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

211.PNG

where 221.PNGThe operator 231.PNG24.PNGwith 25.PNG(t) the Riemann-Liouville non-homogeneous fractional integral process, 26.PNGis the Caputo fractional derivative, 27.PNGis the generator of an isotropic stable process, 301.PNGis the fractional integral operator, and σ : R → R is Lipschitz continuous. The above time fractional stochastic heat type equations may be used to model sequence of catastrophic events 28.PNGfor 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 29.PNGand t0> 1; that is, the result establishes that the energy of the solution grows atleast as c4(t + t0) 18.PNG exp(c5t) and at most as c1t 19.PNGexp(c3t) for different conditions on the initialdata, where c1, c3, c4 and c5 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 σ.