The present paper addresses the following stochastic heat fractional integral equation (SHFIE): ∂ u(x,t)=−(−∆)α/2u(x,t)+σ(u(x,t))N β,ν(t), x∈Rd, t ≥0,
∂t λ
with β > 0, ν ∈ (0,1], α ∈ (0,2]. The operator −(−∆)α/2 is the generator of an isotropic stable process and N β,ν(t) is the
λ
Riemann–Liouville nonhomogeneous fractional integral process. The mean and variance for the process N β,ν(t) for some specific λ
rate functions were computed. Also, the growth moment bounds for the class of heat equation perturbed with the nonhomogeneous fractional time Poisson process were given. In addition, the paper shows that the solution grows exponentially for some small time interval t ∈ [t0,T], T < ∞ and t0 > 1. To explain, the result establishes that the energy of the solution grows at least as
c3(t +t0)−(β+aν) exp(c4t) and at most as c1t−(β+aν) exp(c2t) for different conditions on the initial data, where c1, c2, c3 and c4 are some positive constants depending on T .
