Abstract
Abstract The aim of this paper is to investigate a class of nonlinear stochastic reaction-diffusion systems involving fractional Laplacian in a bounded domain. First, the existence and uniqueness of weak solutions are proved by using Galërkin’s method. Second, the existence of optimal controls for the corresponding stochastic optimal control problem is obtained. Finally, several examples are provided to demonstrate the theoretical results.
Highlights
In this paper, we discuss a class of stochastic fractional reaction-diffusion systems in Ω: du + (−Δ)αudt = f (u)dt + gdt + σdW, t ∈ (0, T], x ∈ Ω, u(t) = 0, t ∈ [0, T], x ∈ d\Ω, (1)u(0) = u0, x ∈ Ω, where Ω is a smooth bounded domain contained in d, α ∈ (0, 1), T ∈ (0, +∞), u is a vector-valued function, σ is an operator-valued function and {W(t)}t∈[0,T] is the space-time noise.Different from Laplacian, the fractional Laplacian is a nonlocal linear operator
U(0) = u0, x ∈ Ω, where Ω is a smooth bounded domain contained in d, α ∈ (0, 1), T ∈ (0, +∞), u is a vector-valued function, σ is an operator-valued function and {W(t)}t∈[0,T] is the space-time noise
A natural question arises: Whether we can extend the results of Laplacian problems to the fractional Laplacian ones or not? these extensions are not always true, see [1,2,3]
Summary
In [7], Ahmed studied on the situation of α = 1 and g ≡ 0, and the well-posedness of weak solution is obtained for p ∈ [1, +∞) We extend this result to the fractional Laplacian case and obtain the existence and uniqueness of weak solutions when p ∈ [1, 2∗α/2). In [10], Wang investigated the following stochastic fractional reaction-diffusion system: dφ + (−Δ)αφdt = f (φ)dt + gdt + cφdW , t > τ, x ∈ Ω, φ(t) = 0, t > τ, x ∈ d\Ω,.
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