This paper is concerned with the following space-time fractional stochastic nonlinear partial differential equation equation* (ₜ^β+ν2 (-Δ) ^α/ 2) u=Iₓ^γ f (t, x, u) -₈=₁^d xᵢ qᵢ (t, x, u) + σ (t, x, u) Fₓ, ₗ equation* for a random field u (t, x): 0, ) ᵈ, where α>0, β (0, 2), γ0, ν>0, Fₓ, ₗ is a Lévy space-time white noise, Iₓ^γ stands for the Riemann-Liouville integral in time, and f, qᵢ, σ: [0, ) ᵈ are measurable functions. Under suitable polynomial growth conditions, we establish the existence and uniqueness of L² (Rᵈ) -valued local solutions when the Lévy white noise Fₓ, ₗ contains Gaussian noise component. Furthermore, for p[1, 2, we derive the existence and uniqueness of Lᵖ (Rᵈ) -valued local solutions for the equation driven by pure jump Lévy white noise. Finally, we obtain certain stronger conditions for the existence and uniqueness of global solutions.
Guo et al. (Sun,) studied this question.