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

Abstract

Consider the following space-time fractional heat equation with Riemann-Liouville derivative of non-homogeneous time-fractional Poisson process \begin{eqnarray*} \partial^β_t u(x,t) =-κ(-Δ)^{α/2} u(x,t) + I_t^{1-β}[σ(u)D_t^\vartheta N^ν_λ(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 $D_t^{\vartheta }{N}_{\lambda }^{\nu }(t)=\frac{\mathrm{d}}{\mathrm{d}t}{I}_t^{1-\vartheta }{N}_{\lambda }^{\nu }(t)=\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal{N}}_{\lambda }^{1-\vartheta ,\nu }(t)$ with ${\mathcal{N}}_{\lambda }^{1-\vartheta ,\nu }(t)$ the Riemann-Liouville non-homogeneous fractional integral process, ${\partial }_t^{\beta }$ is the Caputo fractional derivative, $-(-\Delta {)}^{\alpha /2}$ is the generator of an isotropic stable process, $I_t^{\beta }$ is the fractional integral operator, and $\sigma :\mathbb{R}\to \mathbb{R}$ 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 $\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal{N}}_{\lambda }^{1-\vartheta ,\nu }(t)$ 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 $\sigma$.

18 pages, British Journal of Mathematics and Computer Science, 2017