartheta} N_\lambda^\nu(t) = \frac{\rm d}{\mathrm{d} t} \mathcal{N}_\lambda^{1-\vartheta,\nu}(t)$ with $\mathcal{N}_\lambda^{1-\vartheta,\nu}(t)$ the Riemann-Liouville non-homogeneous fractional integral process, $\partial^\beta_t$ is the Caputo fractional derivative, $-(-\Delta)^{\alpha/2}$ is the generator of an isotropic stable process, $I^\beta_t$ is the fractional integral operator, and $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is Lipschitz continuous. The above time fractional stochastic heat type equations may be used to model sequence of catastrophic 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 some specific rate functions were computed. Consequently, the growth moment bounds for the class of heat equation perturbed with the non-homogeneous fractional time Poisson process were given and we show that the solution grows exponentially 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 grows atleast as $ c_4(t+t_0)^{(\vartheta-a\nu)}\exp(c_5 t)$ and at most as $c_1 t^{(\vartheta- a\nu)}\exp(c_3 t)$ 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 given under linear growth condition on $\sigma$.