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 where The operator with (t) the Riemann-Liouville non-homogeneous fractional integral process, is the Caputo fractional derivative, is the generator of an isotropic stable process, is 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 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 and t0> 1; that is, the result establishes that the energy of the solution grows atleast as c4(t + t0)  exp(c5t) and at most as c1t exp(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.