Space-time fractional stochastic partial differential equations driven by Lévy white noise

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.